The tetranacci numbers are a generalization of the Fibonacci numbers defined by , , , , and the recurrence relation

for . They represent the n = 4 case of the Fibonacci n-step numbers. The first few terms are 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, ... (Sloane's A000078).

The first few prime tetranacci numbers have indices 3, 7, 11, 12, 36, 56, 401, 2707, 8417, 14096, ... (Sloane's A104534), corresponding to 2, 29, 401, 773, 5350220959, ... (Sloane's A104535).

An exact expression for the nth tetranacci number for n > 1 can be given explicitly by

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

This can be written in slightly more concise form as

where is the nth root of the polynomial

and and are in the ordering of Mathematica's Root object.

The tetranacci numbers have the generating function

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

 

Fibonacci n-Step Number, Fibonacci Number, Tetranacci Constant, Tribonacci Number