A296486 Decimal expansion of limiting power-ratio for A295960; see Comments.

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

Offset

1,1

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 = A295960, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

Links

Table of n, a(n) for n=1..86.

Example

limiting power-ratio = 5.859996629844644031286035775860542608816...

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] - 1;
j = 1; While[j < 13, k = a[j] - j - 1;
 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}]; (* A295960 *)
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]   (* A296486 *)

Crossrefs

Cf. A001622, A295960, A296284, A296485.

Sequence in context: A273817 A073212 A059742 * A011424 A011495 A136258

Adjacent sequences: A296483 A296484 A296485 * A296487 A296488 A296489

Keyword

nonn,easy,cons

Author

Clark Kimberling, Apr 13 2018

Status

approved