Hexanacci Number

The hexanacci 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, 32, 63, 125, 248, ... (OEIS A001592).

An exact formula for the th hexanacci number can be given explicitly in terms of the six roots  of

(2)

as

(3)

The ratio of adjacent terms tends to the positive root of , namely 1.98358284342... (OEIS A118427), sometimes called the hexanacci constant.

REFERENCES:

Sloane, N. J. A. Sequence A001592/M1128 and A118427 in "The On-Line Encyclopedia of Integer Sequences."