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A296495 Decimal expansion of ratio-sum for A294414; see Comments.

%I A296495 #7 Jul 25 2021 20:57:48
%S A296495 4,5,8,5,1,9,7,1,7,4,1,8,4,5,6,5,4,0,1,9,8,8,8,6,9,0,5,7,8,1,7,6,5,0,
%T A296495 7,4,6,9,0,1,6,6,0,5,6,9,3,5,9,2,0,5,5,8,2,0,4,1,4,1,6,1,7,4,4,1,3,5,
%U A296495 2,1,8,8,5,7,5,0,6,5,1,7,9,0,7,3,7,5
%N A296495 Decimal expansion of ratio-sum for A294414; see Comments.
%C A296495 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 = A294414, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296495 4.585197174184565401988869057817650746901...
%t A296495 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A296495 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2];
%t A296495 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296495 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296495 Table[a[n], {n, 0, k}]; (* A294414 *)
%t A296495 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296495 Take[RealDigits[s, 10][[1]], 100]  (* A296495 *)
%Y A296495 Cf. A001622, A294414, A296284, A296496.
%K A296495 nonn,easy,cons
%O A296495 1,1
%A A296495 _Clark Kimberling_, Dec 20 2017