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A296479 Decimal expansion of ratio-sum for A295951; see Comments.

%I A296479 #7 Jul 25 2021 20:53:35
%S A296479 2,5,7,1,9,7,1,3,6,9,3,0,7,5,7,8,2,4,5,3,4,0,6,7,2,4,8,9,6,7,2,0,8,7,
%T A296479 6,5,9,8,7,6,8,8,8,2,5,4,5,3,3,3,9,6,3,8,1,9,6,8,6,9,0,7,4,2,8,4,3,0,
%U A296479 5,5,0,2,3,7,4,6,4,3,3,1,8,4,7,3,5,0
%N A296479 Decimal expansion of ratio-sum for A295951; see Comments.
%C A296479 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 = A295951, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296479 ratio-sum = 2.571971369307578245340672489672087659876...
%t A296479 a[0] = 2; a[1] = 3; b[0] = 1; b[1 ] = 4; b[2] = 5;
%t A296479 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
%t A296479 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296479 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296479 Table[a[n], {n, 0, k}]; (* A295951 *)
%t A296479 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296479 Take[RealDigits[s, 10][[1]], 100]  (* A296479 *)
%Y A296479 Cf. A001622, A295951, A296284, A296480.
%K A296479 nonn,easy,cons
%O A296479 1,1
%A A296479 _Clark Kimberling_, Jan 05 2018