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A296471 Decimal expansion of ratio-sum for A295947; see Comments.

%I A296471 #7 Jul 25 2021 22:19:28
%S A296471 2,4,2,7,1,7,9,4,8,8,0,5,6,0,3,9,4,2,4,4,2,3,6,5,3,1,0,3,8,3,1,4,5,2,
%T A296471 2,5,1,7,5,7,9,1,6,7,4,0,4,7,2,5,2,8,1,6,7,7,2,3,6,8,5,3,1,6,1,6,1,1,
%U A296471 0,1,7,9,1,4,9,8,4,2,4,8,6,3,8,9,7,1
%N A296471 Decimal expansion of ratio-sum for A295947; see Comments.
%C A296471 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 = A295947, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296471 ratio-sum = 2.427179488056039424423653103831452251757...
%t A296471 a[0] = 2; a[1] = 4; b[0] = 1; b[1 ] = 3; b[2] = 5;
%t A296471 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
%t A296471 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296471 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296471 Table[a[n], {n, 0, k}]; (* A295947 *)
%t A296471 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296471 Take[RealDigits[s, 10][[1]], 100]  (* A296471 *)
%Y A296471 Cf. A001622, A295947, A296284, A296472.
%K A296471 nonn,easy,cons
%O A296471 1,1
%A A296471 _Clark Kimberling_, Dec 18 2017