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A297012 Decimal expansion of ratio-sum for A297011; see Comments.

%I A297012 #7 Jul 25 2021 20:58:42
%S A297012 2,5,1,8,4,3,9,3,4,6,0,1,2,6,2,3,9,0,0,8,9,6,8,3,7,6,4,1,1,9,1,5,5,0,
%T A297012 6,9,1,0,1,6,3,9,3,9,8,8,1,8,7,7,7,0,4,7,5,8,6,1,5,9,6,2,5,0,9,1,5,1,
%U A297012 5,0,9,0,9,6,8,6,3,8,2,8,2,8,1,9,5,6
%N A297012 Decimal expansion of ratio-sum for A297011; see Comments.
%C A297012 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 = A297011, we have g = 1 + sqrt(2). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.
%e A297012 ratio-sum = 2.518439346012623900896837641191550691016...
%t A297012 a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4;
%t A297012 a[n_] := a[n] = 2 a[n - 1] + a[n - 2] - b[n];
%t A297012 j = 1; While[j < 9, k = a[j] - j - 1;
%t A297012 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A297012 u = Table[a[n], {n, 0, k}]; (* A297011 *)
%t A297012 r = 1 + Sqrt[2]; s = N[Sum[r - a[n]/a[n - 1], {n, 1, 1000}], 200];
%t A297012 StringJoin[StringTake[ToString[s], 41], "..."]
%t A297012 Take[RealDigits[s, 10][[1]], 100] (* A297012 *)
%Y A297012 Cf. A297011.
%K A297012 nonn,easy,cons
%O A297012 1,1
%A A297012 _Clark Kimberling_, Jan 13 2018