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A296472 Decimal expansion of limiting power-ratio for A295947; see Comments.

%I A296472 #4 Dec 18 2017 19:45:41
%S A296472 7,1,7,1,3,5,1,0,4,5,0,4,1,1,3,7,7,9,7,3,9,8,2,0,5,6,7,8,3,8,8,6,9,0,
%T A296472 1,2,5,2,7,3,0,9,1,9,4,5,7,5,2,5,4,8,1,3,3,8,9,9,1,3,2,6,5,4,9,3,0,1,
%U A296472 8,9,2,6,1,7,4,8,7,3,2,3,0,2,9,2,8,4
%N A296472 Decimal expansion of limiting power-ratio for A295947; see Comments.
%C A296472 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 = A295947 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296472 limiting power-ratio = 7.171351045041137797398205678388690125273...
%t A296472 a[0] = 2; a[1] = 4; b[0] = 1; b[1 ] = 3; b[2] = 5;
%t A296472 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
%t A296472 j = 1; While[j < 12, k = a[j] - j - 1;
%t A296472  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296472 Table[a[n], {n, 0, 15}]  (* A295947 *)
%t A296472 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296472 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296472 Take[RealDigits[Last[h], 10][[1]], 120] (* A296472 *)
%Y A296472 Cf. A001622, A296471.
%K A296472 nonn,easy,cons
%O A296472 1,1
%A A296472 _Clark Kimberling_, Dec 18 2017