%I
%S 6,3,5,8,7,1,3,0,2,6,9,8,4,2,9,9,3,5,4,5,4,1,4,7,7,9,6,8,8,9,0,6,0,5,
%T 5,0,4,3,0,2,3,3,0,8,6,8,8,9,4,5,7,0,7,3,2,5,1,6,1,3,3,3,0,1,0,1,5,5,
%U 4,3,0,8,3,2,4,6,4,3,7,3,6,8,1,7,5,9
%N Decimal expansion of ratiosum for A294170; 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(n1) > g. The ratiosum for A is a(1)/a(0)  g + a(2)/a(1)  g + ..., assuming that this series converges. For A = A294170, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e 6.358713026984299354541477968890605504302...
%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] + b[n] + 2 n;
%t j = 1; While[j < 16, k = a[j]  j  1;
%t While[k < a[j + 1]  j + 1, b[k] = j + k + 2; k++]; j++];
%t u = Table[a[n], {n, 0, k}]; (* A294170 *)
%t g = GoldenRatio; s = N[Sum[ g + a[n]/a[n  1], {n, 1, 1000}], 200]
%t Take[RealDigits[s, 10][[1]], 100] (* A296491 *)
%Y Cf. A001622, A294170, A296284, A296492.
%K nonn,easy,cons
%O 1,1
%A _Clark Kimberling_, Dec 20 2017
