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Monday, 13 May 2024

Fibonacci-like Sequences

Consider the following recurrence relation:a(n)=a(n1)+a(n8)with a(i)=1 for i=07The ratio of successive terms approach the golden ratio ϕ just as the terms in the Fibonacci sequence do. Naturally, the terms in the sequence take a while to grow larger. Here are the initial terms:

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 96, 119, 148, 184, 228, 281, 345, 423, 519, 638, 786, 970, 1198, 1479, 1824, 2247, 2766, 3404, 4190, 5160, 6358, 7837, 9661, 11908, 14674, 18078, 22268, 27428, 33786, 41623

These terms form OEIS A005710. The generating function (permalink) for this sequence is:11xx8In general, we have:a(n)=a(n1)+a(nm) with a(n)=1 for n=0m1The generating function is:11xxmIn the case of m=2, we get the terms in the Fibonacci sequence.

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