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A295999 Decimal expansion of the limiting ratio of terms in A296000.

%I A295999 #10 Mar 15 2025 02:37:01
%S A295999 3,6,5,1,1,8,8,3,7,4,4,4,9,6,0,8,4,1,3,6,8,8,2,1,0,4,1,1,9,2,0,1,1,2,
%T A295999 0,2,0,1,4,8,6,0,7,8,3,7,3,3,1,4,0,8,8,2,2,6,4,0,0,7,6,8,7,0,2,2,5,9,
%U A295999 7,3,0,4,1,6,4,1,9,2,0,4,5,4,1,5,5,5
%N A295999 Decimal expansion of the limiting ratio of terms in A296000.
%C A295999 A296000(n)/A296000(n-1) -> 3.65118837444960841368821041192011...
%C A295999 See A296000 for a guide to related sequences and limiting ratios.
%t A295999 $RecursionLimit = Infinity;
%t A295999 mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
%t A295999 a[0] = 1; a[1] = 3; b[0] = 2; a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
%t A295999 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295999 Table[a[n], {n, 0, 100}]  (* A296000 *)
%t A295999 t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
%t A295999 Take[RealDigits[Last[t], 10][[1]], 100]  (* A295999 *)
%Y A295999 Cf. A296000.
%K A295999 nonn,easy,cons
%O A295999 1,1
%A A295999 _Clark Kimberling_, Dec 04 2017