Tetranacci Number

The tetranacci 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 , 1, ... are 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, ... (OEIS A000078).

The first few prime tetranacci numbers have indices 3, 7, 11, 12, 36, 56, 401, 2707, 8417, 14096, 31561, 50696, 53192, 155182, ... (OEIS A104534), corresponding to 2, 29, 401, 773, 5350220959, ... (OEIS A104535), with no others for  (E. W. Weisstein, Mar. 21, 2009).

An exact expression for the th tetranacci number for  can be given explicitly by

(2)

where the three additional terms are obtained by cyclically permuting , which are the four roots of the polynomial

(3)

Alternately,

(4)

This can be written in slightly more concise form as

(5)

where  is the th root of the polynomial

(6)

and  and  are in the ordering of the Wolfram Language's Root object.

The tetranacci numbers have the generating function

(7)

The ratio of adjacent terms tends to the positive real root of , namely 1.92756... (OEIS A086088), which is sometimes known as the tetranacci constant.

REFERENCES:

Sloane, N. J. A. Sequences A000078/M1108, A086088, A104534, and A104535 in "The On-Line Encyclopedia of Integer Sequences."