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As used here  Vajda  Dunlap  Knuth  Definition  Description  

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] 

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  





_{n}C_{r} n choose r; the element in row n column r of Pascal's Triangle the coefficient of x^{r} 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) 
Fibonaccitype series with the rule S(i)=S(i1)+S(i2) 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)  Vajda2, Dunlap5  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)  Vajda4, Dunlap6  Extending the Lucas series 'backwards'  
G(n + 2) = G(n + 1) + G(n)  Vajda3, Dunlap4  Definition of the Generalised Fibonacci series, G(0) and G(1) needed  

Dunlap63 
Phi and –phi are the roots of x^{2} = x + 1  

Dunlap65 
Beware! Dunlap occasionally uses φ to
represent our phi = 0.61803.., but more frequently he uses
φ to represent –0.61803.. !  
2 F(n + 1) = F(n) + √(5 F(n)^{2} + 4(–1)^{n})  F(n+1) from F(n): Problem B42, S Basin, FQ, 2 (1964) page 329 
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), Vajda6, HoggattI8, Dunlap14, Koshy5.14 
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)  HoggattI14 
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)  (Dunlap32) 
F(n + 2) – F(n – 2) = L(n)  Vajda7a, Dunlap15, Koshy5.15 
F(n + 3) – 2 F(n) = L(n)  possible correction for Dunlap31 
F(n + 2) – F(n) + F(n – 1) = L(n)  possible correction for Dunlap31 
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)  Vajda5, Dunlap13, Koshy5.16, B&Q(2003)Identity 34, HoggattI9 
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)  Vajda7b, Dunlap16, B&QIdentity 51 
L(n) + 5 F(n) = 2 L(n + 1)   
3 F(n) + L(n) = 2 F(n + 2)  Vajda26, Dunlap28 
3 L(n) + 5 F(n) = 2 L(n + 2)  Vajda27, Dunlap29 
Phi = 
 ; phi = 

Phi phi = 1  Vajda page 51(3), Dunlap65 
Phi + phi = √5   
Phi / phi = Phi + 1   
phi / Phi = 1 – phi   
Phi – phi = 1   
Phi = phi + 1 = √5 – phi   
phi = Phi – 1 = √5 – Phi   
Phi^{2} = 1 + Phi  Vajda page 51(4), Dunlap64 
phi^{2} = 1 – phi  Vajda page 51(4), Dunlap64 
Phi^{n+2} = Phi^{n+1} + Phi^{n}   
(–phi)^{n+2} = (–phi)^{n+1} + (–phi)^{n}   
phi^{n} = phi^{n+1} + phi^{n+2}   
(–Phi)^{n} = (–Phi)^{n+1} + (–Phi)^{n+2}   

Vajda101  

Vajda101a  
 "Binet's" Formula De Moivre(1718), Binet(1843), Lamé(1844), Vajda58, Dunlap69, Hoggattpage 11, B&Q(2003)Identity 240  
L(n) = Phi^{n} + (–phi)^{n}  Vajda59, Dunlap70, B&Q(2003)Identity 241  
 Vajda62, Dunlap71 corrected, B&Q(2003)Identity 240 Corollary 30  
L(n) = round(Phi^{n}),if n≥2  Vajda63, Dunlap72, B&Q(2003)Corollary 35  
   
L(–n) = round( (–phi)^{–n} ), n≥2    
   
F(n + 1) = round(Phi F(n)),if n≥2  Vajda64, Dunlap73  
L(n + 1) = round(Phi L(n)),if n≥4  Vajda65, Dunlap74  
fract( F(2n) phi ) = 1 – phi^{2n}  Knuth vol 1, Ex 1.2.8 Qu 31 with ψ=phi  
fract( F(2n+1) phi ) = phi^{2n+1}  Knuth vol 1, Ex 1.2.8 Qu 31  

Rabinowitz25, B&Q(2003)Identity 242, Vajda page 125  
Phi^{n} = Phi F(n) + F(n–1)  Rabinowitz28, B&Q(2003)Corrolary 33  
Phi^{n} = F(n+1) + F(n) phi  Rabinowitz28, B&Q(2003)Corollary 33  

I Ruggles (1963) FQ 1.2 pg 80, Rabinowitz25, B&Q(2003)Identity 243, Vajda page 125  

I Ruggles (1963) FQ 1.2 pg 80  
(–phi)^{n} = –phi F(n) + F(n–1)  Rabinowitz28  
(–phi)^{n} = F(n+1) – Phi F(n)  Vajda103b, Dunlap75  
√5 Phi^{n} = Phi L(n) + L(n–1)    
√5 (–phi)^{n} = phi L(n) – L(n–1)   
F(n)^{2} + 2 F(n – 1)F(n) = F(2n)   
F(n + 1)^{2} + F(n)^{2} = F(2n + 1)  Vajda11, Dunlap7, Lucas(1878), B&Q(2003)Identity 13, HoggattI11 
F(n + 1)^{2} – F(n – 1)^{2} = F(2n)  Lucas(1878), B&Q(2003)Identity 14, HoggattI10 
F(n + 1)^{2} – F(n)^{2} = F(n + 2) F(n – 1)  Vajda12, Dunlap8 
F(n + 2)^{2} = 3 F(n + 1)^{2} – F(n)^{2} – 2 (–1)^{n}  V E Hoggatt B208 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)  a generalization of Vajda11,Dunlap7 Melham(1999) 
F( n+p )^{2} – F( n–p )^{2} = F( 2n )F( 2p )  I Ruggles (1963) FQ 1.2 pg 77; HoggattI25 
F(n + 1) F(n – 1) – F(n)^{2} = (–1)^{n} 
Cassini's Formula(1680), Simson(1753), Vajda29, Dunlap9, HoggattI13 special case of Catalan's Identity with r=1 B&Q(2003)Identity 8 
F(n)^{2} – F(n + r)F(n – r) = (1)^{nr}F(r)^{2}  Catalan's Identity(1879) 
F(n)F(m + 1) – F(m)F(n + 1) = (1)^{m}F(n – m) 
d'Ocagne's Identity, special case of Vajda9 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 Dunlap10, B&Q(2003)Identity 3;
variation of R T Hansen FQ (1972) "Generating Identities for Fibonacci and Lucas Triples" p 571578 
F(n) = F(m) F(n + 1 – m) + F(m – 1) F(n – m)  I Ruggles (1963) FQ 1.2 pg 79; Dunlap10, special case of Vajda8 
F(n) F(n + 1) = F(n – 1) F(n + 2) + (–1)^{n1}  Vajda20a special case: i:=1;k:=2;n:=n1; HoggattI19 
F(n + i) F(n + k) – F(n) F(n + i + k) = (–1)^{n} F(i) F(k)  Vajda20a=Vajda18 (corrected) with G:=H:=F 
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) B960 'A Fibonacci Iddentity', solution pg 90 also Johnson7 Cassini, Catalan and D'Ocagne's Identities are all special cases of this formula 
( F(n1)F(n+2) )^{2} + (2 F(n)F(n+1) )^{2} = (F(n+1)F(n+2) – F(n1)F(n))^{2} = F(2n+1)^{2} 
A F Horadam FQ 20 (1982) pgs 121122, B&Q(2003)Identity 19 (corrected) special case of Generalised Fibonacci Pythagorean Triples 
F(nk) is a multiple of F(n)  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 
L(n + 2)^{2} = 3 L(n + 1)^{2} – L(n)^{2} + 10(–1)^{n}  V E Hoggatt B208 FQ 9 (1971) pg 217. 
L(n + 2) L(n – 1) = L(n + 1)^{2} – L(n)^{2}   
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}  Vajda17c, Dunlap12, B&Q(2003)Identity 36 
L(n + m) + (–1)^{m} L(n – m) = L(m) L(n)  Vajda17a, Dunlap11 (special cases: HoggattI15,I18) 
L(4n) + 2 = L(2n)^{2}  HoggattI15, special case of Vajda17a 
2 L(n + 1) = L(n) + √5 √(L(n)^{2} – 4(–1)^{n})  L(n+1) from L(n): Problem B42, S Basin, FQ 2 (1964) page 329 
gcd(L(m),L(n)) = L(gcd(m,n)), if both s/d and t/d are odd integers  Vajda page 86 
L(mn+r) ≡ ± L(r) (mod L(n) )  (Vajda page 87) 
F(2n) = F(n) L(n)  Vajda13, HoggattI7, Koshy5.13, B&Q(2003)Identity 33 
5 F(n) = L(n + 1) + L(n – 1)  
L(n + 1)^{2} + L(n)^{2} = 5 F(2n + 1)  Vajda25a 
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}  Vajda23, Dunlap25 
L(n)^{2} – 4(–1)^{n} = 5 F(n)^{2}  B&Q(2003)Identity 53, HoggattI12 
F(n+k) + F(n–k) = F(n)L(k), k even;  Bergum and Hoggatt (1975) equn (5) 
F(n+k) + F(n–k) = L(n)F(k), k odd;  Bergum and Hoggatt (1975) equn (6) 
F(n+k) – F(n–k) = F(n)L(k), k odd;  Bergum and Hoggatt (1975) equn (7) 
F(n+k) – F(n–k) = L(n)F(k), k even;  Bergum and Hoggatt (1975) equn (8) 
L(n+k) + L(n–k) = L(n)L(k), k even  Bergum and Hoggatt (1975) equn (9) 
L(n+k) + L(n–k) = 5F(n)F(k), k odd  Bergum and Hoggatt (1975) equn (10) 
L(n+k) – L(n–k) = L(n)L(k), k odd  Bergum and Hoggatt (1975) equn (11) 
L(n+k) – L(n–k) = 5F(n)F(k), k even  Bergum and Hoggatt (1975) equn (12) 
F(n + 1) L(n) = F(2n + 1) + (–1)^{n}  Vajda30, Vajda31, Dunlap27, Dunlap30 
L(n + 1) F(n) = F(2n + 1) – (–1)^{n}   
F(2n + 1) = F(n + 1) L(n + 1) – F(n) L(n)  Vajda14, Dunlap18 
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)  R T Hansen FQ (1972) "Generating Identities for Fibonacci and Lucas Triples" p 571578 
L(n)^{2} – 2 L(2n) = –5 F(n)^{2}  Vajda22, Dunlap24 
5 F(n)^{2} – L(n)^{2} = 4 (–1)^{n + 1}  Vajda24, Dunlap26 
F(n)^{2} + L(n)^{2} = 4 F(n + 1)^{2} – 2 F(2n)  FQ (2003)vol 41, B936, M A Rose, page 87 
5 (F(n)^{2} + F(n + 1)^{2}) = L(n)^{2} + L(n + 1)^{2}  Vajda25 
F(n) L(m) = F(n + m) + (–1)^{m} F(n – m)  a recurrence relation for F(n+km): Vajda15a, Dunlap19 
L(n) F(m) = F(n + m) – (–1)^{m} F(n – m)  Vajda15b, Dunlap20 
5 F(m) F(n) = L(n + m) – (–1)^{m} L(n – m)  Vajda17b, Dunlap23, (special cases:HoggattI16,I17) 
2 F(n + m) = L(m) F(n) + L(n) F(m)  Vajda16a, Dunlap2, FQ (1967) B106 H H Ferns pp 466467 
2 L(n + m) = L(m) L(n) + 5 F(n) F(m)  FQ (1967) B106 H H Ferns pp 466467 
F(m) L(n) + F(m – 1) L(n – 1) = L(m + n – 1)  R T Hansen FQ (1972) "Generating Identities for Fibonacci and Lucas Triples" p 571578 
(–1)^{m} 2 F(n – m) = L(m) F(n) – L(n) F(m)  Vajda16b, Dunlap22 
L(n + i) F(n + k) – L(n) F(n + i + k) = (–1)^{n + 1} F(i) L(k)  Vajda19a 
F(n + i) L(n + k) – F(n) L(n + i + k) = (–1)^{n} F(i) L(k)  Vajda19b 
L(n + i) L(n + k) – L(n) L(n + i + k) = (–1)^{n + 1} 5 F(i) F(k)  Vajda20b 
(–1)^{k}F(n)F(m–k) + (–1)^{m}F(k)F(n–m) + (–1)^{n}F(m)F(k–n) = 0  FQ 11 (1973) B228 page 108 
(–1)^{k}L(n)F(m–k) + (–1)^{m}L(k)F(n–m) + (–1)^{n}L(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+v}L(j(ku)+(rv))  FQ 16 General Identities For Linear Fibonacci And Lucas Summations, R T Hansen, 121128 
F(jk+r) L(ju+v) = F(j(k+u)+(r+v)) + (1)^{ju+v}F(j(ku)+(rv))  FQ 16 General Identities For Linear Fibonacci And Lucas Summations, R T Hansen, 121128 
L(jk+r) L(ju+v) = L(j(k+u)+(r+v)) + (1)^{ju+v}L(j(ku)+(rv))  FQ 16 General Identities For Linear Fibonacci And Lucas Summations, R T Hansen, 121128 
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 
Johnson32, special case of Johnson44 
 B&Q(2003)Theorem 2  
 Vajda85  
 Vajda86  
 Vajda87  
L(t) is not a factor of L(kt) for even k  
 Vajda88  
L(t) is not a factor of F(kt) for odd k and t≥3 
F(3n) = F(n + 1)^{3} + F(n)^{3} – F(n – 1)^{3}  B&Q(2003)Identity 232  
F(n + 1)F(n + 2)F(n + 6) – F(n + 3)^{3} = (–1)^{n}F(n)
F(n)F(n + 4)F(n + 5) – F(n + 3)^{3} = (–1)^{n+1}F(n + 6) 
FQ 41 (2003) pg 142, Melham. The second is a variant with n for n and using Vajda2 

F(n–2)F(n–1)F(n+3) – F(n)^{3} = (–1)^{n1}F(n–3) F(n+2)F(n+1)F(n–3) – F(n)^{3} = (–1)^{n}F(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)^{n1} 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(nab)F(n+a)F(n+b) = (–1)^{n+a+b}F(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+c}F(a+b–c)( F(c)F(n+a+b–c) + (–1)^{c}F(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)  
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 155160 
F(n–1)^{2}F(n+1)^{2} – F(n–2)^{2}F(n+2)^{2} = 4(–1)^{n}F(n)^{2}  Melham (2011) 21 
F(n–3)F(n–1)F(n+1)F(n+3) – F(n)^{4} = (–1)^{n}L(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)  Vajda32 
F(n – 2)F(n – 1)F(n + 1)F(n + 2) + 1 = F(n)^{4}  GelinCesàro Identity (1880) (see Dickson page 401) FQ 41 (2003) pg 142, B&Q(2003)Identity 31 HoggattI29, 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(nabc)F(n+a)F(n+b)F(n+c) = (–1)^{n+a+b+c}F(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+c}F(a+b–d)F(a+c–d)F(b+c–d)F(2n+d) 
Melham (2011) Theorem 6 
(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 155160 
[ L(n1)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(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} 
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 243252  
 De Moivre Analogue, S Fisk (1963) FQ 1.2 Problem B10, pg 85. HoggattI44 
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 Vajda2  
√5 G(n) = ( G(0) phi + G(1) ) Phi^{n} + (G(0) Phi – G(1)) ( –phi )^{n}  Vajda55/56, Dunlap77, B&Q(2003)Identity 244  
 Amer Math Monthly (2005) "Fibonacci, Chebyshev and Orthogonal Polynomials" D Aharonov, A Beardam, K Driver, p612630  
2 G(k) = ( 2 G(1) – G(0) ) F(k) + G(0) L(k)  Johnson46  
G(n + m) = F(m – 1) G(n) + F(m) G(n + 1)  Vajda8, Dunlap33, B&Q(2003)Identity 38, Johnson40  
G(n – m) = (–1)^{m} (F(m + 1) G(n) – F(m) G(n + 1))  Vajda9, Dunlap34, B&Q(2003)Identity 47  
G(n + m) + (–1)^{m} G(n – m) = L(m) G(n)  Vajda10a, Dunlap35, 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))  Vajda10b, Dunlap36, 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)  Vajda21a  
G(m) F(n) – G(n) F(m) = (–1)^{m} G(0) F(n – m)  Vajda21b  
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)) 
Vajda18 (corrected), B&Q(2003)Identity 44 a special case of Johnson's: 
G(p)H(q) – G(r)H(s) = (1)^{n}[ G(pn)H(qn) – G(rn)H(sn) ] if p+q = r+s and p,q,r,s,n are integers 
Johnson44 
G(n + 1) G(n – 1) – G(n)^{2} = (–1)^{n} (G(1)^{2} – G(0) G(2))  Vajda28, 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)^{2}G(i+1)^{2} + G(i–1)^{2}G(i+2)^{2} = ( G(i)^{2} + G(i+1)^{2} )^{2}  Generalised Fibonacci Pythagorean Triples A F Horadam Special Properties of the Sequence w_{n}(a,b;p,q) FQ 5 (1967) pgs 424434 
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 

HoggattI1, Lucas(1878), B&Q 2003Identity 1  

B&Q 2003Identity 21  

HoggattI2  

  

  

HoggattI6, Lucas(1878), B&Q(2003)Identity 12  

HoggattI5, Lucas(1878), B&Q(2003)Identity 2  
   

Vajda37a(adapted), Dunlap42(adapted), B&Q(2003)Identity 10  

B&Q(2003)Identity 236  
 B&Q(2003)Identity 23  
 B&Q(2003)Identity 24 (corrected)  
 B&Q(2003)Identity 25 (corrected)  

B&Q 2003Identity 27  

B&Q 2003Identity 26  

B&Q 2003Identity 29  

B&Q 2003Identity 28  

Vajda97, Dunlap54  

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(n1) + b P(n2) for n≥2; P(0) = c; P(1) = d and m and N are defined by B^{2} = m + Ba + b, N = cm + dB + bc, then
provided that abs( (a+√(a^{2}+4b))/(2B) )<1 and  (a–√(a^{2}+4b))/(2B)  < 1  Long (1981) 

Vajda60, Dunlap51  

  

  

  

Vajda61, Dunlap52  

  

Vajda77(corrected), Dunlap53(corrected)  

Vajda89 (corrected)  

R L Graham (1963) FQ 1.1, Problem B9, pg 85, FQ 1.4 page 79  

R L Graham (1963) FQ 1.1, Problem B9, pg 85  

Johnson11, Vajda102 

Vajda45, Dunlap5, HoggattI3, Lucas(1878), Koshy77, B&Q(2003)Identity 9 (Identity 233 variant)  

HoggattI4  

  

Vajda40, Dunlap45  

  

Vajda42, Dunlap47  

  

Vajda93  

Vajda94  

Vajda95, B&Q(2003)Identity 234  

Vajda page 70  

Vajda96, B&Q(2003)Identity 54  

Vajda page 70  

Vajda98, Dunlap55, B&Q(2003)Identity 58  

Vajda99, Dunlap56, B&Q(2003)Identity 57  

Vajda100, Dunlap57, B&Q(2003)Identity 35  

V Hoggatt (1965) Problem B53 FQ 3, pg 157 

adapted from Benjamin, Carnes, Cloitre (2009)  

see A005969  

Ohtsuka and Nakamura (2010) Theorem 1  

Ohtsuka and Nakamura (2010) Theorem 2 

L G Brökling (1964) FQ 2.1 Problem B20 solution, pg76; Vajda33; Dunlap38; B&Q(2003)Identity 39  

  

Vajda34, Dunlap37, B&Q(2003)Identity 61  

Vajda35, Dunlap39, B&Q(2003)Identity 62  

Vajda36, Dunlap40  

Vajda37, Dunlap41, B&Q(2003)Identity 69  

Vajda38, Dunlap43, B&Q(2003)Identity 49  

Vajda39, Dunlap44, B&Q(2003)Identity 41  

Vajda41, Dunlap46  

Vajda43, Dunlap48, B&Q(2003)Identity 64  

Fibonacci with a Golden Ring KungWei Yang Mathematics Magazine 70 (1997), pp. 131135.  

Vajda44, Dunlap49, B&Q(2003)Identity 67  

Stan Rabinowitz, "SecondOrder Linear Recurrences" card, Generating Function special case (x=1/r, P=1, Q=1)  

  
 B&Q(2003)Identity 42 

B&Q(2003) Identity4  

Vajda54(corrected), Dunlap84(corrected)  

B&Q(2003)Identity 165  

B&Q(2003)Identity 166  

S Basin & V Ivanoff (1963) Problem B4, FQ 1.1 pg 74, FQ1.2 pg 79; B&Q(2003)Identity 6  

I Ruggles (1963) FQ 1.2 pg 77  

I Ruggles (1963) FQ 1.2 pg 77  

B&Q(2003)Identity 20  

B&Q(2003)Identity 238, Vajda68  

Vajda50, Dunlap82  

HoggattI41 (special case p=0: Vajda69, Dunlap85)  

Vajda71, Dunlap87  

HoggattI42 (special case p=0: Vajda70, Dunlap86)  

Vajda72, Dunlap88  

Vajda73, Dunlap89,HoggattI45  

Vajda75, Dunlap91, HoggattI46  

Vajda74, Dunlap90, HoggattI47  

Vajda76, Dunlap92  

Vajda91, B&Q(2003)Identity 235, Catalan 1857  

Vajda92, B&Q(2003)Identity 237, Catalan (1857)see Vajda pg 69  

Rabinowitz17 (special case of Vajda66)  

Rabinowitz17 (special case of Vajda66)  

Vajda66  

B&Q(2003) Identity 5  
 Lucas (1878) equns 7476, this form due to Hoggatt and Lindt (1969), see Gould (1977) 

Vajda80  

Vajda81  

Vajda78  

Vajda79  

On a General Fibonacci Identity, J H Halton, Fib Q, 3 (1965), pp 3143 

I Ruggles (1963) FQ 1.2 pg 77; Vajda47; Dunlap80  

B&Q(2003)Identity 239  
 Vajda46, Dunlap79, B&Q(2003)Identity 40  

Vajda49, Dunlap81  

Vajda51, Dunlap83 

D Lind, Problem H93, FQ 4 (1966), page 332 

D Lind, Problem H93, FQ 4 (1966), page 252, corrected page 332 
 from Binet's formula  
 
 from Binet's formula  
 
L( 2n) = 2 cosh( ng )  from Binet's formula  
L( 2n+1 ) = 2 sinh( ng )  from Binet's formula 

D Lind, Problem H64, FQ 3 (1965), page 116  
 from Rabinowitz7 corrected  
 from Rabinowitz7 corrected  
L(n) = 2 i^{–n} cos(–i n ln( i Phi) )  from Rabinowitz7 corrected  
L(n) = 2 i^{–n} cosh( n ln( i Phi) )  from Rabinowitz7 corrected  
√i/√Phi = [1+i, ]  = √Phi+I J Good (1993)  
2 √i) √Phi + (1 – i) √phi  = (1 +I J Good (1993) 
G(x) =  ∞ Σ i=0  S(i) x^{i}  = S(0) + S(1) x + S(2) x^{2} + S(3) x^{3} + ... 
To shift to the right (insert a 0 at the start of the series so all other terms have an index increased by 1),
multiply the GF by x; to shift to the left, divide by x.
There is much more on GFs on my Fibonomials page.
Fibonacci(n) 0,1,1,2,3,... 

Lucas(n) 2,1,3,4,7,... 

G(a,b,n) a,b,a+b,a+2b,... 


Fibonacci(2n) 0,1,3,8,21,... 

Lucas(2n) 2,3,7,18,... 

G(a,b,2n) a,a+b,2a+3b,... 


Fibonacci(2n+1) 1,2,5,13,... 

Lucas(2n+1) 1,4,11,29,... 

G(a,b,2n+1) b,a+2b,3a+5b,... 


Fibonacci(3n) 2,8,34,144,... 

Lucas(3n) 2,4,18,76,... 

G(a,b,3n) a,a+2b,5a+8b,... 


Fibonacci(3n+1) 1,3,13,55,... 

Lucas(3n+1) 3,11,47,199,... 

G(a,b,3n+1) a+b,3a+5b,13a+21b,... 


Fibonacci(3n+2) 1,5,21,89,... 

Lucas(3n+2) 2,4,18,76,... 

G(a,b,3n+2) a,a+2b,5a+8b,... 


Fibonacci(k n) 

Lucas(k n) 

G(a,b,kn) 


Fibonacci(n)^{2} 0^{2},1^{2},1^{2},2^{2},3^{2},... 

Lucas(n)^{2} 2^{2},1^{2},3^{2},4^{2},... 

G(a,b,n)^{2} a^{2},b^{2},(a+b)^{2},... 


Fib(n)Fib(n+1) 1×1,1×2,2×3,3×5,... 

Lucas(n)Lucas(n+1) 2×1,1×3,3×4,4×7,... 

G(a,b,n)G(a,b,n+1) ab,b(a+b), (a+b)(a+2b),... 


Fibonacci(n)^{3} 0^{3},1^{3},1^{3},2^{3},3^{3},... 

Lucas(n)^{3} 2^{3},1^{3},3^{3},4^{3},... 

The GF of 1,2,5,13,... is that of
Fib(2n+1)
which is
(1 – x)/(x^{2} – 3 x + 1)
so 1,0,2,0,5,0,13,... has GF
(1 – x^{2})/(x^{4} – 3 x^{2} + 1)
To insert an extra 0 at the start, multiply the GF by x.
So the GF for the oddindexed Fibonacci numbers only in their correct positions in the Fibonacci series
with Fib(2n+1) is the coefficient of x^{2n+1}
is therefore x (1 – x^{2})/(x^{4}  3 x^{2} + 1)
for the series 0,1,0,2,0,5,0,13,... .
Adding these two GFs, that is, for Fib(2n) as the coefficient of x^{2n}
and Fib(2n+1) as the coefficient of x^{2n+1}
should then give the complete
Fibonacci series GF:
0,0,1,0,3,0,8, 0,21, ... +We can check that x^{2}/(x^{4}  3 x^{2} + 1) + x (1 – x^{2})/(x^{4}  3 x^{2} + 1) = x/(1 – x – x^{2})
0,1,0,2,0,5,0,13, 0, ...
0,1,1,2,3,5,8,13,21,...
Multiplying a GF by a constant k multiples all the members of the series by k.
A series formed by adding two series S and T elementwise to form the series {S(n)+T(n) for n=1,2,3,...},
has a GF which is the sum of the two separate GFs.
Check that a Fib[n1] + b Fib[n] gives the GF of G(a,b).
 See, e.g., Solving Linear Recurrences from Differential Equations in the Exponential Manner and Vice Versa W Oberschelp in Applications of Fibonacci Numbers Vol 6 (1996) pages 365380 
: a book;  
: an article (chapter) in a journal (book);  
: a web resource.  
FQ  : The Fibonacci Quarterly journal 
Fibonacci and Phi in the Arts 
Fibonacci Home Page
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