A296260 -id:A296260 - cp's OEIS frontend

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

Offset: 2

Clark Kimberling, Dec 15 2017

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.

			Limiting power-ratio = 23.10815724358788604144450707514353840694...

Cf. A001622, A296260.

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], "..."]
Take[RealDigits[Last[h], 10][[1]], 120] (* A296455 *)

cp's OEIS Frontend

A296455 Decimal expansion of limiting power-ratio for A296260; see Comments.

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