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A296225 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) + n, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296225 #5 Dec 11 2017 14:30:47
%S A296225 1,3,12,44,161,588,2147,7839,28621,104498,381533,1393015,5086038,
%T A296225 18569636,67799608,247543185,903805055,3299883119,12048205018,
%U A296225 43989207775,160609019998,586399678681,2141004179974,7817021504815,28540731390577,104205079621096
%N A296225 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) + n, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296225 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 A296225 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 A296225 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
%e A296225 a(2) = a(0)*b(1) + a(1)*b(0) + 2 = 12
%e A296225 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, ...)
%t A296225 mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
%t A296225 a[0] = 1; a[1] = 3; b[0] = 2;
%t A296225 a[n_] := a[n] = n + Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
%t A296225 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A296225 Table[a[n], {n, 0, 200}]  (* A296225 *)
%t A296225 Table[b[n], {n, 0, 20}]
%t A296225 N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
%t A296225 RealDigits[Last[t], 10][[1]] (* A296226 *)
%Y A296225 Cf. A296000, A296226.
%K A296225 nonn,easy
%O A296225 0,2
%A A296225 _Clark Kimberling_, Dec 10 2017