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

2,1

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 = 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.

Links

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

Example

Limiting power-ratio = 27.09717363716137318861436923160085309363...

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 *)

Crossrefs

Cf. A001622, A296251.

Sequence in context: A245224 A016638 A199398 * A081544 A011294 A282072

Adjacent sequences: A296450 A296451 A296452 * A296454 A296455 A296456

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 15 2017

Status

approved