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A296847 Decimal expansion of ratio-sum for A296846; see Comments.

%I A296847 #6 Jan 12 2018 23:43:59
%S A296847 2,5,1,3,4,8,8,7,0,3,3,6,6,4,2,1,5,6,2,0,4,4,5,4,9,0,9,4,9,3,9,1,3,9,
%T A296847 1,5,2,1,9,1,7,5,6,9,4,4,3,0,5,3,6,7,3,0,6,5,3,1,7,8,9,8,7,7,2,3,6,5,
%U A296847 3,9,9,9,5,2,4,6,1,8,4,0,4,0,7,2,9,2
%N A296847 Decimal expansion of ratio-sum for A296846; see Comments.
%C A296847 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 = A296846, 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 A296847 ratio-sum = 2.513488703366421562044549094939139152191...
%t A296847 a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4;
%t A296847 a[n_] := a[n] = a[n - 1] + a[n - 2] - b[n - 2];
%t A296847 j = 1; While[j < 16, k = a[j] - j - 1;
%t A296847 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296847 u = Table[a[n], {n, 0, k}];  (* A296846 *)
%t A296847 g = GoldenRatio; s = N[Sum[g - a[n]/a[n - 1], {n, 1, 1000}], 200];  (* A296847 *)
%t A296847 StringJoin[StringTake[ToString[s], 41], "..."]
%t A296847 Take[RealDigits[s, 10][[1]], 100] (* A296847 *)
%Y A296847 Cf. A001622, A296846, A296848.
%K A296847 nonn,easy,cons
%O A296847 1,1
%A A296847 _Clark Kimberling_, Jan 12 2018