A296431 Decimal expansion of ratio-sum for A296278; see Comments.

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

Offset: 2

Author

Clark Kimberling, Dec 14 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 = A296278 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

Examples

			ratio-sum = 34.27679985968295785314965...
		

Crossrefs

Cf. A001622, A296278.

Programs

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 - 2]*b[n - 1]*b[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}]; (* A296278 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296278 *)