A296568 Decimal expansion of limiting power-ratio for A294557; see Comments.

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

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 = A294557, 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 = 3.684140963507143048934998709433983218507...

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] + b[n - 1] + b[n - 2] - n;
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}]; (* A294557 *)
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]   (* A296568 *)

Crossrefs

Cf. A001622, A294557, A296469, A296567.

Sequence in context: A108369 A010621 A258232 * A294095 A306633 A096416

Adjacent sequences: A296565 A296566 A296567 * A296569 A296570 A296571

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 20 2017

Status

approved