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A296454 Decimal expansion of limiting power-ratio for A296257; see Comments.

%I A296454 #6 Dec 18 2017 11:41:52
%S A296454 1,9,9,3,6,8,4,3,0,3,3,5,8,8,2,6,3,7,3,5,2,2,3,0,4,5,3,7,2,2,4,2,3,2,
%T A296454 9,4,5,7,5,1,6,5,4,3,7,7,7,1,9,2,7,9,2,1,0,9,7,6,4,7,3,1,3,5,7,5,6,3,
%U A296454 7,9,9,1,1,7,1,3,0,3,3,3,0,5,5,2,6,6
%N A296454 Decimal expansion of limiting power-ratio for A296257; see Comments.
%C A296454 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 = A296257 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 A296454 Limiting power-ratio = 19.93684303358826373522304537224232945751...
%t A296454 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A296454 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1]^2;
%t A296454 j = 1; While[j < 12, k = a[j] - j - 1;
%t A296454 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296454 Table[a[n], {n, 0, 15}]  (* A296257 *)
%t A296454 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296454 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296454 Take[RealDigits[Last[h], 10][[1]], 120] (* A296454 *)
%Y A296454 Cf. A001622, A296257.
%K A296454 nonn,easy,cons
%O A296454 2,2
%A A296454 _Clark Kimberling_, Dec 15 2017