A296459 Decimal expansion of limiting power-ratio for A296284; see Comments.

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

Offset

2,2

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 = A296284 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 = 16.71583793078729559599070133315659170886...

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*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}]  (* A296284 *)
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] (* A296459 *)

Crossrefs

Cf. A001622, A296284.

Sequence in context: A356562 A011483 A319458 * A354684 A308915 A294644

Adjacent sequences: A296456 A296457 A296458 * A296460 A296461 A296462

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 18 2017

Status

approved