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A296223 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 A296223 #5 Dec 11 2017 03:16:45
%S A296223 1,3,9,34,124,453,1654,6040,22055,80532,294058,1073735,3920679,
%T A296223 14316124,52274468,190877084,696976221,2544966858,9292793804,
%U A296223 33932079081,123900951107,452416889887,1651973131976,6032080786047,22025781112962,80425818360771
%N A296223 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 A296223 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 A296223 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 A296223 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
%e A296223 a(2) = a(0)*b(1) + a(1)*b(0) - 1 = 9
%e A296223 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, ...)
%t A296223 mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
%t A296223 a[0] = 1; a[1] = 3; b[0] = 2;
%t A296223 a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}] - 1;
%t A296223 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A296223 u = Table[a[n], {n, 0, 200}]  (* A296223 *)
%t A296223 Table[b[n], {n, 0, 20}]
%t A296223 N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
%t A296223 RealDigits[Last[t], 10][[1]] (* A296224 *)
%Y A296223 Cf. A296000, A296224.
%K A296223 nonn,easy
%O A296223 0,2
%A A296223 _Clark Kimberling_, Dec 10 2017