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A296427 Decimal expansion of ratio-sum for A296257; see Comments.

%I A296427 #12 Jul 26 2021 01:48:34
%S A296427 7,4,8,6,5,9,8,2,3,8,8,6,1,1,7,5,1,5,0,8,3,0,4,2,2,9,1,1,8,2,1,9,2,9,
%T A296427 6,7,7,6,4,4,6,8,2,9,9,2,3,8,9,5,1,2,3,4,6,7,0,7,0,8,2,3,7,2,0,1,3,6,
%U A296427 8,1,1,7,6,2,5,8,9,6,9,1,6,7,0,6,1,2
%N A296427 Decimal expansion of ratio-sum for A296257; see Comments.
%C A296427 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 = A296257, 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.
%e A296427 7.48659823886117515083042...
%t A296427 a[0] = 1; a[1] = 2; b[0] = 3;
%t A296427 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2]^2;
%t A296427 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296427 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296427 Table[a[n], {n, 0, k}]; (* A296257 *)
%t A296427 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296427 Take[RealDigits[s, 10][[1]], 100]  (* A296427 *)
%Y A296427 Cf. A001622, A296257.
%K A296427 nonn,easy,cons
%O A296427 2,1
%A A296427 _Clark Kimberling_, Dec 14 2017