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Generalized Fibonacci Number

Generalized Fibonacci numbers extend the Fibonacci numbers through the conditions 1=G_1=G_2=...=G_(c-1) and the recurrence relation

G_n=G_(n-1)+G_(n-c).
(1)

These are the sums of elements on successive diagonals of a left-justified Pascal's triangle beginning in the leftmost column and moving in steps of c-1 up and 1 right. The case c=2 equals the usual Fibonacci number. These numbers satisfy the identities

G_1+G_2+G_3+...+G_n=G_(n+3)-1
(2)
G_3+G_6+G_9+...+G_(3k)=G_(3k+1)-1
(3)
G_1+G_4+G_7+...+G_(3k+1)=G_(3k+2)
(4)
G_2+G_5+G_8+...+G_(3k+2)=G_(3k+3)
(5)

(Bicknell-Johnson and Spears 1996). For the special case c=3,

G_(n+w)=G_(w-2)G_n+G_(w-3)G_(n+1)+G_(w-1)G_(n+2).
(6)

Bicknell-Johnson and Spears (1996) give many further identities.

Horadam (1965) defined the generalized Fibonacci numbers {w_n} as w_n=w_n(a,b;p,q), where a, b, p, and q are integers, w_0=a, w_1=b, and w_n=pw_(n-1)-qw_(n-2) for n>=2. They satisfy the identities

w_nw_(n+2r)-eq^nU_r=w_(n+r)^2
(7)
4w_nw_(n+1)^2w_(n+2)+(wq^n)^2=(w_nw_(n+2)+w_(n+1)^2)^2
(8)
w_nw_(n+1)w_(n+3)w_(n+4)=w_(n+2)^4+eq^n(p^2+q)w_(n+2)^2+e^2q^(2n+1)p^2
(9)
4w_nw_(n+1)w_(n+2)w_(n+4)w_(n+5)w_(n+6)+e^2q^(2n)(w_nU_4U_5-w_(n+1)U_2U_6-w_nU_1U_8)^2 =(w_(n+1)w_(n+2)w_(n+6)+w_nw_(n+4)w_(n+5))^2,
(10)

where

e=pab-qa^2-b^2
(11)
U_n=w_n(0,1;p,q)
(12)

(Dujella 1996). The final above result is due to Morgado (1987) and is called the morgado identity.

Another generalization of the Fibonacci numbers is denoted x_n. Given x_1 and x_2, define the generalized Fibonacci number by x_n=x_(n-2)+x_(n-1) for n>=3,

sum_(i=1)^nx_i=x_(n+2)-x_2
(13)
sum_(i=1)^(10)x_i=11x_7
(14)
x_n^2-x_(n-1)x_(n+2)=(-1)^n(x_2^2-x_1^2-x_1x_2),
(15)

where the plus and minus signs alternate.

See also

Fibonacci n-Step Number, Fibonacci Number

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References

Bicknell-Johnson, M. and Spears, C. P. "Classes of Identities for the Generalized Fibonacci Numbers G_n=G_(n-1)+G_(n-c) for Matrices with Constant Valued Determinants." Fib. Quart. 34, 121-128, 1996.Bicknell, M. "A Primer for the Fibonacci Numbers, Part VIII: Sequences of Sums from Pascal's Triangle." Fib. Quart. 9, 74-81, 1971.Dujella, A. "Generalized Fibonacci Numbers and the Problem of Diophantus." Fib. Quart. 34, 164-175, 1996.Horadam, A. F. "Generating Functions for Powers of a Certain Generalized Sequence of Numbers." Duke Math. J. 32, 437-446, 1965.Horadam, A. F. "Generalization of a Result of Morgado." Portugaliae Math. 44, 131-136, 1987a.Horadam, A. F. and Shannon, A. G. "Generalization of Identities of Catalan and Others." Portugaliae Math. 44, 137-148, 1987b.Morgado, J. "Note on Some Results of A. F. Horadam and A. G. Shannon concerning a Catalan's Identity on Fibonacci Numbers." Portugaliae Math. 44, 243-252, 1987.

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Generalized Fibonacci Number

Cite this as:

Weisstein, Eric W. "Generalized Fibonacci Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeneralizedFibonacciNumber.html

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