cp's OEIS Frontend

A296429 Decimal expansion of ratio-sum for A296266; see Comments.

%I A296429 #11 Jul 26 2021 01:17:50
%S A296429 1,0,2,4,2,8,9,3,9,1,0,1,2,1,3,0,3,6,7,9,8,3,4,8,7,2,4,7,4,6,0,2,4,3,
%T A296429 8,2,2,5,2,4,6,5,0,8,6,4,0,9,5,2,2,0,7,1,6,7,7,6,2,4,8,6,4,6,5,3,9,9,
%U A296429 7,1,7,3,6,9,3,7,7,7,9,9,7,0,4,5,3,1
%N A296429 Decimal expansion of ratio-sum for A296266; see Comments.
%C A296429 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 = A296266, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.
%e A296429 10.24289391012130367983487...
%t A296429 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A296429 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2]*b[n];
%t A296429 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296429 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296429 Table[a[n], {n, 0, k}]; (* A296266 *)
%t A296429 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296429 Take[RealDigits[s, 10][[1]], 100]  (* A296429 *)
%Y A296429 Cf. A001622, A296266.
%K A296429 nonn,easy,cons
%O A296429 2,3
%A A296429 _Clark Kimberling_, Dec 14 2017