A296455 - OEIS

A296455

Decimal expansion of limiting power-ratio for A296260; see Comments.

2, 3, 1, 0, 8, 1, 5, 7, 2, 4, 3, 5, 8, 7, 8, 8, 6, 0, 4, 1, 4, 4, 4, 5, 0, 7, 0, 7, 5, 1, 4, 3, 5, 3, 8, 4, 0, 6, 9, 4, 6, 9, 4, 5, 0, 2, 8, 1, 4, 3, 8, 3, 7, 1, 5, 8, 4, 4, 7, 9, 1, 3, 7, 6, 7, 6, 2, 2, 1, 8, 8, 3, 0, 2, 4, 1, 2, 6, 5, 5, 2, 3, 1, 8, 2, 2

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OFFSET

2,1

COMMENTS

Suppose that A = {a(n)}, for n >= 0, is a sequence, and g is a real number such that a(n)/a(n-1) -> g. The limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A296260 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

LINKS

Table of n, a(n) for n=2..87.

EXAMPLE

Limiting power-ratio = 23.10815724358788604144450707514353840694...

MATHEMATICA

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;

a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1]*b[n - 2];

j = 1; While[j < 12, k = a[j] - j - 1;

While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];

Table[a[n], {n, 0, 15}] (* A296260 *)

z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];

StringJoin[StringTake[ToString[h[[z]]], 41], "..."]