%I
%S 2,1,7,4,1,3,0,7,3,5,5,2,3,5,5,8,7,3,5,5,8,1,4,9,8,5,8,5,9,0,8,9,1,5,
%T 8,5,6,8,9,6,3,3,2,1,7,2,8,0,7,1,9,6,3,7,5,6,3,3,6,9,0,1,3,3,8,3,5,5,
%U 4,4,6,2,2,8,6,5,5,8,3,9,8,9,6,2,9,6
%N Decimal expansion of limiting powerratio 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(n1) > g. The limiting powerratio for A is the limit as n>oo of a(n)/g^n, assuming that this limit exists. For A = A296292 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425A296434 for related ratiosums and A296452A296461 for related limiting powerratios.
%e limiting powerratio = 21.74130735523558735581498585908915856896...
%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 < 12, 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, 15}] (* A296292 *)
%t z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t Take[RealDigits[Last[h], 10][[1]], 120] (* A296461 *)
%Y Cf. A001622, A296292.
%K nonn,easy,cons
%O 2,1
%A _Clark Kimberling_, Dec 18 2017
