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A296425 Decimal expansion of ratio-sum for A296245; see Comments.

%I A296425 #12 Jul 26 2021 01:20:09
%S A296425 1,4,9,7,6,3,2,7,1,4,4,8,5,6,3,0,4,1,2,4,1,1,6,8,9,6,3,5,6,2,6,9,8,7,
%T A296425 9,3,6,1,3,5,1,0,5,0,4,8,2,1,7,4,9,2,0,3,2,2,3,6,7,0,3,3,5,7,8,3,0,6,
%U A296425 8,4,9,2,4,3,3,2,4,0,5,8,2,6,9,4,7,2
%N A296425 Decimal expansion of ratio-sum for A296245; see Comments.
%C A296425 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 = A296245, 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 A296425 14.9763271448563041241168963...
%t A296425 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A296425 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n]^2;
%t A296425 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296425 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296425 Table[a[n], {n, 0, k}]; (* A296245 *)
%t A296425 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296425 Take[RealDigits[s, 10][[1]], 100]  (* A296425 *)
%Y A296425 Cf. A001622, A296245.
%K A296425 nonn,easy,cons
%O A296425 2,2
%A A296425 _Clark Kimberling_, Dec 14 2017