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Hexanacci Number


The hexanacci numbers are a generalization of the Fibonacci numbers defined by H_0=0, H_1=1, H_2=1, H_3=2, H_4=4, H_5=8, and the recurrence relation

 H_n=H_(n-1)+H_(n-2)+H_(n-3)+H_(n-4)+H_(n-5)+H_(n-6)
(1)

for n>=6. They represent the n=6 case of the Fibonacci n-step numbers.

The first few terms for n=1, 2, ... are 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, ... (OEIS A001592).

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

 P(x)=x^6-x^5-x^4-x^3-x^2-x-1
(2)

as

 H_n=sum_(i=1)^6(x_i^n)/(-x_i^5+x_i^3+2x_i^2+10x_i-1).
(3)

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


See also

Fibonacci n-Step Number, Fibonacci Number, Heptanacci Number, Hexanacci Constant, Pentanacci Number, Tetranacci Number, Tribonacci Number

Portions of this entry contributed by Tito Piezas III

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References

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

Referenced on Wolfram|Alpha

Hexanacci Number

Cite this as:

Piezas, Tito III and Weisstein, Eric W. "Hexanacci Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HexanacciNumber.html

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