Pentanacci Number

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)

REFERENCES:

Sloane, N. J. A. Sequence A001591/M1122 and A103814 in "The On-Line Encyclopedia of Integer Sequences."