The pentanacci numbers are a generalization of the Fibonacci numbers defined by 
, 
, 
, 
, 
, and the recurrence relation
 |
(1)
|
for 
.
They represent the 
case of the Fibonacci n-step numbers.
The first few terms for 
, 2, ... are 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, ... (OEIS
A001591).
The ratio of adjacent terms tends to the real root of 
, namely 1.965948236645485... (OEIS A103814),
sometimes called the pentanacci constant.
An exact formula for the 
th pentanacci number can be given explicitly in terms of the
five roots 
of
 |
(2)
|
as
 |
(3)
|
The pentanacci numbers have generating function
 |
(4)
|
