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A296483 Decimal expansion of ratio-sum for A295953; see Comments.

%I A296483 #9 Feb 09 2018 17:36:00
%S A296483 4,1,9,4,6,7,8,6,7,9,0,5,6,3,7,1,7,5,8,9,9,1,8,4,0,8,1,8,1,2,3,9,5,4,
%T A296483 4,2,0,9,6,4,7,0,2,4,0,5,8,5,4,7,6,3,3,6,2,6,1,4,0,4,4,6,3,6,5,8,9,0,
%U A296483 0,9,3,9,1,9,0,4,2,3,2,8,2,0,5,9,0,0
%N A296483 Decimal expansion of ratio-sum for A295953; see Comments.
%C A296483 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 = A295953, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296483 ratio-sum = 4.194678679056371758991840818123954420964...
%t A296483 a[0] = 1; a[1] = 3; b[0] = 2; b[1 ] = 4; b[2] = 5;
%t A296483 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 1;
%t A296483 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296483 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296483 Table[a[n], {n, 0, k}]; (* A295953 *)
%t A296483 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296483 Take[RealDigits[s, 10][[1]], 100]  (* A296483 *)
%Y A296483 Cf. A001622, A295953, A296284, A296848.
%K A296483 nonn,easy,cons
%O A296483 1,1
%A A296483 _Clark Kimberling_, Jan 06 2018