Heptanacci Number

The heptanacci 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, 64, 127, 253, ... (OEIS A066178).

An exact formula for the th heptanacci number can be given explicitly in terms of the seven roots  of

(2)

as

(3)

The ratio of adjacent terms tends to the real root of , namely 1.99196419660... (OEIS A118428), sometimes called the heptanacci constant.

REFERENCES:

Sloane, N. J. A. Sequence A066178 and A118428 in "The On-Line Encyclopedia of Integer Sequences."