A generalization of the Fibonacci Numbers defined by
and the Recurrence Relation
![\begin{displaymath}
G_n=G_{n-1}+G_{n-c}.
\end{displaymath}](g_1109.gif) |
(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
up and 1 right. The case
equals the usual Fibonacci Number. These numbers
satisfy the identities
![\begin{displaymath}
G_1+G_2+G_3+\ldots+G_n=G_{n+3}-1
\end{displaymath}](g_1112.gif) |
(2) |
![\begin{displaymath}
G_3+G_6+G_9+\ldots+G_{3k}=G_{3k+1}-1
\end{displaymath}](g_1113.gif) |
(3) |
![\begin{displaymath}
G_1+G_4+G_7+\ldots+G_{3k+1}=G_{3k+2}
\end{displaymath}](g_1114.gif) |
(4) |
![\begin{displaymath}
G_2+G_5+G_8+\ldots+G_{3k+2}=G_{3k+3}
\end{displaymath}](g_1115.gif) |
(5) |
(Bicknell-Johnson and Spears 1996). For the special case
,
![\begin{displaymath}
G_{n+w}=G_{w-2}G_n+G_{w-3}G_{n+1}+G_{w-1}G_{n+2}.
\end{displaymath}](g_1117.gif) |
(6) |
Bicknell-Johnson and Spears (1996) give many further identities.
Horadam (1965) defined the generalized Fibonacci numbers
as
, where
,
,
, and
are Integers,
,
, and
for
. They satisfy the identities
![\begin{displaymath}
w_nw_{n+2r}-eq^nU_r={w_{n+r}}^2
\end{displaymath}](g_1124.gif) |
(7) |
![\begin{displaymath}
4w_n{w_{n+1}}^2w_{n+2}+(wq^n)^2=(w_nw_{n+2}+{w_{n+1}}^2)^2
\end{displaymath}](g_1125.gif) |
(8) |
![\begin{displaymath}
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
\end{displaymath}](g_1126.gif) |
(9) |
|
|
|
(10) |
where
The final above result is due to Morgado (1987) and is called the Morgado Identity.
Another generalization of the Fibonacci numbers is denoted
. Given
and
, define the generalized Fibonacci
number by
for
,
![\begin{displaymath}
\sum_{i=1}^n x_n = x_{n+2}-x_2
\end{displaymath}](g_1136.gif) |
(13) |
![\begin{displaymath}
\sum_{i=1}^{10} x_n =11 x_7
\end{displaymath}](g_1137.gif) |
(14) |
![\begin{displaymath}
{x_n}^2-x_{n-1}x_{n+2}=(-1)^n ({x_2}^2-{x_1}^2-x_1x_2),
\end{displaymath}](g_1138.gif) |
(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
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.
© 1996-9 Eric W. Weisstein
1999-05-25