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A296431 Decimal expansion of ratio-sum for A296278; see Comments.

%I A296431 #8 Jul 26 2021 01:13:13
%S A296431 3,4,2,7,6,7,9,9,8,5,9,6,8,2,9,5,7,8,5,3,1,4,9,6,5,6,7,0,3,6,4,5,8,0,
%T A296431 3,9,5,7,5,2,6,9,8,8,5,8,2,6,1,7,6,8,5,6,2,4,2,6,5,4,7,2,8,3,5,1,0,5,
%U A296431 8,5,0,8,4,6,2,6,3,4,4,6,3,6,0,6,9,9
%N A296431 Decimal expansion of ratio-sum for A296278; see Comments.
%C A296431 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 = A296278 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 A296431 ratio-sum = 34.27679985968295785314965...
%t A296431 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A296431 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2]*b[n - 1]*b[n];
%t A296431 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296431 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296431 Table[a[n], {n, 0, k}]; (* A296278 *)
%t A296431 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296431 Take[RealDigits[s, 10][[1]], 100]  (* A296278 *)
%Y A296431 Cf. A001622, A296278.
%K A296431 nonn,easy,cons
%O A296431 2,1
%A A296431 _Clark Kimberling_, Dec 14 2017