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A296485 Decimal expansion of ratio-sum for A295960; see Comments.

%I A296485 #6 Apr 13 2018 21:52:11
%S A296485 3,5,5,6,9,5,8,4,3,6,6,6,0,9,0,3,9,9,5,0,9,7,5,4,8,4,9,4,4,5,1,0,9,7,
%T A296485 9,3,8,7,2,8,1,4,4,3,9,8,3,5,4,0,8,0,9,1,7,6,4,5,5,7,3,6,9,2,4,6,5,5,
%U A296485 1,9,7,7,2,6,3,7,5,6,7,1,7,9,0,9,3,9
%N A296485 Decimal expansion of ratio-sum for A295960; see Comments.
%C A296485 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 = A295960, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296485 ratio-sum = 3.556958436660903995097548494451097938728...
%t A296485 a[0] = 1; a[1] = 3; b[0] = 2; b[1 ] = 4; b[2] = 5;
%t A296485 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
%t A296485 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296485 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296485 Table[a[n], {n, 0, k}]; (* A295960 *)
%t A296485 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296485 Take[RealDigits[s, 10][[1]], 100]  (* A296485 *)
%Y A296485 Cf. A001622, A295960, A296284, A296486.
%K A296485 nonn,easy,cons
%O A296485 1,1
%A A296485 _Clark Kimberling_, Apr 12 2018