A296470 Decimal expansion of limiting power-ratio for A295862; see Comments.

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

Offset: 1

Author

Clark Kimberling, Dec 18 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 = A295862 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

Crossrefs

Cf. A001622, A295862, A296469.

Programs

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
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}]  (* A295862 *)
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] (* A296470 *)

Formula

limiting power-ratio = 6.514710075055272773358868575343969425373...