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A296500 Decimal expansion of limiting power-ratio for A294546; see Comments.

%I A296500 #7 Jan 04 2018 21:23:39
%S A296500 7,9,2,0,2,0,8,0,7,1,3,1,9,2,9,4,4,3,2,1,2,8,6,9,0,5,2,9,9,8,8,5,7,8,
%T A296500 2,1,4,4,7,6,3,5,7,9,2,8,0,4,4,9,0,6,8,8,7,3,9,1,5,2,1,8,4,1,4,7,4,2,
%U A296500 4,4,6,9,0,6,2,9,1,5,1,4,8,0,9,6,3,4
%N A296500 Decimal expansion of limiting power-ratio for A294546; see Comments.
%C A296500 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 = A294546, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296500 limiting power-ratio = 7.920208071319294432128690529988578214476...
%t A296500 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A296500 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + n;
%t A296500 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296500 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296500 Table[a[n], {n, 0, k}]; (* A294546 *)
%t A296500 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296500 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296500 Take[RealDigits[Last[h], 10][[1]], 120]   (* A296500 *)
%Y A296500 Cf. A001622, A294546, A296284, A296499.
%K A296500 nonn,easy,cons
%O A296500 1,1
%A A296500 _Clark Kimberling_, Dec 20 2017