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A296461 Decimal expansion of limiting power-ratio for A296292; see Comments.

%I A296461 #4 Dec 18 2017 19:46:11
%S A296461 2,1,7,4,1,3,0,7,3,5,5,2,3,5,5,8,7,3,5,5,8,1,4,9,8,5,8,5,9,0,8,9,1,5,
%T A296461 8,5,6,8,9,6,3,3,2,1,7,2,8,0,7,1,9,6,3,7,5,6,3,3,6,9,0,1,3,3,8,3,5,5,
%U A296461 4,4,6,2,2,8,6,5,5,8,3,9,8,9,6,2,9,6
%N A296461 Decimal expansion of limiting power-ratio for A296292; see Comments.
%C A296461 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 = A296292 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 A296461 limiting power-ratio = 21.74130735523558735581498585908915856896...
%t A296461 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A296461 a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n];
%t A296461 j = 1; While[j < 12, k = a[j] - j - 1;
%t A296461  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296461 Table[a[n], {n, 0, 15}]  (* A296292 *)
%t A296461 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296461 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296461 Take[RealDigits[Last[h], 10][[1]], 120] (* A296461 *)
%Y A296461 Cf. A001622, A296292.
%K A296461 nonn,easy,cons
%O A296461 2,1
%A A296461 _Clark Kimberling_, Dec 18 2017