A companion page on Linear Recurrences and their generating Functions for Fibonacci Numbers, Continued Fraction convergents, Pythagorean triples and other series of numbers.
As used here | Vajda | Dunlap | Knuth | Definition | Description | ||||||||||||||
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Phi Φ |
τ | τ | φ, α |
| Koshy uses α (page 78) | ||||||||||||||
phi φ | −σ | −φ | −β |
| 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] |
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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 | ||||||||||||||
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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 ... −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 ... Fibonacci
F(i)... −8 5 −3 2 −1 1 0 1 1 2 3 5 8 ... Lucas
L(i)... 18 −11 7 −4 3 −1 2 1 3 4 7 11 18 ... General Fib
G(a,b,i)... 13a−8b −8a+5b 5a−3b −3a+2b 2a−b −a+b a b a+b a+2b 2a+3b 3a+5b 5a+8b ...
Formula | Refs | Comments | ||||
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-6 | Extending the Lucas series 'backwards' | ||||
G(n + 2) = G(n + 1) + G(n) | Vajda-3, Dunlap-4 | Definition of the Generalised Fibonacci series, G(0) and G(1) needed | ||||
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Dunlap-63 |
Phi and −phi are the roots of x2 = x + 1 | ||||
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Dunlap-65 |
Beware! Dunlap occasionally uses φ to
represent our phi = 0.61803.., but more frequently he uses
φ to represent −0.61803.. ! |
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(*) |
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) | - |
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 |
Φ = Phi = | 1 + √5 | = 1.61803398874989484820458683436... |
2 | ||
−φ = −phi = | 1 − √5 | = −0.61803398874989484820458683436... |
2 |
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 |
Phi = | √5 + 1 | = | 1 | = phi + 1; phi = | √5 −1 | = | 1 | = Phi −1 |
2 | phi | 2 | Phi |
| "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) phi | Rabinowitz-28, B&Q(2003)-Corollary 33 | |||
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Vajda-50b, Rabinowitz-25, B&Q(2003)-Identity 242, I Ruggles (1963) FQ 1.2 pg 80 | |||
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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) | - |
Phi + 2 = √5 Phi | ||||
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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 |
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Vajda-101 | ||||||||
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Vajda-101a | ||||||||
| Vajda-62, Dunlap-71 corrected, B&Q(2003)-Identity 240 Corollary 30 | ||||||||
L(n) = round(Phin),if n≥2 | Vajda-63, Dunlap-72, B&Q(2003)-Corollary 35 | ||||||||
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L(−n) = round( (−phi)−n ), n≥2 | - | ||||||||
F(n + 1) = round(Phi F(n)),if n≥2 | Vajda-64, Dunlap-73 | ||||||||
L(n + 1) = round(Phi L(n)),if n≥4 | Vajda-65, Dunlap-74 | ||||||||
fract( F(2n) phi ) = 1 − phi2n | Knuth vol 1, Ex 1.2.8 Qu 31 with ψ=phi | ||||||||
fract( F(2n+1) phi ) = phi2n+1 | Knuth vol 1, Ex 1.2.8 Qu 31 |
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) | ||||||||||||
| B&Q(2003)-Theorem 2 | ||||||||||||
| Vajda-85 | ||||||||||||
| Vajda-86 | ||||||||||||
| Vajda-87 | ||||||||||||
L(t) is not a factor of L(kt) for even k | |||||||||||||
| 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 |
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 |
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 |
F(2n) = F(n) L(n) | Vajda-13, Hoggatt-I7, Koshy-5.13, B&Q(2003)-Identity 33 | ||||||||||||
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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)2 | Vajda-23, Dunlap-25 | ||||||||||||
L(n)2 − 4(−1)n = 5 F(n)2 | B&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)n | Vajda-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)2 | Vajda-22, Dunlap-24 | ||||||||||||
5 F(n)2 − L(n)2 = 4 (−1)n + 1 | Vajda-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)2 | Vajda-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 |
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 |
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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 |
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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 |
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 |
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 |
Morgado (1987) | ||||||||||
| De Moivre Analogue, S Fisk (1963) FQ 1.2 Problem B-10, pg 85. Hoggatt-I44 |
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:
| = |
| = |
| if n ≥ k ≥ 0 | |||||||||||
= 0, otherwise |
(( | n | )) |
k |
[ |
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] |
n k | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
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0 | 1 | |||||||
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 |
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Vajda page 74, "add the two numbers above" analogy from Pascal's triangle | ||||||||||||||||||||||||||||||||||||||||||||
| Melham (1999).... | ||||||||||||||||||||||||||||||||||||||||||||
| .... 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: | ||||||||||||||||||||||||||||||||||||||||||||
| Knuth AoCP Vol 1 section 1.2.8 Exercise 30, (1997) | ||||||||||||||||||||||||||||||||||||||||||||
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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 | ||||
| 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(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 |
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Hoggatt-I1, Lucas(1878), B&Q 2003-Identity 1 | |||||||||||
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B&Q 2003-Identity 21 | |||||||||||
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Hoggatt-I2 | |||||||||||
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Hoggatt-I6, Lucas(1878), B&Q(2003)-Identity 12 | |||||||||||
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Hoggatt-I5, Lucas(1878), B&Q(2003)-Identity 2 | |||||||||||
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Vajda-37a(adapted), Dunlap-42(adapted), B&Q(2003)-Identity 10 | |||||||||||
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B&Q(2003)-Identity 236 | |||||||||||
| B&Q(2003)-Identity 23 | |||||||||||
| B&Q(2003)-Identity 24 (corrected) | |||||||||||
| B&Q(2003)-Identity 25 (corrected) | |||||||||||
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B&Q 2003-Identity 27 | |||||||||||
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B&Q 2003-Identity 26 | |||||||||||
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B&Q 2003-Identity 29 | |||||||||||
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B&Q 2003-Identity 28 | |||||||||||
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Vajda-97, Dunlap-54 | |||||||||||
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B&Q(2003)-Identity 55 |
1/89 | = | 0.0 1 1 2 3 5 ... |
1/9899 | = | 0.00 01 01 02 03 05 08 13 21 ... |
| 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
provided that |(a+√(a2+4b))/(2B)| < 1 and | (a−√(a2+4b))/(2B) | < 1 | Long (1981) |
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Vajda-60, Dunlap-51 | |||||||||||||
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- | |||||||||||||
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Vajda-61, Dunlap-52 | |||||||||||||
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- | |||||||||||||
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Vajda-77(corrected), Dunlap-53(corrected) | |||||||||||||
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Vajda-89 (corrected) | |||||||||||||
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R L Graham (1963) FQ 1.1, Problem B-9, pg 85, FQ 1.4 page 79 | |||||||||||||
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R L Graham (1963) FQ 1.1, Problem B-9, pg 85 | |||||||||||||
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Johnson-11, Vajda-102 |
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Vajda-45, Dunlap-5, Hoggatt-I3, Lucas(1878), Koshy-77, B&Q(2003)-Identity 9 (Identity 233 variant) | ||||||||||
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Hoggatt-I4 | ||||||||||
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Vajda-40, Dunlap-45 | ||||||||||
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Vajda-42, Dunlap-47 | ||||||||||
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- | ||||||||||
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Vajda-93 | ||||||||||
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Vajda-94 | ||||||||||
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Vajda-95, B&Q(2003)-Identity 234 | ||||||||||
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Vajda page 70 | ||||||||||
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Vajda-96, B&Q(2003)-Identity 54 | ||||||||||
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Vajda page 70 | ||||||||||
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Vajda-98, Dunlap-55, B&Q(2003)-Identity 58 | ||||||||||
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Vajda-99, Dunlap-56, B&Q(2003)-Identity 57 | ||||||||||
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Vajda-100, Dunlap-57, B&Q(2003)-Identity 35 | ||||||||||
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V Hoggatt (1965) Problem B-53 FQ 3, pg 157 |
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adapted from Benjamin, Carnes, Cloitre (2009) | ||||||
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see A005969 | ||||||
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Ohtsuka and Nakamura (2010) Theorem 1 | ||||||
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Ohtsuka and Nakamura (2010) Theorem 2 |
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L G Brökling (1964) FQ 2.1 Problem B-20 solution, pg76; Vajda-33; Dunlap-38; B&Q(2003)-Identity 39 | |||||||||||||||||
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Vajda-34, Dunlap-37, B&Q(2003)-Identity 61 | |||||||||||||||||
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Vajda-35, Dunlap-39, B&Q(2003)-Identity 62 | |||||||||||||||||
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Vajda-36, Dunlap-40 | |||||||||||||||||
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Vajda-37, Dunlap-41, B&Q(2003)-Identity 69 | |||||||||||||||||
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Vajda-38, Dunlap-43, B&Q(2003)-Identity 49 | |||||||||||||||||
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Vajda-39, Dunlap-44, B&Q(2003)-Identity 41 | |||||||||||||||||
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Vajda-41, Dunlap-46 | |||||||||||||||||
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Vajda-43, Dunlap-48, B&Q(2003)-Identity 64 | |||||||||||||||||
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Fibonacci with a Golden Ring Kung-Wei Yang Mathematics Magazine 70 (1997), pp. 131-135. | |||||||||||||||||
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Vajda-44, Dunlap-49, B&Q(2003)-Identity 67 | |||||||||||||||||
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Stan Rabinowitz, "Second-Order Linear Recurrences" card, Generating Function special case (x=1/r, P=1, Q=-1) | |||||||||||||||||
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- | |||||||||||||||||
| B&Q(2003)-Identity 42 |
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B&Q(2003) Identity-4 | ||||||||||||||||
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Vajda-54(corrected), Dunlap-84(corrected) | ||||||||||||||||
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B&Q(2003)-Identity 165 | ||||||||||||||||
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B&Q(2003)-Identity 166 | ||||||||||||||||
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S Basin & V Ivanoff (1963) Problem B-4, FQ 1.1 pg 74, FQ1.2 pg 79; B&Q(2003)-Identity 6 | ||||||||||||||||
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I Ruggles (1963) FQ 1.2 pg 77 | ||||||||||||||||
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I Ruggles (1963) FQ 1.2 pg 77 | ||||||||||||||||
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B&Q(2003)-Identity 20 | ||||||||||||||||
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B&Q(2003)-Identity 238, Vajda-68, Griffiths (2013) 8-corrected | ||||||||||||||||
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Griffiths (2013) page 239-corrected | ||||||||||||||||
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Griffiths (2013) | ||||||||||||||||
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Griffiths (2013) page 239 | ||||||||||||||||
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Vajda-50, Dunlap-82 | ||||||||||||||||
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Hoggatt-I41 (special case p=0: Vajda-69, Dunlap-85) | ||||||||||||||||
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Vajda-71, Dunlap-87 | ||||||||||||||||
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Hoggatt-I42 (special case p=0: Vajda-70, Dunlap-86) | ||||||||||||||||
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Vajda-72, Dunlap-88 | ||||||||||||||||
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Vajda-73, Dunlap-89,Hoggatt-I45 | ||||||||||||||||
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Vajda-75, Dunlap-91, Hoggatt-I46 | ||||||||||||||||
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Vajda-74, Dunlap-90, Hoggatt-I47 | ||||||||||||||||
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Vajda-76, Dunlap-92 | ||||||||||||||||
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Vajda-91, B&Q(2003)-Identity 235, Catalan 1857 | ||||||||||||||||
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Vajda-92, B&Q(2003)-Identity 237, Catalan (1857)-see Vajda pg 69 | ||||||||||||||||
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Rabinowitz-17 (special case of Vajda-66) | ||||||||||||||||
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Rabinowitz-17 (special case of Vajda-66) | ||||||||||||||||
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Vajda-66,B&Q(2003) Identity-11 | ||||||||||||||||
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B&Q(2003) Identity 5 | ||||||||||||||||
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Lucas (1878) equations 74-76, this form due to Hoggatt and Lindt (1969), see Gould (1977) | ||||||||||||||||
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Griffiths (2013) |
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Vajda-80 | |||||||||||||||||||||
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Vajda-81 | |||||||||||||||||||||
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Vajda-78 | |||||||||||||||||||||
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Vajda-79 | |||||||||||||||||||||
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On a General Fibonacci Identity J H Halton, Fib Q, 3 (1965), pp 31-43 |
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I Ruggles (1963) FQ 1.2 pg 77; Vajda-47; Dunlap-80 | ||||||||||
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B&Q(2003)-Identity 239 | ||||||||||
| Vajda-46, Dunlap-79, B&Q(2003)-Identity 40 | ||||||||||
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Vajda-49, Dunlap-81 | ||||||||||
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Vajda-51, Dunlap-83 |
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B&Q(2003) Identity 22 |
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D Lind, Problem H-93, FQ 4 (1966), page 332 | |||||||||||
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D Lind, Problem H-93, FQ 4 (1966), page 252, corrected page 332 |
ln(x) = loge(x) |
| from Binet's formula | ||||||||||||||||
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| from Binet's formula | ||||||||||||||||
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L( 2n) = 2 cosh( ng ) | from Binet's formula | ||||||||||||||||
L( 2n+1 ) = 2 sinh( ng ) | from Binet's formula | ||||||||||||||||
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C. Brown (Jan 2016) private communication | ||||||||||||||||
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Exponential Generating Functions For Fibonacci Identities C Church and Marjorie Bicknell, Fib Q vol 11 (1983) 275-281,with z=1 | ||||||||||||||||
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Exponential Generating Functions For Fibonacci Identities C Church and Marjorie Bicknell, Fib Q vol 11 (1983) 275-281,with z=1 |
sin( π/2 + i ln(Φ) ) = (√5)/2 = Φ + ½ | Schroeder 1986, equation (5.41) page 68 | |||||||||||||||||||||||||
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D Lind, Problem H-64, FQ 3 (1965), page 116 | |||||||||||||||||||||||||
| from Rabinowitz-7 corrected, using Phi2 = (√5 + 1)/(√5 − 1) | |||||||||||||||||||||||||
| 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 | |||||||||||||||||||||||||
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I J Good (1993) | |||||||||||||||||||||||||
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I J Good (1993) |
Fibonacci and Phi in the Arts |
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