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

Offset: 1

Author

Clark Kimberling, Dec 20 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 = A294557, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

Examples

			limiting power-ratio = 3.684140963507143048934998709433983218507...
		

Crossrefs

Cf. A001622, A294557, A296469, A296567.

Programs

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], "..."]
Take[RealDigits[Last[h], 10][[1]], 120]   (* A296568 *)