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A296850 Decimal expansion of ratio-sum for A296849; see Comments.

%I A296850 #6 Jan 12 2018 23:44:14
%S A296850 2,8,9,8,4,3,0,3,7,3,6,0,5,1,8,8,6,7,9,3,6,4,5,5,4,0,4,9,9,7,4,5,3,1,
%T A296850 9,4,1,5,1,7,4,5,4,5,4,5,2,9,5,3,9,2,4,7,3,4,6,9,9,7,5,0,3,3,3,6,3,2,
%U A296850 6,9,2,1,8,1,0,1,7,7,2,8,4,2,9,1,5,0
%N A296850 Decimal expansion of ratio-sum for A296849; see Comments.
%C A296850 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 = A296849, we have g = 1 + sqrt(2). See A296425..A296434 for related ratio-sums and A296452..A296461 for related limiting power-ratios.
%e A296850 ratio-sum = 2.898430373605188679364554049974531941517...
%t A296850 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A296850 a[n_] := a[n] = 2*a[n - 1] + a[n - 2] + b[n];
%t A296850 j = 1; While[j < 8, k = a[j] - j - 1;
%t A296850 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296850 u = Table[a[n], {n, 0, k}]; (* A296849 *)
%t A296850 r = 1 + Sqrt[2]; s = N[Sum[-r + a[n]/a[n - 1], {n, 1, 1000}], 200];
%t A296850 StringJoin[StringTake[ToString[s], 41], "..."]
%t A296850 Take[RealDigits[s, 10][[1]], 100] (* A296850 *)
%Y A296850 Cf. A296849.
%K A296850 nonn,easy,cons
%O A296850 1,1
%A A296850 _Clark Kimberling_, Jan 12 2018