%I #8 Dec 18 2017 11:41:32
%S 8,0,1,2,9,6,8,9,0,3,0,9,5,6,6,1,4,7,2,5,1,5,5,4,1,4,9,9,4,1,6,3,7,7,
%T 2,7,3,1,9,8,3,2,6,4,4,4,4,1,6,2,6,7,6,9,3,1,5,1,4,1,5,0,8,2,0,5,3,7,
%U 5,1,2,3,9,1,3,8,9,6,8,4,6,5,4,7,4,2
%N Decimal expansion of ratio-sum for A296292; see Comments.
%C 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 = A296292 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 Ratio-sum = 8.012968903095661472515541...
%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n];
%t j = 1; While[j < 13, k = a[j] - j - 1;
%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t Table[a[n], {n, 0, k}]; (* A296292 *)
%t g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t Take[RealDigits[s, 10][[1]], 100] (* A296434 *)
%Y Cf. A001622, A296292.
%K nonn,easy,cons
%O 1,1
%A _Clark Kimberling_, Dec 15 2017