%I
%S 5,2,0,4,0,9,1,6,4,9,3,1,3,2,5,1,6,1,1,1,3,0,1,8,7,1,1,5,5,5,8,4,1,3,
%T 0,5,0,1,9,4,0,0,4,2,1,8,2,3,6,3,9,1,9,9,2,8,1,0,8,8,9,1,5,6,5,1,1,2,
%U 1,7,2,8,6,1,3,8,5,5,7,5,0,7,2,4,7,8
%N Decimal expansion of ratiosum for A296555; 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 = A296555, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e 5.204091649313251611130187115558413050194...
%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] + 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}]; (* A296555 *)
%t g = GoldenRatio; s = N[Sum[ g + a[n]/a[n  1], {n, 1, 1000}], 200]
%t Take[RealDigits[s, 10][[1]], 100] (* A296493 *)
%Y Cf. A001622, A296284, A296494, A296555.
%K nonn,easy,cons
%O 1,1
%A _Clark Kimberling_, Dec 19 2017
