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A296433 Decimal expansion of ratio-sum for A296288; see Comments.

%I A296433 #8 Dec 18 2017 11:41:19
%S A296433 7,0,9,3,8,8,3,2,4,4,5,5,8,2,3,3,2,8,2,5,1,8,6,3,2,9,3,3,3,3,8,1,5,1,
%T A296433 2,8,8,8,5,0,3,6,1,6,9,3,0,3,9,2,1,8,1,5,6,0,9,5,1,9,9,8,2,3,1,8,2,1,
%U A296433 8,1,7,8,3,0,2,7,3,2,6,6,5,4,3,0,4,2
%N A296433 Decimal expansion of ratio-sum for A296288; see Comments.
%C A296433 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 = A296288 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 A296433 Ratio-sum = 7.093883244558233282518632...
%t A296433 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A296433 a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n - 2];
%t A296433 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296433 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296433 Table[a[n], {n, 0, k}]; (* A296288 *)
%t A296433 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296433 Take[RealDigits[s, 10][[1]], 100]  (* A296433 *)
%Y A296433 Cf. A001622, A296288.
%K A296433 nonn,easy,cons
%O A296433 1,1
%A A296433 _Clark Kimberling_, Dec 15 2017