A296453 Decimal expansion of limiting power-ratio for A296251; see Comments.

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

Offset: 2

Author

Clark Kimberling, Dec 15 2017

Keywords

• nonn
• easy
• cons

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 = A296251 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.

Examples

			Limiting power-ratio = 27.09717363716137318861436923160085309363...
		

Crossrefs

Cf. A001622, A296251.

Programs

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]^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}]  (* A296251 *)
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] (* A296453 *)