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A296221 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296221 #11 Jun 25 2018 17:38:15
%S A296221 1,3,11,40,146,533,1946,7105,25941,94714,345812,1262601,4609907,
%T A296221 16831321,61453163,224372837,819212023,2991040928,10920647625,
%U A296221 39872588647,145579582824,531528442330,1940673819263,7085631873740,25870488153041,94456241758347
%N A296221 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296221 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.
%H A296221 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.
%e A296221 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
%e A296221 a(2) = a(0)*b(1) + a(1)*b(0) + 1 = 11
%e A296221 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, ...)
%t A296221 mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
%t A296221 a[0] = 1; a[1] = 3; b[0] = 2;
%t A296221 a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}] + 1;
%t A296221 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A296221 u = Table[a[n], {n, 0, 200}];  (* A296221 *)
%t A296221 Table[b[n], {n, 0, 20}]
%t A296221 t = N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
%t A296221 d = RealDigits[Last[t], 10][[1]] (* A296222 *)
%Y A296221 Cf. A296000, A296222.
%K A296221 nonn,easy
%O A296221 0,2
%A A296221 _Clark Kimberling_, Dec 09 2017
%E A296221 Conjectured g.f. removed by _Alois P. Heinz_, Jun 25 2018