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%I #14 Jul 26 2021 01:13:36
%S 8,8,3,9,4,4,2,6,2,1,2,7,6,4,5,1,5,3,9,7,4,9,4,4,7,3,4,0,8,4,4,1,3,0,
%T 8,1,5,6,0,3,3,1,5,6,9,0,4,1,4,6,9,1,2,9,7,0,5,3,8,5,3,9,3,9,5,9,9,9,
%U 1,4,0,5,3,9,6,9,1,3,9,1,9,5,2,1,2,1
%N Decimal expansion of ratio-sum for A295367; see Comments.
%C 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 = A295367, 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 8.83944262127645153974944...
%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2]*b[n - 1];
%t j = 1; While[j < 13, k = a[j] - j - 1;
%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t Table[a[n], {n, 0, k}]; (* A295367 *)
%t g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t Take[RealDigits[s, 10][[1]], 100] (* A296428 *)
%Y Cf. A001622, A295367.
%K nonn,easy,cons
%O 1,1
%A _Clark Kimberling_, Dec 14 2017
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