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A296475 Decimal expansion of ratio-sum for A295949; see Comments.

%I A296475 #7 Jul 25 2021 20:57:15
%S A296475 3,9,7,5,7,8,9,5,5,7,9,7,1,0,7,3,4,7,3,4,5,4,9,6,5,3,2,6,8,4,2,9,1,7,
%T A296475 0,7,3,1,4,4,1,2,4,7,5,0,0,4,2,6,5,2,1,5,9,1,9,2,3,7,4,8,6,6,6,7,0,5,
%U A296475 9,2,3,2,0,2,3,6,5,0,2,4,1,3,6,7,6,3
%N A296475 Decimal expansion of ratio-sum for A295949; see Comments.
%C A296475 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 = A295949, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296475 ratio-sum = 3.975789557971073473454965326842917073144...
%t A296475 a[0] = 1; a[1] = 2; b[0] = 3; b[1 ] = 4; b[2] = 5;
%t A296475 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
%t A296475 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296475 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296475 Table[a[n], {n, 0, k}]; (* A295949 *)
%t A296475 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296475 Take[RealDigits[s, 10][[1]], 100]  (* A296475 *)
%Y A296475 Cf. A001622, A295949, A296284, A296476.
%K A296475 nonn,easy,cons
%O A296475 1,1
%A A296475 _Clark Kimberling_, Dec 19 2017