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Generalized Fibonacci Number

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

\end{displaymath} (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
\end{displaymath} (2)

\end{displaymath} (3)

\end{displaymath} (4)

\end{displaymath} (5)

(Bicknell-Johnson and Spears 1996). For the special case $c=3$,
\end{displaymath} (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

\end{displaymath} (7)

\end{displaymath} (8)

\end{displaymath} (9)

$ = (w_{n+1}w_{n+2}w_{n+6}+w_nw_{n+4}w_{n+5})^2,\quad$ (10)
$\displaystyle e$ $\textstyle \equiv$ $\displaystyle pab-qa^2-b^2$ (11)
$\displaystyle U_n$ $\textstyle \equiv$ $\displaystyle w_n(0,1; p,q).$ (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
\end{displaymath} (13)

\sum_{i=1}^{10} x_n =11 x_7
\end{displaymath} (14)

{x_n}^2-x_{n-1}x_{n+2}=(-1)^n ({x_2}^2-{x_1}^2-x_1x_2),
\end{displaymath} (15)

where the plus and minus signs alternate.

See also Fibonacci Number


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.

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© 1996-9 Eric W. Weisstein