A296569 Decimal expansion of ratio-sum for A294558; see Comments.

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

Offset: 1

Author

Clark Kimberling, Jan 06 2018

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 = A294558, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

Examples

			ratio-sum = 2.820339618753543689299555329148441281565...
		

Crossrefs

Cf. A001622, A294558, A296469, A296570.

Programs

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] + 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}]; (* A294558 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296569 *)