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A296503 Decimal expansion of ratio-sum for A294553; see Comments.

%I A296503 #6 Apr 17 2018 18:32:08
%S A296503 3,9,7,4,1,2,5,9,5,1,3,2,5,3,7,0,8,1,2,5,3,5,5,2,2,0,3,9,0,0,9,5,3,7,
%T A296503 6,3,6,7,6,8,3,4,2,8,2,8,6,3,6,4,1,7,1,1,7,5,5,9,7,9,3,0,1,1,0,7,7,5,
%U A296503 9,5,2,0,3,4,8,5,3,9,8,3,0,5,8,0,2,1
%N A296503 Decimal expansion of ratio-sum for A294553; see Comments.
%C A296503 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 = A294553, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296503 ratio-sum = 3.974125951325370812535522039009537636768...
%t A296503 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A296503 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] - n;
%t A296503 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296503 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296503 Table[a[n], {n, 0, k}]; (* A294553 *)
%t A296503 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296503 Take[RealDigits[s, 10][[1]], 100]  (* A296503 *)
%Y A296503 Cf. A001622, A294553, A296469, A296504.
%K A296503 nonn,easy,cons
%O A296503 1,1
%A A296503 _Clark Kimberling_, Apr 14 2018