A296846 -id:A296846 - cp's OEIS frontend

A296847 Decimal expansion of ratio-sum for A296846; see Comments.

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

Offset: 1

Clark Kimberling, Jan 12 2018

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 = A296846, 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.

			ratio-sum = 2.513488703366421562044549094939139152191...

Cf. A001622, A296846, A296848.

a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] - b[n - 2];
j = 1; While[j < 16, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}];  (* A296846 *)
g = GoldenRatio; s = N[Sum[g - a[n]/a[n - 1], {n, 1, 1000}], 200];  (* A296847 *)
StringJoin[StringTake[ToString[s], 41], "..."]
Take[RealDigits[s, 10][[1]], 100] (* A296847 *)

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A296847 Decimal expansion of ratio-sum for A296846; see Comments.

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