Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #5 Dec 11 2017 14:30:40
%S 3,6,5,1,4,4,0,0,0,7,8,8,4,2,6,3,1,0,2,8,6,0,9,0,8,8,3,8,8,3,6,6,1,1,
%T 5,6,6,2,1,0,9,9,1,7,9,0,0,6,0,5,3,2,3,8,8,7,1,7,8,0,2,9,0,3,3,0,3,1,
%U 8,2,0,0,4,6,7,1,8,8,0,6,1,5,1,3,6,5
%N Decimal expansion of the limiting ratio of terms in A296223.
%C A296223(n)/A296223(n-1)) -> 3.651440004884...
%C See A296000 for a guide to related sequences and limiting ratios.
%t mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
%t a[0] = 1; a[1] = 3; b[0] = 2;
%t a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}] - 1;
%t b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t Table[a[n], {n, 0, 200}] (* A296223 *)
%t Table[b[n], {n, 0, 20}]
%t N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
%t RealDigits[Last[t], 10][[1]] (* A296224 *)
%Y Cf. A296000, A296223.
%K nonn,easy,cons
%O 1,1
%A _Clark Kimberling_, Dec 10 2017