A296455 Decimal expansion of limiting power-ratio for A296260; see Comments.

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

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 = A296260 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 = 23.10815724358788604144450707514353840694...

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]*b[n - 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}]  (* A296260 *)
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] (* A296455 *)

Crossrefs

Cf. A001622, A296260.

Sequence in context: A048994 A344172 A121434 * A137329 A265604 A171996

Adjacent sequences: A296452 A296453 A296454 * A296456 A296457 A296458

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 15 2017

Status

approved