Fibonacci and Golden Ratio Formulae

Here are almost 300 formula involving the Fibonacci numbers and the golden ratio together with the Lucas numbers and the General Fibonacci series (the G series). This forms a major reference page for Ron Knott's Fibonacci Web site (http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html) where there are many more details and explanations with applications, puzzles and investigations aimed at secondary school students and teachers as well as interested mathematical enthusiasts.
Note that it is easy to search for a named formula on this page since it is an HTML page and the formulae are not images. In your browser main menu, under the Edit menu look for Find... and type Vajda-N or Dunlap-N for the relevant formula. Full references are at the foot of this document.

A companion page on Linear Recurrences and their generating Functions for Fibonacci Numbers, Continued Fraction convergents, Pythagorean triples and other series of numbers.

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Definitions and Notation

Beware of different golden ratio symbols used by different authors!
Where a formula below (or a simple re-arrangement of it) occurs in either Vajda or Dunlap's book, the reference number they use is given here. Dunlap's formulae are listed in his Appendix A3. Hoggatt's formula are from his "Fibonacci and Lucas Numbers" booklet. Full bibliographic details are at the end of this page in the References section.
Phi
Φ
τ τφ, α
 √5 + 1 2
= 1.6180339...
Koshy uses α (page 78)
phi
φ
−σ−φ−β
 √5 − 1 2
= 0.6180339...
Koshy uses −β (page 78)
abs(x)
|x|
|x||x||x|abs(x) = x if x≥0;
abs(x) = −x if x<0
the absolute value of a number, its magnitude; ignore the sign;
floor(x)
x
[x]trunc(x), not used for x<0 x the nearest integer ≤ x. When x>0, this is "the integer part of x" or "truncate x" i.e. delete any fractional part after the decimal point.
3=floor(3)=floor(3.1)=floor(3.9), -4=floor(-4)=floor(-3.1)=floor(-3.9)
round(x)
[x]
 [ x + 1 ] 2
trunc(x + 1/2)   the nearest integer to x; trunc(x+0.5) 3=round(3)=round(3.1), 4=round(3.9),
-4=round(-4)=round(-3.9), -3=round(-3.1)
4=round(3.5), -3=round(-3.5)
ceil(x)
x
-- x the nearest integer ≥ x. 3=ceil(3), 4=ceil(3.1)=ceil(3.9), -3=ceil(-3)=ceil(-3.1)=ceil(-3.9)
fract(x)
frac(x)
-- x mod 1 x − floor(x) the fractional part of x, i.e. the part of abs(x) after the decimal point
 ( nr )
 ( nr )
 ( nr )
 ( nr )
 n! r! (n − r)!
nCr
n choose r;
the element in row n column r of Pascal's Triangle
the coefficient of xr in (1+x)n
the number of ways of choosing r objects from a set of n different objects.
n≥0 and r≥0 (otherwise value is 0)

Fibonacci-type series with the rule S(i)=S(i-1)+S(i-2) for all integers i:
 i FibonacciF(i) LucasL(i) General FibG(a,b,i) ... −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 ... ... −8 5 −3 2 −1 1 0 1 1 2 3 5 8 ... ... 18 −11 7 −4 3 −1 2 1 3 4 7 11 18 ... ... 13a−8b −8a+5b 5a−3b −3a+2b 2a−b −a+b a b a+b a+2b 2a+3b 3a+5b 5a+8b ...
F(0) = 0, F(1) = 1,
F(n+2) = F(n + 1) + F(n)
-Definition of the Fibonacci series
F(−n) = (−1)n + 1 F(n)Vajda-2, Dunlap-5 Extending the Fibonacci series 'backwards'
L(0) = 2, L(1) = 1,
L(n + 2) = L(n + 1) + L(n)
-Definition of the Lucas series
L(−n) = (−1)n L(n)Vajda-4, Dunlap-6Extending the Lucas series 'backwards'
G(n + 2) = G(n + 1) + G(n)Vajda-3, Dunlap-4Definition of the Generalised Fibonacci series, G(0) and G(1) needed
Phi = 1.618... =
 √5 + 1 2
Dunlap-63 Phi and −phi are the roots of x2 = x + 1
phi = 0.618... =
 √5 − 1 2
Dunlap-65 Beware! Dunlap occasionally uses φ to represent our phi = 0.61803.., but more frequently he uses φ to represent −0.61803.. !

Linear Formulae

Linear relationships involve only sums or differences of Fibonacci numbers or Lucas numbers or their multiples.

Linear Sums of Fibonacci numbers

F(n + 2) + F(n) + F(n − 2) = 4 F(n) B&Q(2003)-Identity 18
F(n + 2 ) + F(n) = L(n + 1)by Definition of L(n), Vajda-6, Hoggatt-I8,
B&Q(2003) Identity 32, Dunlap-14, Koshy-5.14
F(n + 2) − F(n) = F(n + 1) by Definition of F(n)
F(n + 3) + F(n) = 2 F(n + 2) B&Q(2003)-Identity 16
F(n + 3) − F(n) = 2 F(n + 1)-
F(n + 4) + F(n) = 3 F(n + 2) B&Q(2003)-Identity 17
F(n + 2) + F(n − 2) = 3 F(n) B&Q(2003)-Identity 7
F(n + 2) − F(n − 2) = L(n) Hoggatt-I14
F(n + 4) − F(n) = L(n + 2)-
F(n + 5) + F(n) = F(n + 2) + L(n + 3)-
F(n + 5) − F(n) = L(n + 2) + F(n + 3)-
F(n + 6) + F(n) = 2 L(n + 3)-
F(n + 6) − F(n) = 4 F(n + 3)-
F(n) + 2 F(n − 1) = L(n)(Dunlap-32), B&Q(2003) Identity 50
F(n + 2) − F(n − 2) = L(n)Vajda-7a, Dunlap-15,
Koshy-5.15
F(n + 3) − 2 F(n) = L(n)possible correction for Dunlap-31
F(n + 2) − F(n) + F(n − 1) = L(n)possible correction for Dunlap-31
F(n) + F(n + 1) + F(n + 2) + F(n + 3) = L(n + 3)C Hyson(*)

Linear Sums of Lucas numbers

 L(n − 1) + L(n + 1) = 5 F(n) Vajda-5, Dunlap-13,Koshy-5.16,B&Q(2003)-Identity 34, Hoggatt-I9 L(n) + L(n + 3) = 2 L(n + 2) - L(n) + L(n + 4) = 3 L(n + 2) - 2 L(n) + L(n + 1) = 5 F(n + 1) B&Q(2003)-Identity 52 L(n + 2) − L(n − 2) = 5 F(n) - L(n + 3) − 2 L(n) = 5 F(n) -

Linear Sum of a Fibonacci and a Lucas number

 F(n) + L(n) = 2 F(n + 1) Vajda-7b, Dunlap-16,B&Q-Identity 51 L(n) + 5 F(n) = 2 L(n + 1) - 3 F(n) + L(n) = 2 F(n + 2) Vajda-26, Dunlap-28 3 L(n) + 5 F(n) = 2 L(n + 2) Vajda-27, Dunlap-29

Golden Ratio Formulae

Defining relations: The roots of x2 = x + 1 are
 Φ = Phi = 1 + √5 = 1.61803398874989484820458683436... 2 −φ = −phi = 1 − √5 = −0.61803398874989484820458683436... 2

Basic Phi Formulae

 Phi phi = 1 Vajda page 51(3), Dunlap-65 Phi + phi = √5 - Phi / phi = Phi + 1 - phi / Phi = 1 − phi - Phi = phi + 1 = √5 − phi - phi = Phi − 1 = √5 − Phi - Phi2 = Phi + 1 Vajda page 51(4), Dunlap-64 Phin+2 = Phin+1 + Phin ∀n∈ℤ Phin× Vajda page 51(4) phi2 = 1 − phi Vajda page 51(4), Dunlap-64 phin+2 = phin − phin+1 ∀n∈ℤ phin×Vajda page 51(4) phin = phin+1 + phin+2 ∀n∈ℤ from line above

Golden Ratio with Fibonacci and Lucas

 Phi = √5 + 1 = 1 = phi + 1; phi = √5 −1 = 1 = Phi −1 2 phi 2 Phi
 F(n) = Phin − (−phi)n √5
"Binet's" Formula
De Moivre(1718), Binet(1843), Lamé(1844),
Vajda-58, Dunlap-69, Hoggatt-page 11, B&Q(2003)-Identity 240
L(n) = Phin + (−phi)n Vajda-59, Dunlap-70, B&Q(2003)-Identity 241
Phin = Phi F(n) + F(n−1) Vajda-50a, Rabinowitz-28, B&Q(2003)-Corrolary 33
Phin = F(n+1) + F(n) phiRabinowitz-28, B&Q(2003)-Corollary 33
 Phin = L(n) + F(n)√5 2
Vajda-50b, Rabinowitz-25, B&Q(2003)-Identity 242,
I Ruggles (1963) FQ 1.2 pg 80
 (−phi)n = L(n) − F(n)√5 2
Vajda-50c, I Ruggles (1963) FQ 1.2 pg 80,
Rabinowitz-25, B&Q(2003)-Identity 243
(−phi)n = −phi F(n) + F(n−1)Rabinowitz-28
(−phi)n = F(n+1) − Phi F(n)Vajda-103b, Dunlap-75
√5 Phin = Phi L(n) + L(n−1)-
√5 (−phi)n = phi L(n) − L(n−1)-

Some useful special cases

These follow simply from Vajda-50a and the basic definitions of Phi above.
Phi + 2 = √5 Phi
 Phi2 + 1 = 2 + Phi = 5 + √5 = √5 Phi 2
Phi3 + 1 = 2 + 2 Phi = 2 Phi2 = 3 + √5
Phi6 = 4 Phi3 + 1 = 5 + 8 Phi = 9 + 4√5
Phi8 = 7 Phi4 − 1 = 13 + 21 Phi

Golden Ratio with Fibonacci and Lucas - Approximations

 Lim n→∞

 F( n+1 ) F( n )
= Phi
Vajda-101
 Lim n→∞

 F( n+m ) F( n )
= Phim
Vajda-101a
 F(n) = round ( Phin ) ,if n≥0 √5
Vajda-62, Dunlap-71 corrected, B&Q(2003)-Identity 240 Corollary 30
L(n) = round(Phin),if n≥2Vajda-63, Dunlap-72, B&Q(2003)-Corollary 35
F(−n) = round (
 −(−phi)−n √5
) ,if n≥0
-
L(−n) = round( (−phi)−n ), n≥2-
F(n + 1) = round(Phi F(n)),if n≥2Vajda-64, Dunlap-73
L(n + 1) = round(Phi L(n)),if n≥4Vajda-65, Dunlap-74
fract( F(2n) phi ) = 1 − phi2nKnuth vol 1, Ex 1.2.8 Qu 31 with ψ=phi
fract( F(2n+1) phi ) = phi2n+1Knuth vol 1, Ex 1.2.8 Qu 31

Fibonacci and Lucas Factors

F(nk) is a multiple of F(n)
F(nk) ≡ 0 (mod F(k))
B&Q(2003)-Theorem 1, Vajda Theorem I page 82
gcd(F(m),F(n)) = F(gcd(m,n))Lucas (1878)
B&Q(2003)-Theorem 6,Vajda Theorem II page 83
F(mn+r) ≡ ± F(r) (mod F(n) )Knuth Vol 1 Ex 1.2.8 Qu. 32, Vajda page 86
gcd(L(m),L(n)) = L(gcd(m,n)),
if both L(m)/gcd(L(m),L(n)) and
L(n)/gcd(L(m),L(n)) are odd integers
Vajda page 86
L(mn+r) ≡ ± L(r) (mod L(n) ) (Vajda page 87)
F(m q) = F(m)
 q ∑ j = 1
F(m - 1) j-1 F( m(q - j) + 1 )
B&Q(2003)-Theorem 2
 F(kt) F(t)
=
 (k−3)/2 ∑ i = 0
(−1)itL( (k−2i−1)t )
+ (−1)(k−1)t/2 for ODD k ≥ 3
Vajda-85
 F(kt) F(t)
=
 k/2−1 ∑ i = 0
(−1)itL( (k−2i−1)t )
for EVEN k ≥ 2
Vajda-86
 L(kt) L(t)
=
 (k−3)/2 ∑ i = 0
(−1)i(t+1)L( (k−2i−1)t )
+ (−1)(k−1)(t+1)/2    for ODD k ≥ 3
Vajda-87
L(t) is not a factor of L(kt) for even k
 F(kt) L(t)
=
 k/2−1 ∑ i = 0
(−1)i(t+1)F( (k−2i−1)t )
for EVEN k ≥ 2
Vajda-88
L(t) is not a factor of F(kt) for odd k and t≥3
p prime ⇒ p is a factor of L(p) − 1 B&Q(2003) Identity 228
p prime ⇒ p is a factor of L(2p) − 3 B&Q(2003) Identity 229

Order 2 Formulae

Order 2 means these formulae have terms involving the product of at most 2 Fibonacci or Lucas numbers.

Fibonacci numbers

 F(2n) = F(n + 1)2 − F(n − 1)2 Lucas(1878), B&Q(2003)-Identity 14, Hoggatt-I10 F(2n) = F(n) ( F(n+1) + F(n−1) ) Vajda-13, Hoggatt-I7,Koshy-5.13, B&Q(2003)-Identity 33 with Vajda-6 F(2n) = F(n) (2F(n+1) − F(n)) simple alternative to Vajda-13 F(2n+1) = F(n + 1)2 + F(n)2 Vajda-11, Dunlap-7, Lucas(1878), B&Q(2003)-Identity 13, Hoggatt-I11 F(n+2)2 + F(n)2 = 3 F(n + 1)2 − 2 (−1)n V E Hoggatt B-208 FQ 9 (1971) pg 217. F(n+3)2 + F(n)2 = 2 ( F(n+1)2 + F(n+2)2 ) B&Q(2003)-Identity 30 F(n + k + 1)2 + F(n − k)2 = F(2k + 1)F(2n + 1) Sharpe(1965), a generalization of Vajda-11,Dunlap-7 Melham(1999) F(n + k)2 + F(n − k)2 =F(n + k −2)F(n + k + 1) + F(2k − 1)F(2n − 1) Sharpe (1965) F(n + 1)2 − F(n)2 = F(n + 2) F(n − 1) Vajda-12, Dunlap-8 F(n + k + 1)2 − F(n − k)2 = F(n − k − 1)F(n − k + 2) + F(2 k)F(2n + 2) Sharpe (1965) F( n+p )2 − F( n−p )2 = F( 2n )F( 2p ) I Ruggles (1963) FQ 1.2 pg 77; Hoggatt-I25, Sharpe (1965) F(n + 1) F(n − 1) − F(n)2 = (−1)n Cassini's Formula(1680), Simson(1753), Vajda-29, Dunlap-9, Hoggatt-I13 special case of Catalan's Identity with r=1 B&Q(2003)-Identity 8 F(n)2 − F(n + r)F(n − r) = (-1)n-rF(r)2 Catalan's Identity(1879) F(n)F(m + 1) − F(m)F(n + 1) = (-1)mF(n − m) d'Ocagne's Identity,special case of Vajda-9 with G=F F(n + m) = F(n + 1)F(m + 1) − F(n − 1)F(m − 1) B&Q(2003)-Identity 231 F(n + m) = F(m) F(n + 1) + F(m − 1) F(n) alternative to Dunlap-10, B&Q(2003)-Identity 3; variation of Hansen (1972) Vorob'ev (1951) pages 9-10 proof, attributed to I S Sominskii F(n) = F(m) F(n + 1 − m) + F(m − 1) F(n − m) I Ruggles (1963) FQ 1.2 pg 79; Dunlap-10, special case of Vajda-8 F(n) F(n + 1) = F(n − 1) F(n + 2) + (−1)n-1 Vajda-20a special case: i:=1;k:=2;n:=n-1; Hoggatt-I19 F(n + i) F(n + k) − F(n) F(n + i + k) = (−1)n F(i) F(k) Vajda-20a=Vajda-18 (corrected) with G:=H:=F 2 F(n + 1) = F(n) + √(5 F(n)2 + 4(−1)n) F(n+1) from F(n): Problem B-42, S Basin, FQ, 2 (1964) page 329 F(a)F(b) − F(c)F(d) = (−1)r( F(a − r)F(b − r) − F(c − r)F(d − r) ) a+b=c+d for any integers a,b,c,d,r Johnson FQ 42 (2004) B-960 'A Fibonacci Iddentity', solution pg 90 also Johnson-7 Cassini, Catalan and D'Ocagne's Identities are all special cases of this formula

Lucas numbers

 L(n + 2)2 = 3 L(n + 1)2 − L(n)2 + 10(−1)n V E Hoggatt B-208 FQ 9 (1971) pg 217. L(n + 2) L(n − 1) = L(n + 1)2 − L(n)2 from Vajda-17a L(n + 1)2 + L(n)2 = L(2n) + L(2n + 2) from Vajda-17a L(n + 1)2 − L(n − 1)2 = L(2n + 1) + L(2n − 1) from Vajda-17a L(n + 1) L(n − 1) − L(n)2 = −5 (−1)n B&Q(2003)-Identity 60 L(2n) + 2 (−1)n = L(n)2 Vajda-17c, Dunlap-12, B&Q(2003)-Identity 36 L(n + m) + (−1)m L(n − m) = L(m) L(n) Vajda-17a, Dunlap-11 (special cases: Hoggatt-I15,I18) L(4n) + 2 = L(2n)2 Hoggatt-I15, special case of Vajda-17a 2 L(n + 1) = L(n) + √5 √(L(n)2 − 4(−1)n) L(n+1) from L(n): Problem B-42, S Basin, FQ 2 (1964) page 329

Fibonacci and Lucas Numbers

F(2n) = F(n) L(n)Vajda-13, Hoggatt-I7,
Koshy-5.13,
B&Q(2003)-Identity 33
 F(4n) + 1 = F(2n−1) L(2n+1) F(4n+1) + 1 = F(2n+1) L(2n) F(4n+2) + 1 = F(2n+2) L(2n) F(4n+3) + 1 = F(2n+1) L(2n+2)
F(n)+1 is a product of a FIbonacci and a Lucas number:
A001611 F(n)+1, Formula by R K Guy (2003)
5 F(n) = L(n + 1) + L(n − 1)
L(n + 1)2 + L(n)2 = 5 F(2n + 1)Vajda-25a
L(n + 1)2 − L(n − 1)2 = 5 F(2n)-
L(n + 1)2 − 5 F(n)2 = L(2n + 1)-
L(2n) − 2 (−1)n = 5 F(n)2Vajda-23, Dunlap-25
L(n)2 − 4(−1)n = 5 F(n)2B&Q(2003)-Identity 53, Hoggatt-I12
F(n+k) + (−1)k F(n−k) = F(n)L(k)Bro U Alfred (1964), Bergum and Hoggatt (1975) equns (5),(7)
F(n+k) − (−1)k F(n−k) = L(n)F(k)Bro U Alfred (1964), Bergum and Hoggatt (1975) equns (6),(8)
L(n+k) + (−1)k L(n−k) = L(n)L(k)Bro U Alfred (1964), Bergum and Hoggatt (1975) equns (9),(11)
L(n+k) − (−1)k L(n−k) = 5F(n)F(k)Bro U Alfred (1964), Bergum and Hoggatt (1975) equns (10),(12)
F(n + 1) L(n) = F(2n + 1) + (−1)nVajda-30, Vajda-31,
Dunlap-27, Dunlap-30
L(n + 1) F(n) = F(2n + 1) − (−1)n-
F(2n + 1) = F(n + 1) L(n + 1) − F(n) L(n)Vajda-14, Dunlap-18
L(2n + 1) = F(n + 1) L(n + 1) + F(n) L(n)-
L(m) L(n) + L(m − 1) L(n − 1) = 5 F(m + n − 1) Hansen 1972
L(n)2 − 2 L(2n) = −5 F(n)2Vajda-22, Dunlap-24
5 F(n)2 − L(n)2 = 4 (−1)n + 1Vajda-24, Dunlap-26
F(n)2 + L(n)2 = 4 F(n + 1)2 − 2 F(2n)FQ (2003)vol 41, B-936, M A Rose, page 87
5 (F(n)2 + F(n + 1)2) = L(n)2 + L(n + 1)2Vajda-25
F(n) L(m) = F(n + m) + (−1)m F(n − m)a recurrence relation for F(n+km):
Vajda-15a, Dunlap-19
L(n) F(m) = F(n + m) − (−1)m F(n − m)Vajda-15b, Dunlap-20
5 F(m) F(n) = L(n + m) − (−1)m L(n − m)Vajda-17b, Dunlap-23, (special cases:Hoggatt-I16,I17)
2 F(n + m) = L(m) F(n) + L(n) F(m)Vajda-16a, Dunlap-2, FQ (1967) B106 H H Ferns pp 466-467
2 L(n + m) = L(m) L(n) + 5 F(n) F(m)FQ (1967) B106 H H Ferns pp 466-467
F(m) L(n) + F(m − 1) L(n − 1) = L(m + n − 1) Hansen (1972)
(−1)m 2 F(n − m) = L(m) F(n) − L(n) F(m)Vajda-16b, Dunlap-22
L(n + i) F(n + k) − L(n) F(n + i + k) =
(−1)n + 1 F(i) L(k)
Vajda-19a
F(n + i) L(n + k) − F(n) L(n + i + k) = (−1)n F(i) L(k)Vajda-19b
L(n + k + 1)2 + L(n − k)2 = 5 F(2n + 1)F(2k + 1) Melham (1999) Theorem 1
L(n + i) L(n + k) − L(n) L(n + i + k)
= (−1)n + 1 5 F(i) F(k)
Vajda-20b
(−1)kF(n)F(m−k) + (−1)mF(k)F(n−m) + (−1)nF(m)F(k−n) = 0 FQ 11 (1973) B228 page 108
(−1)kL(n)F(m−k) + (−1)mL(k)F(n−m) + (−1)nL(m)F(k−n) = 0 FQ 11 (1973) B229 page 108
5 F(jk+r) F(ju+v) = L(j(k+u)+(r+v)) - (-1)ju+vL(j(k-u)+(r-v)) Hansen (1978)
F(jk+r) L(ju+v) = F(j(k+u)+(r+v)) + (-1)ju+vF(j(k-u)+(r-v)) Hansen (1978)
L(jk+r) L(ju+v) = L(j(k+u)+(r+v)) + (-1)ju+vL(j(k-u)+(r-v)) Hansen (1978)
5F(a)F(b) − L(c)L(d) = (−1)r( 5F(a − r)F(b − r) − L(c − r)L(d − r) )
a+b=c+d for any integers a,b,c,d,r
Johnson
F(a) L(b) − F(c) L(d) = (−1)r( F(a−r) L(b−r) − F(c−r) L(d−r)
with a+b=c+d
Johnson-32, special case of Johnson-44

Higher Order Fibonacci and Lucas

Fibonacci and Lucas cubed

 F(3n) = F(n + 1)3 + F(n)3 − F(n − 1)3 Lucas (1876), B&Q(2003)-Identity 232 5 L(3n) = L(n + 1)3 + L(n)3 − 3 L(n − 1)3 Long (1986) equation (45) 3 F(3n) = F(n+2)3 − 3 F(n)3 + F(n−2)3 J Ginsburg "A Relationship Between Cubes of Fibonacci Numbers." Scripta Mathematica (1953) page 242. L(3n) = L(n+1)F(n+1)2 + L(n)F(n)2 − L(n-2)F(n-1)2) Long (1986) equation (43) 5 F(3n) = F(n+1)L(n+1)2 + F(n)L(n)2 − F(n-1)L(n-1)2 Long (1986) equation (44) F(n + 1)F(n + 2)F(n + 6) − F(n + 3)3 = (−1)nF(n) F(n)F(n + 4)F(n + 5) − F(n + 3)3 = (−1)n+1F(n + 6) FQ 41 (2003) pg 142, Melham. The second is a variant with -n for n and using Vajda-2 F(n−2)F(n−1)F(n+3) − F(n)3 = (−1)n-1F(n−3) F(n+2)F(n+1)F(n−3) − F(n)3 = (−1)nF(n+3) Fairgrieve and Gould (2005) versions of the above two formulae of Melham F(n−2)F(n+1)2 − F(n)3 = (−1)n-1 F(n−1) F(n+2)F(n−1)2 − F(n)3 = (−1)n F(n+1) Fairgrieve and Gould (2005) F(n+a+b)F(n−a)F(n−b) − F(n-a-b)F(n+a)F(n+b) = (−1)n+a+bF(a)F(b)F(a+b)L(n) Melham (2011) Theorem 1 F(n+a+b−c)F(n−a+c)F(n−b+c) − F(n−a−b+c)F(n+a)F(n+b) = (−1)n+a+b+cF(a+b−c)( F(c)F(n+a+b−c) + (−1)cF(a−c)F(b−c)L(n) ) Melham (2011) Theorem 5 F(i+j+k) =F(i+1)F(j+1)F(k+1) + F(i)F(j)F(k) − F(i−1)F(j−1)F(k−1) for any integers i,j,k Johnson's (6) F(3n) = F(n+1)3 + F(n)3 − F(n−1)3 From Johnson's (6) with i=j=k F(n)3 = F(n−1)3 + F(n−2)3 + 3 F(n)F(n−1)F(n−2) G Gelatti (2020, private communication) L(5n) = L(n) (L(2n) + 5F(n) + 3)( L(2n) − 5F(n) + 3), n odd Aurifeuille's Identity (1879) FQ 42 (2004) R S Melham, pgs 155-160

Fibonacci and Lucas to the fourth

 F(4n) = F(n+1)4 + 2F(n)4 − F(n−1)4 + 4F(n)3F(n−1) Khomovsky (2018) A42 F(n−1)2F(n+1)2 − F(n−2)2F(n+2)2 = 4(−1)nF(n)2 Melham (2011) 21 F(n−3)F(n−1)F(n+1)F(n+3) − F(n)4 = (−1)nL(n)2 Melham (2011) 22 F(n)2 F(m + 1) F(m − 1) − F(m)2 F(n + 1) F(n − 1) = (−1)n − 1 F(m + n) F(m − n) Vajda-32 F(n − 2)F(n − 1)F(n + 1)F(n + 2) + 1 = F(n)4 Gelin-Cesàro Identity (1880) (see Dickson page 401) FQ 41 (2003) pg 142, B&Q(2003)-Identity 31 Hoggatt-I29, Simson(1753) L(n − 2)L(n − 1)L(n + 1)L(n + 2) + 25 = L(n)4 B&Q(2003)-Identity 56 F(n+a+b+c)F(n−a)F(n−b)F(n−c) − F(n-a-b-c)F(n+a)F(n+b)F(n+c) = (−1)n+a+b+cF(a+b)F(a+c)F(b+c)F(2n) Melham (2011) Theorem 2 F(n+a+b+c−d)F(n−a+d)F(n−b+d)F(n−c+d) − F(n−a−b−c+2d)F(n+a)F(n+b)F(n+c) = (−1)n+a+b+cF(a+b−d)F(a+c−d)F(b+c−d)F(2n+d) Melham (2011) Theorem 6 ( F(n-1)F(n+2) )2 + (2 F(n)F(n+1) )2 = (F(n+1)F(n+2) − F(n-1)F(n))2 = F(2n+1)2 A F Horadam FQ 20 (1982) pgs 121-122, B&Q(2003)-Identity 19 (corrected) special case of Generalised Fibonacci Pythagorean Triples ( F(n)2 + F(n+1)2 + F(n+2)2 )2 = 2 ( F(n)4 + F(n+1)4 + F(n+2)4 ) Candido's Identity (1951) FQ 42 (2004) R S Melham, pgs 155-160 ( L(n-1)L(n+2) )2 + ( 2L(n)L(n+1) )2 = ( 5F(2n+1) ) 2 Wulczyn FQ 18 (1980) pg 188 special case of Generalised Fibonacci Pythagorean Triples

Fibonacci and Lucas Higher Powers

F(5n) = F(n+1)5 + 4F(n)5 − F(n-1)5 + 10F(n+1)F(n)3F(n−1) Falcon, Plaza (2007)
F(n)F(n+1)F(n+2)F(n+4)F(n+5)F(n+6) + L(n+3)2
= ( F(n+3)( 2F(n+2)F(n+4) − F(n+3)2) )2
(
 L(n) + √5 F(n) 2
) k  =
 L(kn) + √5 F(kn) 2
De Moivre Analogue,
S Fisk (1963) FQ 1.2 Problem B-10, pg 85.
Hoggatt-I44

Fibonomial formulae

The Fibonomials are defined using Fibonacci numbers instead of integers in binomial coefficients and Fibonacci factorials instead of normal factorials. There are many analagous results to those using binomial coefficients but using Fibonomials instead.

We define F!(n) = F(n)F(n-1)...F(2)F(1), n>0; F!(0)=1 for which some authors use n!F, to compare with n! = n(n-1)...3.2.1.
There is no universal notation for the Fibonomial. The fibonomial "Fibonacci n choose k" is defined as:

 ( n ) F k
=
 F!(n) F!(k) F!(n − k)
=
 F(n)F(n−1)...F(n−k+1) F(n−k)F(n−k−1)...F(2)F(1) F(k)F(k−1)...F(2)F(1)   F(n−k)F(n−k−1)...F(2)F(1)
if n ≥ k ≥ 0
= 0, otherwise
Vajda (page 74) uses J(n,k). D Knuth and others use double brackets:
 (( n )) k
while Melham (1999) and others use square brackets:
[
 n k
]

A simple alternative is to write fibonomial(n,k).
Here is a table of some values of the fibonomial (A010048)
n k 0 1 2 3 4 5 6 7
01
1 1 1
2 1 1 1
3 1 2 2 1
4 1 3 6 3 1
5 1 5 15 15 5 1
6 1 8 40 60 40 8 1
7 1 13 104 260 260 104 13 1
(
 m + n n
) F   =  F(m − 1) (
 m + n − 1 n−1
) F + F(n + 1) (
 m + n − 1 n
) F
Vajda page 74,
"add the two numbers above" analogy from Pascal's triangle
 m ∑ j = 0
(-1)j(j+3)/2
 ( m ) F j
F(n+m−j)m+1 = F!(m) F((m+1)n+m(m+1) /2))
Melham (1999)....
 1 F(n+1)2 + 1 F(n)2 = 1 F(2n+1) 1 F(n+2)3 + 1 F(n+1)3 −1 F(n)3 = 1.1 F(3n+3) 1 F(n+3)4 + 2 F(n+2)4 −2 F(n+1)4 − 1 F(n)4 = 1.1.2 F(4n+6) 1 F(n+4)5 + 3 F(n+3)5 −6 F(n+2)5 −3 F(n+1)5 + 1 F(n)5 = 1.1.2.3 F(5n+10) 1 F(n+5)6 + 5 F(n+4)6 −15 F(n+3)6 −15 F(n+2)6 + 5 F(n+1)6 + 1 F(n)6 = 1.1.2.3.5 F(6n+15)
.... examples
0 = F(n) − F(n−1) − F(n−2)
0 = F(n)2 − 2 F(n−1)2 − 2 F(n−2)2 + F(n−3)2
0 = F(n)3 − 3 F(n−1)3 − 6 F(n−2)3 + 3 F(n−3)3 + F(n−4)3
0 = F(n)4 − 5 F(n−1)4 − 15 F(n−2)4 + 15 F(n−3)4 + 5 F(n−4)4 − F(n−5)4
...
Brousseau (1968)...but the general formula was not given.
For this see next line:
 p ∑ k=0
 ( p )F k
(−1)k/2F(n − k)p−1
= 0, if p>0
Knuth AoCP Vol 1 section 1.2.8 Exercise 30, (1997)
F(k)
 ( n )F k
=
 F( n − k + 1)
 ( n )F k − 1
compare with
 ( n ) k
=
 n − k + 1 k
 ( n ) k − 1
F(k)
 ( n )F k
=
 F( n )
 ( n − 1 )F k − 1
compare with
k
 ( n ) k
=   n
 ( n − 1 ) k − 1
F(n − k)
 ( n )F k
=  F( n )
 ( n − 1 )F k
compare with
n − k
 ( n ) k
=  n
 ( n − 1 ) k
 ( n )F k
 ( k )F j
=
 ( n )F j
 ( n − j )F k − j
compare with
 ( n ) k
 ( k ) j
=
 ( n ) j
 ( n − j ) k − j

G Formulae

G(i) is the General Fibonacci series. It has the same recurrence relation as Fibonacci and Lucas, namely G(n+2) = G(n+1) + G(n) for all integers n (i.e. n can be negative) Vajda-3,Dunlap 4, but the "starting values" of G(0)=a and G(1)=b can be specified. To make it clear which starting values for G(0)=a and G(1)=b are being used, we write G(a,b,i) for G(i). G(n) is an abbreviation for G(a,b,n) when a and b are understood from the context.
Special cases are the Fibonacci and Lucas series since F(n) = G(0,1,n) and L(n)=G(2,1,n):
• If G(0)=0 and G(1)=1 we have 0,1,1,2,3,5,8,13,.. the Fibonacci series, i.e. G(0,1,n) = F(n);
• G(0)=2 and G(1)=1 gives 2,1,3,4,7,11,18,.. the Lucas series, i.e. G(2,1,n) = L(n);

Basic G Formulae

Two independent G series are here denoted G(n) and H(n), i.e. G(0) and G(1) are independent of H(0) and H(1).
G(n) = G(0) F(n − 1) + G(1) F(n)B&Q(2003)-Identity 37
G(−n) = (−1)n (G(0) F(n + 1) − G(1) F(n))ditto - applying Vajda-2 or
Vajda-9 with n=0
√5 G(n) = ( G(0) phi + G(1) ) Phin + (G(0) Phi − G(1)) ( −phi )n Vajda-55/56, Dunlap-77, B&Q(2003)-Identity 244
F(n) =
 G(0) G(n+1) − G(1) G(n) G(0)G(2) − G(1)2
Amer Math Monthly (2005) "Fibonacci, Chebyshev and
Orthogonal Polynomials"
D Aharonov, A Beardam, K Driver, p612-630
2 G(k) = ( 2 G(1) − G(0) ) F(k) + G(0) L(k) Johnson-46
G(n + m) = F(m − 1) G(n) + F(m) G(n + 1)Vajda-8, Dunlap-33, B&Q(2003)-Identity 38,
Johnson-40
G(n − m) = (−1)m (F(m + 1) G(n) − F(m) G(n + 1))Vajda-9, Dunlap-34, B&Q(2003)-Identity 47
G(n + m) + (−1)m G(n − m) = L(m) G(n) Vajda-10a, Dunlap-35, B&Q(2003)-Identity 45,
Bergum & Hoggatt (1975) (36) and (38)
G(n + m) − (−1)m G(n − m) = F(m) ( G(n−1) + G(n+1)) Vajda-10b, Dunlap-36, B&Q(2003)-Identity 48,
Bergum & Hoggatt (1975) (37) and (39)
G(m) F(n) − G(n) F(m) = (−1)n+1 G(0) F(m − n)Vajda-21a
G(m) F(n) − G(n) F(m) = (−1)m G(0) F(n − m)Vajda-21b
G(m+k) F(n+k) + (−1)k+1 G(m) F(n) = F(k) G(m + n + k)Howard(2003)

G Formulae of Order 2 or more

These formulae include terms which are a product of two G numbers either from the same G series of from two different G series i.e. with different index 0 and 1 values. Where the series may be different they are denoted G and H e.g. special cases include G = F (i.e. Fibonacci) and H = L (i.e. Lucas), or they could also be the same series G=H.
 G(n + i) H(n + k) − G(n) H(n + i + k) = (−1)n (G(i) H(k) − G(0) H(i + k)) Vajda-18 (corrected), B&Q(2003)-Identity 44 (also Identity 68) a special case of Johnson-44: G(p)H(q) − G(r)H(s) = (-1)n[ G(p-n)H(q-n) − G(r-n)H(s-n) ] if p+q = r+s and p,q,r,s,n are integers Johnson-44 G(n + 1) G(n − 1) − G(n)2 = (−1)n (G(1)2 − G(0) G(2)) Vajda-28, B&Q(2003)-Identity 46 4 G(n−1)G(n) + G(n−2)2 = G(n+1)2 B&Q(2003)-Identity 65 G(n + 3)2 + G(n)2 = 2( G(n+1)2 + G(n+2)2 ) B&Q(2003)-Identity 70 G(i+j+k) = F(i+1)F(j+1)G(k+1) + F(i)F(j)G(k) − F(i−1)F(j−1)G(k−1) for any integers i,j,k Johnson (39a) 4G(i)2G(i+1)2 + G(i−1)2G(i+2)2 = ( G(i)2 + G(i+1)2 )2 Generalised Fibonacci Pythagorean Triples Horadam (1967) G(n + 2)G(n + 1)G(n − 1)G(n − 2) + ( G(2)G(0) − G(1)2 )2 = G(n)4 B&Q(2003)-Identity 59

Summations

This section has formulae that sum a variable number of terms.

Fibonacci and Lucas Summations

These formulae involve a sum of Fibonacci or Lucas numbers only.
 n ∑ i = 0
F(i) = F(n + 2) − 1
Hoggatt-I1, Lucas(1878), B&Q 2003-Identity 1
 n ∑ i = 0
(-1) i F(i) = (-1)n F(n − 1) − 1
B&Q 2003-Identity 21
 n ∑ i = 0
L(i) = L(n + 2) − 1
Hoggatt-I2
 n ∑ i = a
F(i) = F(n + 2) − F(a + 1)
-
 n ∑ i = a
L(i) = L(n + 2) − L(a + 1)
-
 n ∑ i = 0
F(2i) = F(2n + 1) − 1, n≥0
Hoggatt-I6, Lucas(1878), B&Q(2003)-Identity 12
 n ∑ i = 1
F(2i − 1) = F(2n), n≥1
Hoggatt-I5, Lucas(1878), B&Q(2003)-Identity 2
 n ∑ i = 1
L(2i − 1) = L(2n) − 2
-
 n ∑ i = 1
2n − i F(i − 1) = 2n  − F(n + 2)
B&Q(2003)-Identity 10
 n ∑ i = 0
2i L(i) = 2n+1 F(n + 1)
B&Q(2003)-Identity 236
 n ∑ i = 0
F(3i + 1)
=
 F(3n + 3) 2
B&Q(2003)-Identity 23
 n ∑ i = 0
F(3i + 2)
=
 F(3n + 4) − 1 2
B&Q(2003)-Identity 24 (corrected)
 n ∑ i = 0
F(3i)
=
 F(3n + 2) − 1 2
B&Q(2003)-Identity 25 (corrected)
 n ∑ i = 0
F(4i) = F(2n + 1)2 − 1
B&Q 2003-Identity 27
 n ∑ i = 0
F(4i + 1) = F(2n + 1)F(2n + 2)
B&Q 2003-Identity 26
 n ∑ i = 0
F(4i + 2) = F(2n + 1)F(2n + 3) − 1
B&Q 2003-Identity 29
 n ∑ i = 0
F(4 i + 3) = F(2n + 3)F(2n + 2)
B&Q 2003-Identity 28
 n ∑ i = 0
(−1)i L(n − 2i) = 2 F(n + 1)
Vajda-97, Dunlap-54
 n ∑ i = 0
(−1)i L(2n − 2i + 1) = F(2 n + 2)
B&Q(2003)-Identity 55

Decimal (and other bases) fractions

We saw in The Fibonacci Series as a Decimal Fraction that the Fibonacci series occurs naturally as the decimal expansion of a simple fraction in several ways:
 1/89 = 0.0 1 1 2 3 5 ... 1/9899 = 0.00 01 01 02 03 05 08 13 21 ...
with a varying number of decimal digits before the Fibonacci numbers overlap and the series is obscured. This section gives formulae for these fractions for various subsequences of Fibonacci and General Fibonacci series.
 ∞ ∑ k = 1
10−n(k+1)F(ak)
=
 F(a) 102n − 10nL(a) − (−1)a
Hudson and Winans (1981)
If P(n) = a P(n-1) + b P(n-2) for n≥2; P(0) = c; P(1) = d and
m and N are defined by B2 = m + Ba + b, N = cm + dB + bc,
then
N
Bm
=
 ∞ ∑ i = 1
 P(i−1) Bi

provided that |(a+√(a2+4b))/(2B)| < 1 and
| (a−√(a2+4b))/(2B) | < 1
Long (1981)

Summations with fractions

 ∞ ∑ i = 0
 F(i) 2i
= 2
Vajda-60, Dunlap-51
 ∞ ∑ i = 0
 L(i) 2i
= 6
-
 ∞ ∑ i = 0
 F(i) ri
=
 r r2 − r − 1
-
 ∞ ∑ i = 0
 L(i) ri
= 2 +
 r +2 r2 − r − 1
-
 ∞ ∑ i = 1
 i F(i) 2i
= 10
Vajda-61, Dunlap-52
 ∞ ∑ i = 1
 i L(i) 2i
= 22
-
 ∞ ∑ i = 0
 1 F(2i)
= 4 − Phi = 3 − phi
Vajda-77(corrected), Dunlap-53(corrected)
 n ∑ i = 1
 (−1)2i-1r F(2ir)
=
 (−1)r F( r(2n−1) ) F(r) F(2n r)
Vajda-89 (corrected)
 ∑ k ≥ 2
 1 F(k−1)F(k+1)
= 1
R L Graham (1963) FQ 1.1, Problem B-9, pg 85, FQ 1.4 page 79
 ∑ k ≥ 2
 F(k) F(k−1)F(k+1)
= 2
R L Graham (1963) FQ 1.1, Problem B-9, pg 85
 ∑ k ≥ 2
 (−1)k F(k)F(k−1)
= phi
Johnson-11, Vajda-102

Order 2 summations

 n ∑ i = 1
F(i)2 = F(n) F(n + 1)
Vajda-45, Dunlap-5,
Hoggatt-I3, Lucas(1878),
Koshy-77,
B&Q(2003)-Identity 9 (Identity 233 variant)
 n ∑ i = 1
L(i)2 = L(n) L(n + 1) − 2
Hoggatt-I4
 2n-1 ∑ i = 0
L(i)2 = 5 F(2n) F(2n - 1)
-
 2n ∑ i = 1
F(i) F(i − 1) = F(2n)2
Vajda-40, Dunlap-45
 2n ∑ i = 1
L(i) L(i − 1) = L(2n)2 − 4
-
 2n+1 ∑ i = 1
F(i) F(i − 1) = F(2n +1)2 − 1
Vajda-42, Dunlap-47
 2n+1 ∑ i = 1
L(i) L(i − 1) = L(2n +1)2 − 1
-
 5 n ∑ k = 0
(−1)r(1+k) F(r(1+k))2 = (−1)r(n+1)
 F((2n+3)r) F(r)
− 2n − 3
Vajda-93
 n ∑ k = 0
(−1)r(1+k) L(r(1+k))2 = (−1)r(n+1)
 F((2n+3)r) F(r)
+ 2n + 1
Vajda-94
 n−1 ∑ i=0
F(2i + 1)2 =
 F(4n) + 2n 5
Vajda-95, B&Q(2003)-Identity 234
 n ∑ i=0
F(2i)2 =
 F(4n + 2) − 2n − 1 5
Vajda page 70
 n−1 ∑ i = 0
L(2i + 1)2 = F(4n) − 2n
Vajda-96, B&Q(2003)-Identity 54
 n ∑ i = 1
L(2i)2 = F(4n + 2) + 2n − 1
Vajda page 70
5
 n ∑ i = 0
F(i) F(n − i)
 = (n + 1) L(n) − 2 F(n + 1) = n L(n) − F(n)
Vajda-98, Dunlap-55, B&Q(2003)-Identity 58
 n ∑ i = 0
L(i) L(n − i)
 = (n + 1) L(n) + 2 F(n + 1) = (n + 2) L(n) + F(n)
Vajda-99, Dunlap-56, B&Q(2003)-Identity 57
 n ∑ i = 0
F(i) L(n − i) = (n + 1) F(n)
Vajda-100, Dunlap-57, B&Q(2003)-Identity 35
 2n−1 ∑ k = 1
(2n − k) F(k)2 = F(2n)2
V Hoggatt (1965) Problem B-53 FQ 3, pg 157

Summations of order > 2

10
 n ∑ i = 1
F(i)3 = F(3n+2) + 6(-1)n+1F(n−1) +5
adapted from Benjamin, Carnes, Cloitre (2009)
25
 n ∑ i = 1
F(i)4 = F(4n+2) + 4(-1)n+1F(2n + 1) +6n + 3
see A005969
4
 n ∑ k = 1
F(k)6 = F(n)5F(n+3) + F(2n)
Ohtsuka and Nakamura (2010) Theorem 1
4
 n ∑ k = 1
L(k)6 = L(n)5L(n+3) + 125 F(2n) − 128
Ohtsuka and Nakamura (2010) Theorem 2

G Summations

Two independent G series are denoted G(n) and H(n).
 n ∑ i = 1
G(i) = G(n + 2) − G(2)
L G Brökling (1964) FQ 2.1 Problem B-20 solution, pg76;
Vajda-33; Dunlap-38; B&Q(2003)-Identity 39
 n ∑ i = a
G(i) = G(n + 2) − G(a + 1)
-
 n ∑ i = 1
G(2i − 1) = G(2n) − G(0)
Vajda-34, Dunlap-37, B&Q(2003)-Identity 61
 n ∑ i = 1
G(2i) = G(2n + 1) − G(1)
Vajda-35, Dunlap-39, B&Q(2003)-Identity 62
 n ∑ i = 1
G(2i) −
 n ∑ i = 1
G(2i − 1) =
 2n ∑ i = 1
(−1)iG(i) = G(2n − 1) + G(0) − G(1)
Vajda-36, Dunlap-40
 n ∑ k = 1
G(k − 1) 2−k = ( G(0) + G(3) )/2 − G(n + 2) 2−n
Vajda-37, Dunlap-41,
B&Q(2003)-Identity 69
 4n+2 ∑ i = 1
G(i) = L(2n + 1) G(2n + 3)
Vajda-38, Dunlap-43, B&Q(2003)-Identity 49
 2n ∑ i = 1
G(i) G(i − 1) = G(2n)2 − G(0)2
Vajda-39, Dunlap-44, B&Q(2003)-Identity 41
 2n+1 ∑ i = 1
G(i) G(i − 1) = G(2 n + 1)2 − G(0)2 − G(1)2 + G(0)G(2)
Vajda-41, Dunlap-46
 n ∑ i = 1
G(i + 2) G(i − 1) = G(n + 1)2 − G(1)2
Vajda-43, Dunlap-48, B&Q(2003)-Identity 64
(1 + (−1)r − L(r) )
 n ∑ k = 0
G(m + kr) =
G(m) − G(m+(n+1)r) + (−1)r(G(m+nr) − G(m−r))
Fibonacci with a Golden Ring
Kung-Wei Yang Mathematics Magazine 70 (1997),
pp. 131-135.
 n ∑ i = 1
G(i)2 = G(n) G(n + 1) − G(0) G(1)
Vajda-44, Dunlap-49, B&Q(2003)-Identity 67
 ∞ ∑ i = 0
 G(a, b, i) ri
 = a + a + b r r2 − r − 1
Stan Rabinowitz,
"Second-Order Linear Recurrences" card,
Generating Function
special case (x=1/r, P=1, Q=-1)
 ∞ ∑ i = 0
 i G(a, b, i) ri
 r (b r2 − 2 a r + b − a) = (r2 − r − 1)2
-
 2n − 1 ∑ i = 1
G( i ) H( i )
= G ( 2n ) H( 2n − 1) − G(0) H(1)
B&Q(2003)-Identity 42

Summations with Binomial Coefficients

 n ∑ i = 1
 ( n−ii−1 )
= F(n)
B&Q(2003) Identity-4
 ∞ ∑ i = 0
 ( n−i−1i )
= F(n)
Vajda-54(corrected),
Dunlap-84(corrected)
 n ∑ i = 0
 ( n+i2i )
= F(2n + 1)
B&Q(2003)-Identity 165
 n−1 ∑ i = 0
 ( n+i2i+1 )
= F(2n)
B&Q(2003)-Identity 166
 n ∑ k = 0
 ( nk )
F(k) = F(2n)
S Basin & V Ivanoff (1963) Problem B-4, FQ 1.1 pg 74, FQ1.2 pg 79; B&Q(2003)-Identity 6
 n ∑ k = 0
 ( nk )
(−1)k+1 F(k) = F(n)
I Ruggles (1963) FQ 1.2 pg 77
 n ∑ k = 0
 ( nk )
(−1)k L(k) = L(n)
I Ruggles (1963) FQ 1.2 pg 77
 n ∑ k = 0
 ( nk )
F(p−k) = F(p+n)
B&Q(2003)-Identity 20
 n ∑ k = 1
 ( nk )
2kF(k) = F(3n)
B&Q(2003)-Identity 238, Vajda-68, Griffiths (2013) 8-corrected
 n ∑ k = 0
(−1)n−k
 ( nk )
2kF(2 k) = F(3 n)
Griffiths (2013) page 239-corrected
 n ∑ k = 0
 ( nk ) F(3k + m)
= 2nF(2n + m)
Griffiths (2013)
 n ∑ k = 0
(−1)n−k
 ( nk )
F(3 k) = 2n F(n)
Griffiths (2013) page 239
 n ∑ i = 0
 ( n+1i+1 )
F(i) = F(2n + 1) − 1
Vajda-50, Dunlap-82
 2n ∑ i = 0
 ( 2ni )
F(2i + p) = 5n F(2n + p)
Hoggatt-I41 (special case p=0: Vajda-69, Dunlap-85)
 2n ∑ i = 0
 ( 2ni )
L(2i) = 5n L(2n)
Vajda-71, Dunlap-87
 2n+1 ∑ i = 0
 ( 2n+1i )
F(2i + p) = 5n L(2n + 1 + p)
Hoggatt-I42 (special case p=0: Vajda-70, Dunlap-86)
 2n+1 ∑ i = 0
 ( 2n+1i )
L(2i) = 5n + 1 F(2n + 1)
Vajda-72, Dunlap-88
 2n ∑ i = 0
 ( 2ni )
F(i)2 = 5n − 1 L(2n)
Vajda-73, Dunlap-89,Hoggatt-I45
 2n ∑ i = 0
 ( 2ni )
L(i)2 = 5n L(2n)
Vajda-75, Dunlap-91, Hoggatt-I46
 2n+1 ∑ i = 0
 ( 2n+1i )
F(i)2 = 5n F(2n + 1)
Vajda-74, Dunlap-90, Hoggatt-I47
 2n+1 ∑ i = 0
 ( 2n+1i )
L(i)2 = 5n + 1 F(2n + 1)
Vajda-76, Dunlap-92
 ∞ ∑ i = 0
5i
 ( n2i+1 )
= 2n − 1 F(n)
Vajda-91, B&Q(2003)-Identity 235, Catalan 1857
 ∞ ∑ i = 0
5i
 ( n2i )
= 2n − 1 L(n)
Vajda-92, B&Q(2003)-Identity 237, Catalan (1857)-see Vajda pg 69
 k ∑ i = 0
 ( ki )
F(n)iF(n−1)k − iF(i) = F( kn )
Rabinowitz-17 (special case of Vajda-66)
 k ∑ i = 0
 ( ki )
F(n)iF(n−1)k − iL(i) = L( kn )
Rabinowitz-17 (special case of Vajda-66)
 p ∑ i = 0
 ( pi )
F(t)iF(t−1)p − iG( m+i ) = G( m+tp )
Vajda-66,B&Q(2003) Identity-11
 ∑ i ≥ 0
 ∑ j ≥ 0
 ( n − ij ) ( n − ji )
= F( 2n + 3 )
B&Q(2003) Identity 5
 F(r n) F(r)
=
 [(n − 1)/2] ∑ k = 0
(−1)k(r−1)
 ( n − k − 1k )
L(r)n − 1 − 2k
Lucas (1878) equations 74-76,
this form due to Hoggatt and Lindt (1969), see Gould (1977)
 n ∑ k = 0
(−1)k
 ( 2n+1k )
L(2n + 1 − 2k)  = 1
Griffiths (2013)

Powers of Fibonacci and Lucas as Sums

5k/2 F(t)k =
 (k−1)/2 ∑ i = 0
 ( ki )
(−1)i(t+1) √5 F( (k−2i)t ), k odd
Vajda-80
5k/2 F(t)k =
 k/2 −1 ∑ i = 0
 ( ki )
(−1)i(t+1) L( (k−2i)t ) +
 ( kk/2 )
(−1)(t+1)k/2, k even
Vajda-81
L(t)k =
 (k−1)/2 ∑ i = 0
 ( ki )
(−1)it L( (k−2i)t ) , k odd
Vajda-78
L(t)k =
 k/2 −1 ∑ i = 0
 ( ki )
(−1)it L( (k−2i)t ) +
 ( kk/2 )
(−1)tk/2, k even
Vajda-79
Fk
m
F
n
= (−1)kr

 k ∑ h = 0
( k
h
) (−1)h

Fh
r
Fk−h
r+m
F
n+kr+hm
On a General Fibonacci Identity
J H Halton, Fib Q, 3 (1965), pp 31-43

Summations with Binomials and G Series

 n ∑ i = 0
 ( ni )
G(i) = G(2n)
I Ruggles (1963) FQ 1.2 pg 77; Vajda-47; Dunlap-80
 n ∑ i = 0
 ( ni )
2i G(i) = G(3n)
B&Q(2003)-Identity 239
 n ∑ i = 0
 ( ni )
G(p − i) = G(p + n)
Vajda-46, Dunlap-79, B&Q(2003)-Identity 40
 n ∑ i = 0
 ( ni )
G(p + i) = G(p + 2n)
Vajda-49, Dunlap-81
 p ∑ i = 0
(−1)p−i
 ( pi )
G( n+i ) = G( n−p )
Vajda-51, Dunlap-83

Products

n
 F(n+1) = F(n+2)
 1 − (−1)k F(k+1)2
, n ≥ 0
k = 1
B&Q(2003) Identity 22

Trigonometric Formulae

(n-1)/2
F(n) =
 3 + 2 cos 2kπ n
, n ≥ 1
k = 1
D Lind, Problem H-93,
FQ 4 (1966), page 332
(n-2)/2
L(n) =
 3 + 2 cos (2k+1)π n
, n ≥ 2
k = 0
D Lind, Problem H-93, FQ 4 (1966),
page 252, corrected page 332

E and Logs

 ln(x) = loge(x)
Here we use g for ln(Phi), the natural log of Phi so that cosh(g) = √5 / 2.
 F( 2n ) = 2 sinh( 2ng ) √5
from Binet's formula
 = sinh( 2ng ) cosh( g )
 F( 2n+1 ) = 2 cosh( (2n+1)g ) √5
from Binet's formula
 = cosh( (2n+1)g ) cosh( g )
L( 2n) = 2 cosh( ng )from Binet's formula
L( 2n+1 ) = 2 sinh( ng )from Binet's formula
 ∞ ∑ k = 1
 Φ F(k) − F(k+1) k
 = ∞ ∑ k = 1
 √5 F(k) − L(k) 2 k
= g
C. Brown (Jan 2016) private communication
 ∞ ∑ n = 0
 F(n) n!
=
 eΦ − e−φ √5
Exponential Generating Functions For Fibonacci Identities
C Church and Marjorie Bicknell, Fib Q vol 11 (1983) 275-281,with z=1
 ∞ ∑ n = 0
 L(n) n!
= eΦ + e−φ
Exponential Generating Functions For Fibonacci Identities
C Church and Marjorie Bicknell, Fib Q vol 11 (1983) 275-281,with z=1

Complex Numbers

i = −1
sin( π/2 + i ln(Φ) ) = (√5)/2 = Φ + ½ Schroeder 1986, equation (5.41) page 68
F(n) =
 n−1 ∏ k = 1
 ( 1 − 2 i cos k πn )
D Lind, Problem H-64, FQ 3 (1965), page 116
 F(n) = 2 i 1−n sin(−i n ln( i Phi) ) √5
from Rabinowitz-7 corrected, using Phi2 = (√5 + 1)/(√5 − 1)
 F(n) = 2 i −n sinh(n ln( i Phi) ) √5
from Rabinowitz-7 corrected
L(n) = 2 i −n cos(−i n ln( i Phi) ) from Rabinowitz-7 corrected
L(n) = 2 i −n cosh( n ln( i Phi) ) from Rabinowitz-7 corrected
 √ 1 + 2i = √Phi + i √phi = [ 1 + i ; 2 + 2i ]
I J Good (1993)
1 + i/2 = ( √5 + 2 + i √5 − 2  ) /2
= ( Phi3/2 + i phi3/2 ) /2
 = [ 1 + i ; 1 + i ] 2
I J Good (1993)

Generating Functions

This section is now part of the following reference page on Linear Recurrence Relations and Generating Functions

References

• : a book
• : an article (paper) in an academic journal
The Fibonacci Quarterly journal: all papers older than 7 years are freely available online as PDFs; those published within the last 7 years are only available online to subscribers.
Arranged in alphabetical order of author:

• B&Q (2003) Proofs That Really Count A T Benjamin, J J Quinn, Mathematical Association of America, 2003, ISBN 0-88385-333-7, hardback, 194 pages.
Art Benjamin and Jennifer Quinn have a wonderful knack of presenting proofs that involve counting arrangements of dominoes and tiling patterns in two ways that convince you that a formula really is true and not just "proved"! The identities proved mainly involve Fibonacci, Lucas and the General Fibonacci series with chapters on continued fractions, binomial identites, the Harmonic and Stirling numbers too. There is so much here to inspire students to find proofs for themselves and to show that proofs can be fun too!
Important notation difference: Benjamin and Quinn use fn for the Fibonacci number F(n+1)
• Bergum and Hoggatt (1975)
G. E. Bergum and V. E. Hoggatt, Jr. "Sums and Products for Recurring Sequences" Fib Q 13 (1975), pages 115-120 free pdf
• Benjamin, Carnes, Cloitre (2009)
"Recounting the Sums of Cubes of Fibonacci Numbers" A T Benjamin, T A. Carnes, B Cloitre, Congressus Numerantium, Proceedings of the Eleventh International Conference on Fibonacci Numbers and their Applications, William Webb (ed.), Vol 194, pp. 45-51, 2009.
• Binet (1843) J P M Binet, page 563 of Comptes Rendus, Vol 17, Paris, 1843
• Bro A Brousseau (1968) A Sequence of Power Formulas The Fibonacci Quarterly vol 6 (1968) pages 81-83 as pdf
gives the recurrence relations of powers of Fibonacci's in terms of Fibonomials, as developed at the start of this page, but without explicitly stating the general formula and without recognizing the Fibonomials.
• Bro U Alfred (1964) Continued Fractions and Fibonacci and Lucas Ratios The Fibonacci Quarterly vol 4 (1964) pages 269-276 pdf
Formulae for F(n)/F(n-a), L(n)/F(n-a), L(n)/L(n-a) and G(a,b,n)/G(a,b,n-k) are developed into CFs
• L E Dickson History of the Theory of Numbers: Vol 1 Divisibility and Primality
is a classic and monumental reference work on all aspects of Number Theory in 3 volumes (volume II is on Diophantine Analysis and volume III on Quadratic and Higher Forms). Although not up-to-date (the original edition was 1952) it is still a comprehensive summary of useful historical and early references on all aspects of Number Theory. The link is to a new cheap Dover paperback edition (2005) of Volume 1 which contains the most about Fibonacci Numbers, Lucas numbers and the golden section: see Chapter XV11 on Recurring Series, Lucas' un, vn where he uses the series of Pisano for what we now call the Fibonacci numbers.
• Dunlap (1997), The Golden Ratio and Fibonacci Numbers R A Dunlap, World Scientific Press, 1997, 162 pages.
An introductory book strong on the geometry and natural aspects of the golden section but it does not include much on the mathematical detail. Beware - some of the formulae in the Appendix are wrong! Dunlap has copied them from Vajda's book (see below) and he has faithfully preserved all of the original errors! The formulae on this page (that you are now reading) are corrected versions and have been verified.
• Fairgrieve and Gould (2005)
"Product Difference Fibonacci Identities of Simson, Gelin-Cesáro, Tagiuri and Generalizations" S Fairgrieve and H W Gould, The Fibonacci Quarterly vol 43 (2005), 137-141. free pdf
• Falcon, Plaza (2007) "On the Fibonacci k-numbers" S Falcon, A Plaza Chaos, Solitons, Fractals 32 (2007) pages 1615-1624
• Gould (1977) "A Fibonacci Formula of Lucas and its Subsequent Manifestations and Rediscoveries" H W Gould, Fibonacci Quarterly vol 15 (1977) pages 25-29 free pdf
• R L Graham, D E Knuth, O Patashnik Concrete Mathematics Second Edition (1994), hardback, Addison-Wesley.
No - this is not a book about proportions of sand to cement . The title is meant as an antidote to the "Abstract Mathematics" courses so often found in the curricullum of a university maths degree.
As such, it is the book to dip into if you want to go really deeply into any part of the mathematics covered on this Fibonacci and Phi site. However, it quickly gets to an advanced mathematics undergraduate level after some nice introductions to every topic.
There are notes left in the margins which were inserted by students taking the original courses based on this book at Stanford university and they are interesting, often useful and sometimes quite funny.
• Griffiths (2013)
"From golden ratio equalities to Fibonacci and Lucas Identities" M Griffiths Math. Gaz. 97 (2013) pages 234-241
Some errors and typos in the paper have been corrected on this webpage.
• Hansen (1972)
"Generating Identities for Fibonacci and Lucas Triples" Rodney T Hansen, FQ (1972), pages 571-578 pdf
• V E Hoggatt Jr "Fibonacci and Lucas Numbers" published by The Fibonacci Association, 1969 (Houghton Mifflin) (free online)
A very good introduction to the Fibonacci and Lucas Numbers written by a founder of the Fibonacci Quarterly.
• Hoggatt and Lind (1969) "Compositions and Fibonacci Numbers" V E Hoggatt Jr, D A Lind, The Fibonacci Quarterly, Vol. 7, No. 3 (Oct., 1969), pp. 253-266. free pdf
• Howard (2003) "The Sum of the Squares of Two Generalized Fibonacci Numbers", F T Howard, FQ vol 41 pages 80-84, pdf
A F Horadam "Special Properties of the Sequence wn(a,b;p,q)" FQ 5 (1967) pgs 424-434 pdf
• Hudson and Winans (1981)
"A Complete Characterization of the Decimal Fractions That Can Be Represented as Σ 10k(a + 1)Fai , where Fai is the aith Fibonacci Number" R H Hudson, C F Winans The Fibonacci Quarterly 19, no. 5 (1981) pages 414-421. free pdf
A Complete Characterization Of B-Power Fractions That Can Be Represented As Series Of General N-Bonacci Numbers J-Z Lee, J-S Lee Fibonacci Quarterly 25 (1987) pages 72-75. free pdf
• I J Good "Complex Fibonacci And Lucas Numbers, Continued Fractions and The Square Root Of The Golden Ratio", Fib Q 31 (1993) pages 7-19 free pdf and corrections
• Khomovsky (2018) "A Method for Obtaining Fibonacci Identities" D Khomovsky, #A42 in INTEGERS vol 18 (2018) link
A useful paper on generalising many Fibonacci identities to General Fibonacci series.
• Knuth (1997) The Art of Computer Programming: Vol 1 Fundamental Algorithms D E Knuth, hardback, Addison-Wesley third edition (1997).
The paperback is now out of print but the title link is to hardcover, second hand and Kindle versions. This is the first of three volumes and an absolute must for all computer scientist/mathematicians.
• Koshy (2001) Fibonacci and Lucas Numbers with Applications, T Koshy, Wiley-Interscience, 2001, 648 pages.
This book is packed full of an amazing number of Fibonacci and related equations, mostly culled from the pages of the Fibonacci Quarterly. Although initially impressive in its size and breadth, be aware that there are far too many typos, errors and missing or irrelevant conditions in many of its formulae as well as some glaring omissions and misattributions particularly with respect to the original references for a number of the formulae. Although Fibonacci representations of integers are included Zeckendorf himself is never even mentioned! There are lots of exercises with answers to the odd-numbered questions.
• Long (1981) The Decimal Expansion Of 1/89 And Related Results, C Long Fibonacci Quarterly vol 19 (1985) pages 53-55 free pdf
• Long (1986)
Discovering Fibonacci Identities, FQ 24 (1986), pages 160-166 pdf
• Lucas (1876)
E Lucas, in Nuov. Corresp. Math. 2 (1876) , pages 74-75
See Dickson Vol 1 page 395
• E Lucas, "Théorie des Fonctions Numériques Simplement Périodiques" in American Journal of Mathematics vol 1 (1878) pages 184-240 and 289-321.
Reprinted in English translation as The Theory of Simply Periodic Functions, the Fibonacci Association, 1969 free pdf
• R S Melham (1999) "Families of Identities Involving Sums of Powers of the Fibonacci and Lucas Numbers" FQ vol 37 (1999), pages 315-319 free pdf
• R S Melham (2011), On Product Difference Fibonacci Identities Article A10, Integers, vol 11
J Morgado "Note on some results of A F Horadam and A G Shannon concerning Catalan's Identity on Fibonacci Numbers" Portugaliae Math. 44 (1987) pgs 243-252 pdf
• Ohtsuka and Nakamura (2010)
"A New Formula For The Sum Of The Sixth Powers Of Fibonacci Numbers" H Ohtsuka, S Nakamura, Congressus Numerantium Vol. 201 (2010), Proceedings of the Thirteenth Conference on Fibonacci Numbers and their Applications , pp.297-300.
• S Rabinowitz (1996) "Algorithmic Manipulation of Fibonacci Identities" in Applications of Fibonacci Numbers: Proceedings of the Sixth International Research Conference on Fibonacci Numbers and their Applications, editors G E Bergum, A N Philippou, A F Horodam; Kluwer Academic (1996), pages 389 - 408.
• Schroeder (1986)
Number Theory in Science and Communication, With Applications in Cryptography, M R Schroeder, Springer-Verlag (2nd enlarged edition) 1986.
• B Sharpe (1965)
On Sums Fx2 ± Fy2 Fib Quart (1965) 3.1 page 63 pdf
• S Vajda, Fibonacci and Lucas numbers, and the Golden Section: Theory and Applications, Dover Press (2008).
This is a wonderful book, a classic, originally published in 1989 and now back in print in this Dover edition. This book is full of formulae on the Fibonacci numbers and Phi and the Lucas numbers. The whole book develops the formulae step by step, proving each from earlier ones or occasionally from scratch. It has a few errors in its formulae and all of them have been dutifully and exactly copied by authors such as Dunlap (see above) and others! Where I have identified errors, they have been corrected here on this page.
• Vorob'ev (1951) Fibonacci Numbers N N Vorob'ev (2013 Dover paperback of the 1961 English version which itself was translated from the Russian 1951 edition)
An excellent and compact source book but only 66 pages long.