%I
%S 7,4,8,6,5,9,8,2,3,8,8,6,1,1,7,5,1,5,0,8,3,0,4,2,2,9,1,1,8,2,1,9,2,9,
%T 6,7,7,6,4,4,6,8,2,9,9,2,3,8,9,5,1,2,3,4,6,7,0,7,0,8,2,3,7,2,0,1,3,6,
%U 8,1,1,7,6,2,5,8,9,6,9,1,6,7,0,6,1,2
%N Decimal expansion of ratiosum for A296257; 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 = A296257, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425A296434 for related ratiosums and A296452A296461 for related limiting powerratios.
%e ratiosum = 7.48659823886117515083042...
%t a[0] = 1; a[1] = 2; b[0] = 3;
%t a[n_] := a[n] = a[n  1] + a[n  2] + b[n  2]^2;
%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}]; (* A296257 *)
%t g = GoldenRatio; s = N[Sum[ g + a[n]/a[n  1], {n, 1, 1000}], 200]
%t Take[RealDigits[s, 10][[1]], 100] (* A296427 *)
%Y Cf. A001622, A296257.
%K nonn,easy,cons
%O 2,1
%A _Clark Kimberling_, Dec 14 2017
