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A296481 Decimal expansion of ratio-sum for A295952; see Comments.

%I A296481 #7 Jul 25 2021 20:57:29
%S A296481 4,8,4,5,8,5,3,6,8,3,5,1,4,3,6,2,0,7,1,3,5,0,0,2,0,5,6,7,3,7,2,5,0,1,
%T A296481 7,8,9,0,3,4,8,4,3,5,6,2,3,5,7,9,0,5,1,6,3,2,0,5,9,9,3,0,5,7,2,8,9,5,
%U A296481 5,2,9,0,7,4,0,0,5,7,1,0,7,9,6,9,5,0
%N A296481 Decimal expansion of ratio-sum for A295952; see Comments.
%C A296481 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 = A295952, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296481 ratio-sum = 4.845853683514362071350020567372501789034...
%t A296481 a[0] = 1; a[1] = 5; b[0] = 2; b[1 ] = 3; b[2] = 4;
%t A296481 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
%t A296481 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296481 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296481 Table[a[n], {n, 0, k}]; (* A295952 *)
%t A296481 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296481 Take[RealDigits[s, 10][[1]], 100]  (* A296481 *)
%Y A296481 Cf. A001622, A296284, A296482.
%K A296481 nonn,easy,cons
%O A296481 1,1
%A A296481 _Clark Kimberling_, Jan 05 2018