Tetranacci Number

The tetranacci numbers are a generalization of the Fibonacci numbers defined by T_0=0, T_1=1, T_2=1, T_3=2, and the recurrence relation


for n>=4. They represent the n=4 case of the Fibonacci n-step numbers. The first few terms for n=0, 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 n<=236965 (E. W. Weisstein, Mar. 21, 2009).

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 (alpha,beta,gamma,delta), which are the four roots of the polynomial




This can be written in slightly more concise form as


where r_n is the nth root of the polynomial


and (alpha,beta,gamma,delta) and (r_1,r_2,r_3,r_4) are in the ordering of the Wolfram Language'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... (OEIS A086088), which is sometimes known as the tetranacci constant.

See also

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

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

Referenced on Wolfram|Alpha

Tetranacci Number

Cite this as:

Weisstein, Eric W. "Tetranacci Number." From MathWorld--A Wolfram Web Resource.

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