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A296457 Decimal expansion of limiting power-ratio for A296272; see Comments.

%I A296457 #6 Dec 18 2017 11:42:19
%S A296457 3,1,1,4,5,0,3,2,6,8,6,2,1,2,6,7,8,0,6,4,4,2,2,4,2,0,6,2,6,3,1,4,4,7,
%T A296457 3,3,0,7,3,4,1,5,3,7,2,2,5,0,8,3,8,8,0,5,8,5,3,2,6,5,1,4,0,4,5,2,0,4,
%U A296457 8,0,9,5,4,5,6,4,5,2,4,4,6,1,6,0,2,1
%N A296457 Decimal expansion of limiting power-ratio for A296272; see Comments.
%C A296457 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 = A296272 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 A296457 Limiting power-ratio = 31.14503268621267806442242062631447330734...
%t A296457 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A296457 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n]*b[n - 1];
%t A296457 j = 1; While[j < 12, k = a[j] - j - 1;
%t A296457 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296457 Table[a[n], {n, 0, 15}]  (* A296272 *)
%t A296457 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296457 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296457 Take[RealDigits[Last[h], 10][[1]], 120] (* A296457 *)
%Y A296457 Cf. A001622, A296272.
%K A296457 nonn,easy,cons
%O A296457 2,1
%A A296457 _Clark Kimberling_, Dec 15 2017