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A296458 Decimal expansion of limiting power-ratio for A296278; see Comments.

%I A296458 #6 Dec 18 2017 11:42:26
%S A296458 1,9,0,0,6,0,7,5,3,0,9,3,3,0,1,5,2,3,8,8,6,9,6,8,0,8,3,8,2,9,4,1,3,8,
%T A296458 5,8,9,0,0,0,5,8,2,8,5,9,6,0,5,6,9,7,6,1,7,7,8,4,8,0,3,1,4,4,0,4,3,7,
%U A296458 0,9,1,6,2,4,3,5,8,6,4,6,6,6,1,6,1,9
%N A296458 Decimal expansion of limiting power-ratio for A296278; see Comments.
%C A296458 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 limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A296278 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 A296458 Limiting power-ratio = 190.0607530933015238869680838294138589000...
%t A296458 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A296458 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n]*b[n - 1]*b[n - 2];
%t A296458 j = 1; While[j < 12, k = a[j] - j - 1;
%t A296458 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296458 Table[a[n], {n, 0, 15}]  (* A296278 *)
%t A296458 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296458 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296458 Take[RealDigits[Last[h], 10][[1]], 120] (* A296458 *)
%Y A296458 Cf. A001622, A296278.
%K A296458 nonn,easy,cons
%O A296458 3,2
%A A296458 _Clark Kimberling_, Dec 15 2017