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Lucas n-Step Number

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A Lucas n-step number is a term of the sequence {L_k^((n))}_(k=0)^infty, where n>=2 is an integer. The initial conditions are L_k^((n))=-1 for 1-n<=k<0, L_0^((n))=n, and subsequent terms are given by the linear recurrence equation

L_k^((n))=sum_(i=1)^nL_(k-i)^((n))

for k>0 (Noe and Post 2005).

The first few sequences of n-step Lucas numbers are summarized in the table below.

nOEISL_1^((n)), L_2^((n)), ...
2A0002041, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...
3A0016441, 3, 7, 11, 21, 39, 71, 131, 241, 443, ...
4A0016481, 3, 7, 15, 26, 51, 99, 191, 367, 708, ...
5A0234241, 3, 7, 15, 31, 57, 113, 223, 439, 863, ...

See also

Fibonacci n-Step Number, Lucas Number

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References

Noe, T. D. and Post, J. V. "Primes in Fibonacci n-Step and Lucas n-Step Sequences." J. Integer Seq. 8, Article 05.4.4, 2005. https://cs.uwaterloo.ca/journals/JIS/VOL8/Noe/noe5.html.Sloane, N. J. A. Sequences A000204/M2341, A001644/M2625, A001648/M2648, and A023424 in "The On-Line Encyclopedia of Integer Sequences."

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Lucas n-Step Number

Cite this as:

Weisstein, Eric W. "Lucas n-Step Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Lucasn-StepNumber.html

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