A296568 - OEIS

A296568

Decimal expansion of limiting power-ratio for A294557; see Comments.

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

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OFFSET

1,1

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 = A294557, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

LINKS

Table of n, a(n) for n=1..86.

EXAMPLE

limiting power-ratio = 3.684140963507143048934998709433983218507...

MATHEMATICA

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;

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

j = 1; While[j < 13, k = a[j] - j - 1;

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

Table[a[n], {n, 0, k}]; (* A294557 *)

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

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