A296460 Decimal expansion of limiting power-ratio for A296288; see Comments.

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

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 = A296288 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 = 19.20919895224276906694327100124445652349...

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n - 1];
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}]  (* A296288 *)
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] (* A296460 *)

Crossrefs

Cf. A001622, A296288.

Sequence in context: A096388 A153463 A240985 * A379587 A319533 A010160

Adjacent sequences: A296457 A296458 A296459 * A296461 A296462 A296463

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 18 2017

Status

approved