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A296499 Decimal expansion of ratio-sum for A294546; see Comments.

%I A296499 #7 Jul 25 2021 20:58:04
%S A296499 4,7,3,7,0,5,7,9,9,8,7,9,2,2,1,7,3,6,5,4,2,8,3,7,5,6,2,1,2,9,5,7,5,9,
%T A296499 3,2,6,1,5,9,9,7,6,5,8,7,3,4,7,0,8,6,3,7,1,7,7,5,2,1,0,2,1,6,1,5,6,0,
%U A296499 8,0,6,4,2,9,2,6,8,7,9,3,8,1,7,1,5,8
%N A296499 Decimal expansion of ratio-sum for A294546; see Comments.
%C A296499 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 = A294546, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296499 4.737057998792217365428375621295759326159...
%t A296499 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A296499 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + n;
%t A296499 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296499 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296499 Table[a[n], {n, 0, k}]; (* A294549 *)
%t A296499 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296499 Take[RealDigits[s, 10][[1]], 100]  (* A296499 *)
%Y A296499 Cf. A001622, A294546, A296284, A296500.
%K A296499 nonn,easy,cons
%O A296499 1,1
%A A296499 _Clark Kimberling_, Dec 20 2017