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A296434 Decimal expansion of ratio-sum for A296292; see Comments.

%I A296434 #8 Dec 18 2017 11:41:32
%S A296434 8,0,1,2,9,6,8,9,0,3,0,9,5,6,6,1,4,7,2,5,1,5,5,4,1,4,9,9,4,1,6,3,7,7,
%T A296434 2,7,3,1,9,8,3,2,6,4,4,4,4,1,6,2,6,7,6,9,3,1,5,1,4,1,5,0,8,2,0,5,3,7,
%U A296434 5,1,2,3,9,1,3,8,9,6,8,4,6,5,4,7,4,2
%N A296434 Decimal expansion of ratio-sum for A296292; see Comments.
%C A296434 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 = A296292 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 A296434 Ratio-sum = 8.012968903095661472515541...
%t A296434 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A296434 a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n];
%t A296434 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296434 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296434 Table[a[n], {n, 0, k}]; (* A296292 *)
%t A296434 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296434 Take[RealDigits[s, 10][[1]], 100]  (* A296434 *)
%Y A296434 Cf. A001622, A296292.
%K A296434 nonn,easy,cons
%O A296434 1,1
%A A296434 _Clark Kimberling_, Dec 15 2017