A296496 Decimal expansion of limiting power-ratio for A294414; see Comments.

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

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 ratio-sum for A is |a(1)/a(0) - g| + |a(2)/a(1) - g| + ..., assuming that this series converges. For A = A294414, 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 = 8.814104632202565527925188322585412678508...

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2];
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}]; (* A294414 *)
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]   (* A296496 *)

Crossrefs

Cf. A001622, A294414, A296284, A296495.

Sequence in context: A091648 A135707 A021923 * A065465 A265308 A375153

Adjacent sequences: A296493 A296494 A296495 * A296497 A296498 A296499

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 20 2017

Status

approved