A296457 Decimal expansion of limiting power-ratio for A296272; see Comments.

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

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 = A296272 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 = 31.14503268621267806442242062631447330734...

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]*b[n - 1];
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}]  (* A296272 *)
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] (* A296457 *)

Crossrefs

Cf. A001622, A296272.

Sequence in context: A271644 A262191 A334844 * A124020 A124234 A135226

Adjacent sequences: A296454 A296455 A296456 * A296458 A296459 A296460

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 15 2017

Status

approved