A296461 Decimal expansion of limiting power-ratio for A296292; see Comments.

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

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 = A296292 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 = 21.74130735523558735581498585908915856896...

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] + n*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}]  (* A296292 *)
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] (* A296461 *)

Crossrefs

Cf. A001622, A296292.

Sequence in context: A194797 A255138 A115629 * A144696 A072248 A317360

Adjacent sequences: A296458 A296459 A296460 * A296462 A296463 A296464

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 18 2017

Status

approved