%I #4 Nov 18 2017 09:05:58
%S 1,2,7,9,15,18,26,31,40,46,56,63,75,83,97,106,121,131,147,158,175,188,
%T 206,220,239,255,275,292,313,331,353,372,395,416,440,462,487,510,537,
%U 561,589,614,643,669,699,726,757,786,818,848,881,912,946,979,1014
%N Solution of the complementary equation a(n) = a(n2) + b(n2) + 3, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294860 for a guide to related sequences.
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 113.
%e a(0) = 1, a(1) = 2, b(0) = 3
%e b(1) = 4 (least "new number")
%e a(2) = a(0) + b(0) + 3 = 7
%e Complement: (b(n)) = (3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 16, ...)
%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t a[0] = 1; a[1] = 2; b[0] = 3;
%t a[n_] := a[n] = a[n  2] + b[n  2] + 3;
%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, 18}] (* A294863 *)
%t Table[b[n], {n, 0, 10}]
%Y Cf. A294860, A294864.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Nov 16 2017
