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%I #4 Dec 08 2017 09:49:39
%S 2,9,6,1,8,4,4,3,2,4,2,0,3,0,8,0,2,7,4,4,1,2,5,3,8,5,8,4,2,9,3,9,3,1,
%T 3,8,1,5,9,8,5,8,8,3,7,0,9,1,9,8,4,1,4,5,0,0,4,1,0,4,8,8,0,3,3,2,3,9,
%U 6,2,0,3,9,8,9,3,3,1,3,9,4,1,6,6,2,4
%N Decimal expansion of the limiting ratio of terms in A296003.
%C A296003(n)/A296003(n-1)) -> 2.961844324.....
%C See A296000 for a guide to related sequences and limiting ratios.
%t mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
%t a[0] = 2; a[1] = 4; b[0] = 1;
%t a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 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, 100}]; (* A296003 *)
%t t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
%t Take[RealDigits[Last[t], 10][[1]], 100] (* A296004 *)
%Y Cf. A296000, A296003.
%K nonn,easy,cons
%O 1,1
%A _Clark Kimberling_, Dec 07 2017