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A296566 Decimal expansion of limiting power-ratio for A294556; see Comments.

%I A296566 #7 Jan 04 2018 21:23:47
%S A296566 9,0,6,5,4,5,0,2,2,6,7,3,2,3,3,2,9,2,8,9,8,9,9,7,5,9,9,2,6,2,6,5,1,5,
%T A296566 9,4,6,1,1,0,3,9,2,9,9,7,0,4,9,3,2,5,4,2,7,2,7,0,5,6,1,1,3,1,4,2,6,8,
%U A296566 9,2,5,1,9,5,2,2,2,6,6,4,7,1,0,5,9,8
%N A296566 Decimal expansion of limiting power-ratio for A294556; see Comments.
%C A296566 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 = A294556, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296566 limiting power-ratio = 9.065450226732332928989975992626515946110...
%t A296566 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
%t A296566 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + n;
%t A296566 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296566 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296566 Table[a[n], {n, 0, k}]; (* A294556 *)
%t A296566 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296566 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296566 Take[RealDigits[Last[h], 10][[1]], 120]   (* A296566 *)
%Y A296566 Cf. A001622, A294556, A296469, A296565.
%K A296566 nonn,easy,cons
%O A296566 1,1
%A A296566 _Clark Kimberling_, Dec 20 2017