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A296498 Decimal expansion of limiting power-ratio for A294541; see Comments.

%I A296498 #7 Jan 04 2018 21:23:33
%S A296498 5,4,1,6,4,3,7,4,3,0,1,3,8,4,8,6,3,1,3,8,4,8,4,9,4,9,5,0,8,8,0,8,8,8,
%T A296498 0,0,8,4,8,6,7,6,6,5,9,9,0,1,5,3,5,0,6,5,3,2,7,4,6,4,2,8,1,0,9,3,6,4,
%U A296498 1,1,5,9,4,1,6,6,6,0,3,7,2,9,1,4,5,2
%N A296498 Decimal expansion of limiting power-ratio for A294541; see Comments.
%C A296498 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.
%C A296498 For A = A294541, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296498 limiting power-ratio = 5.416437430138486313848494950880888008486...
%t A296498 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A296498 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2];
%t A296498 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296498 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296498 Table[a[n], {n, 0, k}]; (* A294541 *)
%t A296498 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296498 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296498 Take[RealDigits[Last[h], 10][[1]], 120]   (* A296498 *)
%Y A296498 Cf. A001622, A294541, A296284, A296497.
%K A296498 nonn,easy,cons
%O A296498 1,1
%A A296498 _Clark Kimberling_, Dec 20 2017