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

Author

Clark Kimberling, Dec 20 2017

Keywords

• nonn
• easy
• cons

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.

Examples

			limiting power-ratio = 8.814104632202565527925188322585412678508...
		

Crossrefs

Cf. A001622, A294414, A296284, A296495.

Programs

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 *)