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A296432 Decimal expansion of ratio-sum for A296284; see Comments.

%I A296432 #6 Dec 18 2017 11:41:12
%S A296432 6,2,1,0,3,2,7,1,0,9,4,6,6,1,8,4,9,4,2,2,7,9,6,7,9,0,4,8,4,0,2,4,2,2,
%T A296432 4,6,0,5,4,5,3,6,8,4,1,5,7,0,9,5,7,9,1,2,3,4,0,6,9,2,7,3,5,8,7,0,5,4,
%U A296432 0,4,4,9,1,7,0,1,8,9,8,8,8,9,6,2,7,1
%N A296432 Decimal expansion of ratio-sum for A296284; see Comments.
%C A296432 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 = A296284 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 A296432 Ratio-sum = 6.21032710946618494227967...
%t A296432 a[0] = 1; a[1] = 2; b[0] = 3;
%t A296432 a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n - 2];
%t A296432 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296432 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296432 Table[a[n], {n, 0, k}]; (* A296284 *)
%t A296432 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296432 Take[RealDigits[s, 10][[1]], 100]  (* A296432 *)
%Y A296432 Cf. A001622, A296284.
%K A296432 nonn,easy,cons
%O A296432 1,1
%A A296432 _Clark Kimberling_, Dec 15 2017