Fibonacci and Golden Ratio Formulae

Here are almost 200 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.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/) 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.

Contents of This Page


Definitions and Notation

Beware of different golden ratio symbols used by different authors!
At this web site Phi is 1.618033... and phi is 0.618033.. but Vajda (see below) and Dunlap (see below) use a symbol for -0.618033.. .
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. 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.
As used hereVajdaDunlapDescription
floor(x)[x]trunc(x), not used for x<0 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 +  1 ]
--
2
trunc(x+1/2) the nearest integer to x, equivalent to 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)-- 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)-- the fractional part of x, i.e. the part of abs(x) after the decimal point
Knuth writes this a x mod 1 defined as x–floor(x)
(n
r
)
(n
r
)
(n
r
)
 = 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.

F(i) is the Fibonacci series 0,1,1,2,3,5,... and L(i) is the Lucas series 2,1,3,4,7,11,....

FormulaRefsComments
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-63Vajda and Dunlap use tau (τ) and Koshy uses alpha (α).
Phi and –phi are the roots of x2 = x + 1
phi = 0.618... =
√5 – 1
--
2
Dunlap-65Vajda uses –σ, and Dunlap uses –φ and Koshy uses –β
Beware! Dunlap occasionally uses φ to represent our phi = 0.61803.., but more frequently he uses φ to represent -0.618033..

Linear Formulae

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

Linear Sums of Fibonacci numbers

by Definition of L(n)
F(n + 2) + F(n) + F(n – 2) = 4 F(n) B&Q(2003)-Identity 18
F(n + 2 ) + F(n) = L(n + 1)
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 + 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 + 1) + F(n – 1) = L(n)Vajda-6, Hoggatt-18,
Dunlap-14, Koshy-5.14
F(n) + 2 F(n – 1) = L(n)(Dunlap-32)
F(n + 2) + F(n – 2) = 3 F(n) B&Q(2003)-Identity 7
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
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

Here Phi (see Definitions above) is Vajda's and Dunlap's τ)
and –phi (see Definitions above) is Vajda's σ, Dunlap's φ and Koshy's β.
Basic Phi Formulae
Phi phi = 1Vajda page 51(3), Dunlap-65
Phi + phi = √5-
Phi / phi = Phi + 1-
phi / Phi = 1 – phi-
Phi – phi = 1-
Phi = phi + 1 = √5 – phi-
phi = Phi – 1 = √5 – Phi -
Phi2 = 1 + Phi Vajda page 51(4), Dunlap-64
phi2 = 1 – phiVajda page 51(4), Dunlap-64
Phin+2 = Phin+1 + Phin-
(–phi)n+2 = (–phi)n+1 + (–phi)n-
phin = phin+1 + phin+2-
(–Phi)n = (–Phi)n+1 + (–Phi)n+2-

Golden Ratio with Fibonacci and Lucas

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)nVajda-59, Dunlap-70, B&Q(2003)-Identity 241
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 ) ,if n≥0
--
√5
-
L(–n) = round( (–phi)–n ), n≥3-
F(–n) = (–1)n+1round( Phin ) ,if n≥0
--
√5
-
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
fract( F(2n+1) Phi ) = phi2n–1Knuth vol 1, Ex 1.2.8 Qu 31
Phin = Phi F(n) + F(n–1)Rabinowitz-28, B&Q(2003)-Corrolary 33
Phin = L(n) + F(n)√5
--
2
Rabinowitz-25, B&Q(2003)-Identity 242, Vajda page 125
Phin = F(n+1) + F(n) phiRabinowitz-28, B&Q(2003)-Corollary 33
(–phi)n = L(n) – F(n)√5
--
2
Rabinowitz-25, B&Q(2003)-Identity 243, Vajda page 125
(–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)-

Order 2 Formulae

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

Fibonacci numbers

F(n)2 + 2 F(n – 1)F(n) = F(2n)-
F(n + 1)2 + F(n)2 = F(2n + 1)Vajda-11, Dunlap-7, Lucas(1876), B&Q(2003)-Identity 13
F(n + 1)2 – F(n – 1)2 = F(2n)Lucas(1876), B&Q(2003)-Identity 14
F(n + 1)2 – F(n)2 = F(n + 2) F(n – 1) Vajda-12, Dunlap-8
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 Vajda-11,Dunlap-7
Melham(1999)
F(n + 1) F(n – 1) – F(n)2 = (–1)n Cassini's Formula(1680), Simson(1753), Vajda-29, Dunlap-9,
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 + 1)F(m + 1) – F(n – 1)F(m – 1) = F(n + m) B&Q(2003)-Identity 231
F(n) = F(m) F(n + 1 – m) + F(m – 1) F(n – m) Dunlap-10
F(n + m) = F(m) F(n + 1) + F(m – 1) F(n) alternative to Dunlap-10, B&Q(2003)-Identity 3
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
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
F(a)(Fb) – 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 41 (2003) B-960, pg 182.
Cassini, Catalan and D'Ocagne's Identities
are all special cases of this formula
( 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(nk) is a multiple of F(n) B&Q(2003)-Theorem 1
gcd(F(m),F(n)) = F(gcd(m,n))Lucas (1876)
B&Q(2003)-Theorem 6
F(m) mod F(n) = F(k)Knuth Vol 1 Ex 1.2.8 Qu. 32

Lucas numbers

L(2n) = L(n)2 – 2 (–1)nB&Q(2003)-Identity 36
L(n + 2) L(n – 1) = L(n + 1)2 – L(n)2-
L(n + 1) L(n – 1) – L(n)2 = –5 (–1)nB&Q(2003)-Identity 60
L(2n) + 2 (–1)n = L(n)2Vajda-17c, Dunlap-12
L(n + m) + (–1)m L(n – m) = L(m) L(n)Vajda-17a, Dunlap-11
[ 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 Numbers

F(2n) = F(n) L(n)Vajda-13, Hoggatt-17,
Koshy-5.13,
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)Vajda-25a
L(n + 1)2 – L(n)2 = 5 F(2n)-
L(n + 1)2 – 5 F(n) = L(2n + 1)2-
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
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(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)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
2 F(n + m) = L(m) F(n) + L(n) F(m)Vajda-16a, Dunlap-21
2 L(n + m) = L(m) L(n) + 5 F(n) F(m)-
(–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 + i) L(n + k) – L(n) L(n + i + k)
= (–1)n + 1 5 F(i) F(k)
Vajda-20b
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

Higher Order Fibonacci and Lucas

F(3n) = F(n + 1)3 + F(n)3 – F(n – 1)3 B&Q(2003)-Identity 232
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 + 1)F(n + 2)F(n + 6) – F(n + 3)3 = (–1)nF(n) FQ 41 (2003) pg 142, Melham
F(n – 2)F(n – 1)F(n + 1)F(n + 2) + 1 = F(n)4 Gelin-Cesàro Identity (1880)
FQ 41 (2003) pg 142, B&Q(2003)-Identity 31
L(n – 2)L(n – 1)L(n + 1)L(n + 2) + 25 = L(n)4B&Q(2003)-Identity 56
F(n)F(n+2)F(n+3)F(n+5) + 1 = [ F(n+4)2 – 2F(n+3)2 ]2 -
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(n) + √5 F(n)) k=L(kn) + √5 F(kn)
----
22
De Moivre Analogue
( L(n) – √5 F(n)) k=L(kn) – √5 F(kn)
----
22
De Moivre Analogue
(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(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(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 Fibonaci Numbers

Portugaliae Math. 44 (1987) pgs 243-252

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), but the "starting values" of G(0)=a and G(1)=b can be specified. It therefore includes both series them both as special cases. 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). Hoggatt and others use the letter H for series G. For example:

Basic G Formulae

Two independent G series are denoted G(n) and H(n).
G(n) =
( G(0) phi + G(1) ) Phin + (G(0) Phi – G(1)) ( –phi )n
sqrt--
5
Vajda-55/56, Dunlap-77, B&Q(2003)-Identity 244
G(n + 2) = G(n + 1) + G(n)Vajda-3, Dunlap-4
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))-
G(n + m) = F(m – 1) G(n) + F(m) G(n + 1)Vajda-8, Dunlap-33, B&Q(2003)-Identity 38
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
G(n + m) – (–1)m G(n – m) = F(m) ( G(n–1) + G(n+1))B&Q(2003)-Identity 48
F(m) (G(n – 1) + G(n + 1)) = G(n + m) – (–1)m G(n – m)Vajda-10b, Dunlap-36
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
a special case of Johnson's:
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 (see ref below)
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)2B&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's (39a)
4G(i)2G(i+1)2 + G(i–1)2G(i+2)2 = ( G(i)2 + G(i+1)2 )2Generalised Fibonacci Pythagorean Triples
A F Horadam Special Properties of the Sequence wn(a,b;p,q) FQ 5 (1967) pgs 424-434
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
sum
i=0
F(i) = F(n + 2) – 1
Hoggatt-11, Lucas(1876), B&Q 2003-Identity 1
n
sum
i=0
(-1) i F(i) = (-1)n F(n – 1) – 1
B&Q 2003-Identity 21
n
sum
i=0
L(i) = L(n + 2) – 1
Hoggatt-12
n
sum
i=a
F(i) = F(n + 2) – F(a + 1)
-
n
sum
i=a
L(i) = L(n + 2) – L(a + 1)
-
n
sum
i=0
F(2i) = F(2n + 1) – 1, n≥0
Hoggatt-16, Lucas(1876), B&Q(2003)-Identity 12
n
sum
i=1
F(2i – 1) = F(2n), n≥1
Hoggatt-15, Lucas(1876), B&Q(2003)-Identity 2
n
sum
i=1
L(2i – 1) = L(2n) – 2
-
n
sum
i=1
 2n – i F(i – 1) = 2n  – F(n + 2)
Vajda-37a(adapted),
Dunlap-42(adapted),
B&Q(2003)-Identity 10
n
sum
i=0
 2i L(i) = 2n+1 F(n + 1)
B&Q(2003)-Identity 236
n
Sum
i = 0
F(3i - 1)
=
F(3n + 1) + 1
2
B&Q(2003)-Identity 24
n
Sum
i = 0
F(3i)
=
F(3n + 2) – 1
2
B&Q(2003)-Identity 25
n
Sum
i = 0
F(3i + 1)
=
F(3n + 3)
2
B&Q(2003)-Identity 23
n
sum
i=0
F(4i) = F(2n + 1)2 – 1
B&Q 2003-Identity 27
n
sum
i=0
F(4i + 1) = F(2n + 1)F(2n + 2)
B&Q 2003-Identity 26
n
sum
i=0
F(4i + 2) = F(2n + 1)F(2n + 3)– 1
B&Q 2003-Identity 29
n
sum
i=0
F(4 i + 3) = F(2n + 3)F(2n + 2)
B&Q 2003-Identity 28
n
sum
i=0
(–1)i L(n – 2i) = 2 F(n + 1)
Vajda-97, Dunlap-54
n
sum
i=0
(–1)i L(2n – 2i + 1) = F(2 n + 2)
B&Q(2003)-Identity 55

Summations with fractions

Sum
i=0
F(i
--
2i
 = 2
Vajda-60, Dunlap-51
Sum
i=0
L(i)
--
2i
 = 6
-
Sum
i=0
F(i)
--
ri
=
r
--
r2 – r – 1
-
Sum
i=0
L(i)
--
ri
= 2 +
r +2
--
r2 – r – 1
-
Sum
i=1
i F(i)
--
2i
 = 10
Vajda-61, Dunlap-52
Sum
i=1
i L(i)
--
2i
 = 22
-
Sum
i=1
1
--
F(2i)
 = 4 – Phi = 3 – phi
Vajda-77(corrected), Dunlap-53(corrected)

Order 2 summations

2n
sum
i=1
F(i) F(i – 1) = F(2n)2
Vajda-40, Dunlap-45
2n
sum
i=1
L(i) L(i – 1) = L(2n)2 – 4
-
2n+1
sum
i=1
F(i) F(i – 1) = F(2n +1)2 – 1
Vajda-42, Dunlap-47
2n+1
sum
i=1
L(i) L(i – 1) = L(2n +1)2 – 5
-
n–1
Sum
i=0
F(2i + 1)2 =
F(4n) + 2n
--
5
Vajda-95, B&Q(2003)-Identity 234
n–1
Sum
i=0
L(2i + 1)2 = F(4n) – 2n
Vajda-96, B&Q(2003)-Identity 54
n
sum
i=1
F(i)2 = F(n) F(n + 1)
Vajda-45, Dunlap-5,
Hoggatt-13, Lucas(1876),
Koshy-77,
B&Q(2003)-Identity 9 (Identity 233 variant)
n
sum
i=1
L(i)2 = L(n) L(n + 1) – 2
Hoggatt-14
2n-1
sum
i=1
L(i)2 = 5 F(2n) F(2n - 1)
-
5
n
sum
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
sum
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
sum
i=0
F(i) L(n – i) = (n + 1) F(n)
Vajda-100, Dunlap-57, B&Q(2003)-Identity 35
n
Sum
i=1
L(2i)2
= F(4n + 2) + 2n – 1
Vajda page 70

Summations of order > 2

F(m q) = F(m)
q
Sum
j = 1
F(m - 1) j-1 F( m(q - j) + 1 )
B&Q(2003)-Theorem 2

G Summations

Two independent G series are denoted G(n) and H(n).
n
sum
i=1
G(i) = G(n + 2) – G(2)
Vajda-33, Dunlap-38, B&Q(2003)-Identity 39
n
sum
i=a
G(i) = G(n + 2) – G(a + 1)
-
n
sum
i=1
G(2i – 1) = G(2n) – G(0)
Vajda-34, Dunlap-37, B&Q(2003)-Identity 61
n
sum
i=1
G(2i) = G(2n + 1) – G(1)
Vajda-35, Dunlap-39, B&Q(2003)-Identity 62
n
sum
i=1
G(2i) –
n
sum
i=1
G(2i – 1) = G(2n – 1) + G(0) – G(1)
Vajda-36, Dunlap-40
n
sum
i=1
2n – i G(i – 1) = 2n – 1( G(0) + G(3) ) – G(n + 2)
= 2n ( G(0) + G(1) ) – G(n + 2)
Vajda-37, Dunlap-41,
B&Q(2003)-Identity 69
4n+2
sum
i=1
G(i) = L(2n + 1) G(2n + 3)
Vajda-38, Dunlap-43, B&Q(2003)-Identity 49
2n
sum
i=1
G(i) G(i – 1) = G(2n)2 – G(0)2
Vajda-39, Dunlap-44, B&Q(2003)-Identity 41
2n+1
sum
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
sum
i=1
G(i + 2) G(i – 1) = G(n + 1)2 – G(1)2
Vajda-43, Dunlap-48, B&Q(2003)-Identity 64
n
sum
i=1
G(i)2 = G(n) G(n + 1) – G(0) G(1)
Vajda-44, Dunlap-49, B&Q(2003)-Identity 67
Sum
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)
Sum
i=0
i G(a, b, i)
--
ri
r (b r2 – 2 a r + b – a)
=
--
(r2 – r – 1)2
-
2n – 1
Sum
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
Sum
i=1
(n–i
i–1
)
= F(n)
B&Q(2003) Identity-4
Sum
i=0
(n–i–1
i
)
= F(n)
Vajda-54(corrected),
Dunlap-84(corrected)
n
Sum
i = 0
(n + i)
2 i
= F(2n + 1)
B&Q(2003)-Identity 165
n - 1
Sum
i = 0
(n + i)
2 i + 1
= F(2n)
B&Q(2003)-Identity 166
n
Sum
k = 0
(n)
k
F(k)
= F(2 n)
B&Q(2003)-Identity 6
n
Sum
k = 0
(n)
k
F(p – k)
= F(p + n)
B&Q(2003)-Identity 20
n
Sum
k=1
(n)
k
2 kF(k)
= F(3n)
B&Q(2003)-Identity 238
n
Sum
i=0
(n+1
i+1
)
F(i) = F(2n + 1) – 1
Vajda-50, Dunlap-82
2n
Sum
i=0
(2n
i
)
F(2i) = 5n F(2n)
Vajda-69, Dunlap-85
2n
Sum
i=0
(2n
i
)
L(2i) = 5n L(2n)
Vajda-71, Dunlap-87
2n+1
Sum
i=0
(2n+1
i
)
F(2i) = 5n L(2n + 1)
Vajda-70, Dunlap-86
2n+1
Sum
i=0
(2n+1
i
)
L(2i) = 5n + 1 F(2n + 1)
Vajda-72, Dunlap-88
2n
Sum
i=0
(2n
i
)
F(i)2 = 5n – 1 L(2n)
Vajda-73, Dunlap-89
2n
Sum
i=0
(2n
i
)
L(i)2 = 5n L(2n)
Vajda-75, Dunlap-91
2n+1
Sum
i=0
(2n+1
i
)
F(i)2 = 5n F(2n + 1)
Vajda-74, Dunlap-90
2n+1
Sum
i=0
(2n+1
i
)
L(i)2 = 5n + 1 F(2n + 1)
Vajda-76, Dunlap-92
Sum
i=0
5i 
(n
2i+1
)
 = 2n-1 F(n)
Vajda-91, B&Q(2003)-Identity 235
Sum
i=0
5i 
(n
2i
)
 = 2n-1 L(n)
Vajda-92, B&Q(2003)-Identity 237
k
Sum
i=0
(k
i
)
F(n)iF(n–1)k–iF(i) = F( kn )
Rabinowitz-17
k
Sum
i=0
(k
i
)
F(n)iF(n–1)k–iL(i) = L( kn )
Rabinowitz-17
 
Sum
i ≥ 0
 
 
Sum
j ≥ 0
(n - i)
j
(n - j)
i
= F(2 n + 3)
B&Q(2003) Identity 5

Summations with Binomials and G Series

n
Sum
i=0
(n
i
)
G(i) = G(2n)
Vajda-47, Dunlap-80
n
Sum
i=0
(n
i
)
2i G(i) = G(3n)
B&Q(2003)-Identity 239
n
Sum
i=0
(n
i
)
G(p – i) = G(p + n)
Vajda-46, Dunlap-79, B&Q(2003)-Identity 40
n
Sum
i=0
(n
i
)
G(p + i) = G(p + 2n)
Vajda-49, Dunlap-81
n
Sum
i=0
(–1)i
(n
i
)
G(n + p – i) = G(p – n)
Vajda-51, Dunlap-83

Other Formulae

F(n) =
floor((n-1)/2)
Product
k=0
( 3 + 2 cos  2kπ )
--
n
-

Hyperbolic Functions

Here we use g for ln(Phi), the natural log of Phi. cosh(g)=√5 / 2. There are several derivations of formulae above using hyperbolic functions in chapter XI of Vajda.
F( 2n ) = 2sinh( 2ng )
--
√5
from Binet's formula
         = sinh( 2ng )
--
cosh( g )
F( 2n+1 ) = 2cosh( (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

Complex Numbers

F(n) = 2 i1–n sin(–i n ln( i Phi) )
--
√5
from Rabinowitz-7 corrected
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

References

(*) above indicates a private communication.
Book: : a book;
Article: : an article (chapter, paper) in a book (journal);
WWW: : a web resource.
FQ : The Fibonacci Quarterly
Arranged in alphabetical order of author:

Book: A T Benjamin, J J Quinn Proofs That Really Count Mathematical Association of America, 2003, ISBN 0-88385-333-7, hardback, 194 pages. shown as B&Q(2003) in the Table above
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)
Book: R A Dunlap, The Golden Ratio and Fibonacci Numbers 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.
Book: V E Hoggatt Jr "Fibonacci and Lucas Numbers" published by The Fibonacci Association, 1969 (Houghton Mifflin).
A very good introduction to the Fibonacci and Lucas Numbers written by a founder of the Fibonacci Quarterly.
Article: F T Howard (2003) "The Sum of the Squares of Two Generalized Fibonacci Numbers" FQ vol 41 pages 80-84.
WWW: R Johnson (Durham university) has an excellent web page
on the power of matrix methods to establish many Fibonacci formula with ease (but it does rely on at least undergraduate level matrix mathematics). See the Matrix methods for Fibonacci and Related Sequences link to a Postscript and PDF version on his Fibonacci Resources web page.
The latest version (Nov 12, 2004) contains an appendix showing how formulae developed in Johnson's paper can prove almost all the identities here in my table above.
Book: D E Knuth The Art of Computer Programming: Vol 1 Fundamental Algorithms hardback, Addison-Wesley third edition (1997).
The paperback is now out of print and hard to find. This is the first of three volumes and an absolute must for all computer scientist/mathematicians.
Book: T Koshy Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, 2001, 648 pages.
This is a new book packed full of an amazing number of Fibonacci and related equations, 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.
Article: E Lucas, "Théorie des Fonctions Numériques Simplement Périodiques" in American Journal of Mathematics vol 1 (1878) pages 184-240 and 289-321.
Reprinted as The Theory of Simply Periodic Functions, the Fibonacci Association, 1969.
Article: R S Melham (1999) "Families of Identities Involving Sums of Powers of the Fibonacci and Lucas Numbers" FQ vol 37, pages 315-319.
Article: S Rabinowitz "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.
Book: S Vajda, "Fibonacci and Lucas numbers, and the Golden Section: Theory and Applications", Halsted Press (1989).
This is a wonderful book, a classic but now unfortunately out of print. Vajda packs the book full of formulae on the Fibonacci numbers and Phi and the Lucas numbers. The whole book develops these 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 above! Where I have identified errors, they have been corrected above.

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