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A296455 Decimal expansion of limiting power-ratio for A296260; see Comments.

%I A296455 #6 Dec 18 2017 11:41:59
%S A296455 2,3,1,0,8,1,5,7,2,4,3,5,8,7,8,8,6,0,4,1,4,4,4,5,0,7,0,7,5,1,4,3,5,3,
%T A296455 8,4,0,6,9,4,6,9,4,5,0,2,8,1,4,3,8,3,7,1,5,8,4,4,7,9,1,3,7,6,7,6,2,2,
%U A296455 1,8,8,3,0,2,4,1,2,6,5,5,2,3,1,8,2,2
%N A296455 Decimal expansion of limiting power-ratio for A296260; see Comments.
%C A296455 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 = A296260 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 A296455 Limiting power-ratio = 23.10815724358788604144450707514353840694...
%t A296455 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A296455 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1]*b[n - 2];
%t A296455 j = 1; While[j < 12, k = a[j] - j - 1;
%t A296455 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296455 Table[a[n], {n, 0, 15}]  (* A296260 *)
%t A296455 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296455 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296455 Take[RealDigits[Last[h], 10][[1]], 120] (* A296455 *)
%Y A296455 Cf. A001622, A296260.
%K A296455 nonn,easy,cons
%O A296455 2,1
%A A296455 _Clark Kimberling_, Dec 15 2017