The pentanacci numbers are a generalization of the Fibonacci numbers defined by 
, 
, 
, 
, 
, and the recurrence
relation
 |
(1)
|
for integers 
. 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.
For a nonnegative integer 
, 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)
|
for 
, where 
is the pentanacci constant.
