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A296456 Decimal expansion of limiting power-ratio for A296266; see Comments.

%I A296456 #6 Dec 18 2017 11:42:06
%S A296456 2,6,3,8,7,6,2,8,9,9,1,3,8,5,1,1,7,7,5,3,8,3,3,2,0,7,8,1,2,3,2,0,7,3,
%T A296456 6,6,5,0,3,0,3,9,3,2,0,1,8,0,4,5,5,2,4,4,6,6,5,6,4,2,7,1,8,5,2,5,0,7,
%U A296456 9,7,3,0,4,8,2,8,9,5,0,3,0,7,8,7,2,4
%N A296456 Decimal expansion of limiting power-ratio for A296266; see Comments.
%C A296456 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 = A296266 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 A296456 Limiting power-ratio = 26.38762899138511775383320781232073665030...
%t A296456 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A296456 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n]*b[n - 2];
%t A296456 j = 1; While[j < 12, k = a[j] - j - 1;
%t A296456 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296456 Table[a[n], {n, 0, 15}]  (* A296266 *)
%t A296456 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296456 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296456 Take[RealDigits[Last[h], 10][[1]], 120] (* A296456 *)
%Y A296456 Cf. A001622, A296266.
%K A296456 nonn,easy,cons
%O A296456 2,1
%A A296456 _Clark Kimberling_, Dec 15 2017