Generalized Fibonacci Number

A generalization of the Fibonacci Numbers defined by $1=G_1=G_2=\ldots=G_{c-1}$ 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 left-most 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

$$G_1+G_2+G_3+\ldots+G_n=G_{n+3}-1$$ (2)

$$G_3+G_6+G_9+\ldots+G_{3k}=G_{3k+1}-1$$ (3)

$$G_1+G_4+G_7+\ldots+G_{3k+1}=G_{3k+2}$$ (4)

$$G_2+G_5+G_8+\ldots+G_{3k+2}=G_{3k+3}$$ (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 $n\geq 2$. They satisfy the identities

$$w_nw_{n+2r}-eq^nU_r={w_{n+r}}^2$$ (7)

$$4w_n{w_{n+1}}^2w_{n+2}+(wq^n)^2=(w_nw_{n+2}+{w_{n+1}}^2)^2$$ (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)

$4w_nw_{n+1}w_{n+2}w_{n+4}w_{n+5}w_{n+6}+e^2q^{2n}(w_nU_4U_5-w_{n+1}U_2U_6-w_nU_1U_8)^2$
$= (w_{n+1}w_{n+2}w_{n+6}+w_nw_{n+4}w_{n+5})^2,\quad$	(10)

where

$$e \equiv pab-qa^2-b^2$$ (11)

$$U_n \equiv w_n(0,1; p,q).$$ (12)

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 $x_n\equiv x_{n-2}+x_{n-1}$ for $n\geq 3$,

$$\sum_{i=1}^n x_n = x_{n+2}-x_2$$ (13)

$$\sum_{i=1}^{10} x_n =11 x_7$$ (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 Number

References

Bicknell, M. ``A Primer for the Fibonacci Numbers, Part VIII: Sequences of Sums from Pascal's Triangle.'' Fib. Quart. 9, 74-81, 1971.

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.

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, 1987.

Horadam, A. F. and Shannon, A. G. ``Generalization of Identities of Catalan and Others.'' Portugaliae Math. 44, 137-148, 1987.

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.