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A296487 Decimal expansion of ratio-sum for A293076; see Comments.

%I A296487 #7 Jul 25 2021 20:57:36
%S A296487 3,0,9,2,2,6,2,2,8,5,7,7,3,1,0,6,3,3,0,1,8,3,5,3,4,6,5,5,2,0,2,7,1,6,
%T A296487 1,6,2,4,2,5,9,4,5,8,5,3,6,9,4,2,4,6,2,4,5,5,0,6,7,2,9,0,8,0,6,9,5,8,
%U A296487 3,5,9,6,3,1,8,2,6,8,5,5,6,2,4,7,7,3
%N A296487 Decimal expansion of ratio-sum for A293076; see Comments.
%C A296487 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 = A293076, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296487 ratio-sum = 3.092262285773106330183534655202716162425...
%t A296487 a[0] = 1; a[1] = 3; b[0] = 2; b[1 ] = 4;
%t A296487 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2];
%t A296487 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296487 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296487 Table[a[n], {n, 0, k}]; (* A293076 *)
%t A296487 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296487 Take[RealDigits[s, 10][[1]], 100]  (* A296487 *)
%Y A296487 Cf. A001622, A293076, A296284, A296488.
%K A296487 nonn,easy,cons
%O A296487 1,1
%A A296487 _Clark Kimberling_, Dec 19 2017