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A296453 Decimal expansion of limiting power-ratio for A296251; see Comments.

%I A296453 #6 Dec 18 2017 11:41:45
%S A296453 2,7,0,9,7,1,7,3,6,3,7,1,6,1,3,7,3,1,8,8,6,1,4,3,6,9,2,3,1,6,0,0,8,5,
%T A296453 3,0,9,3,6,3,8,6,6,0,5,9,5,9,1,4,1,6,1,9,8,9,1,8,8,7,3,5,6,8,0,2,8,5,
%U A296453 4,7,4,3,7,7,7,1,9,4,2,7,0,9,9,0,4,4
%N A296453 Decimal expansion of limiting power-ratio for A296251; see Comments.
%C A296453 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 = A296251 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 A296453 Limiting power-ratio = 27.09717363716137318861436923160085309363...
%t A296453 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A296453 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1]^2;
%t A296453 j = 1; While[j < 12, k = a[j] - j - 1;
%t A296453 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296453 Table[a[n], {n, 0, 15}]  (* A296251 *)
%t A296453 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296453 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296453 Take[RealDigits[Last[h], 10][[1]], 120] (* A296453 *)
%Y A296453 Cf. A001622, A296251.
%K A296453 nonn,easy,cons
%O A296453 2,1
%A A296453 _Clark Kimberling_, Dec 15 2017