General Summations

n
i=1
G(i) = G(n + 2) – G(2) Vajda-33, Dunlap-38
n
G(i) = G(n + 2) – G(a + 1) -
Fibonacci and Golden Ratio Equations http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibForm...
10 of 12 6/4/03 1:49 PM
i=a
n
i=1
G(2 i – 1) = G(2 n) – G(0) Vajda-34, Dunlap-37
n
i=1
G(2 i) = G(2 n + 1) – G(1) Vajda-35, Dunlap-39
n
i=1
G(2 i) –
n
i=1
G(2 i – 1) = G(2 n – 1) + G(0) – G(1) Vajda-36, Dunlap-40
n
i=1
2n – i G(i – 1) = 2n – 1( G(0) + G(3) ) – G(n + 2) Vajda-37(variant), Dunlap-41(variant)
4 n + 2
i=1
G(i) = L(2 n + 1) G(2 n + 3) Vajda-38, Dunlap-43
2 n
i=1
G(i) G(i – 1) = G(2 n)2 – G(0)2 Vajda-39, Dunlap-44
2 n +
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
n
i=1
G(i)2 = G(n) G(n + 1) – G(0) G(1) Vajda-44, Dunlap-49
i = 0
G(a, b, i)
ri
a + b r
= a +
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 -
n
i = 1
n – i
i – 1
= F(n) -
Fibonacci and Golden Ratio Equations http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibForm...
11 of 12 6/4/03 1:49 PM
i = 0
n – i – 1
i = F(n) Vajda-54(corrected), Dunlap-84(corrected)
n
i = 0
n + 1
i + 1 F(i) = F(2 n + 1) – 1 Vajda-50, Dunlap-82
2 n
i = 0
2 n
i F(2 i) = 5n F(2 n) Vajda-69, Dunlap-85
2 n
i = 0
2 n
i L(2 i) = 5n L(2 n) Vajda-71, Dunlap-87
2 n + 1
i = 0
2 n + 1
i F(2 i) = 5n L(2 n + 1) Vajda-70, Dunlap-86
2 n + 1
i = 0
2 n + 1
i L(2 i) = 5n + 1 F(2 n + 1) Vajda-72, Dunlap-88
2 n
i = 0
2 n
i F(i)2 = 5n – 1 L(2 n) Vajda-73, Dunlap-89
2 n
i = 0
2 n
i L(i)2 = 5n L(2 n) Vajda-75, Dunlap-91
2 n + 1
i = 0
2 n + 1
i F(i)2 = 5n F(2 n + 1) Vajda-74, Dunlap-90
2 n + 1
i = 0
2 n + 1
i L(i)2 = 5n + 1 F(2 n + 1) Vajda-76, Dunlap-92
i=0
5i n
2 i + 1 = 2n-1 F(n) Vajda-91
5i n
2 i = 2n-1 L(n) Vajda-92
Fibonacci and Golden Ratio Equations http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibForm...
12 of 12 6/4/03 1:49 PM
i=0
With Generalised Fibonacci
n
i = 0
ni
G(i) = G(2 n) Vajda-47, Dunlap-80
n
i = 0
ni
G(p – i) = G(p + n) Vajda-46, Dunlap-79
n
i = 0
n
i
G(p + i) = G(p + 2 n) Vajda-49, Dunlap-81
n
i = 0
(–1)i n
i G(n + p – i) = G(p – n) Vajda-51, Dunlap-83

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