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A296430 Decimal expansion of ratio-sum for A296272; see Comments.

%I A296430 #8 Jul 26 2021 01:13:22
%S A296430 1,2,5,8,3,1,8,6,1,0,0,5,5,6,0,9,5,7,1,8,9,0,9,6,6,0,8,2,7,9,6,6,1,1,
%T A296430 9,8,7,5,4,5,9,4,1,1,2,9,8,2,6,3,1,7,9,2,5,1,5,2,0,0,3,8,0,0,0,8,1,2,
%U A296430 9,4,3,5,1,5,9,8,0,7,3,0,7,0,3,1,1,9
%N A296430 Decimal expansion of ratio-sum for A296272; see Comments.
%C A296430 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 = A296272 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 A296430 ratio-sum = 12.5831861005560957189096...
%t A296430 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A296430 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1]*b[n];
%t A296430 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296430 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296430 Table[a[n], {n, 0, k}]; (* A296272 *)
%t A296430 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296430 Take[RealDigits[s, 10][[1]], 100]  (* A296430 *)
%Y A296430 Cf. A001622, A296272.
%K A296430 nonn,easy,cons
%O A296430 2,2
%A A296430 _Clark Kimberling_, Dec 14 2017