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A294863 Solution of the complementary equation a(n) = a(n-2) + b(n-2) + 3, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A294863 #4 Nov 18 2017 09:05:58
%S A294863 1,2,7,9,15,18,26,31,40,46,56,63,75,83,97,106,121,131,147,158,175,188,
%T A294863 206,220,239,255,275,292,313,331,353,372,395,416,440,462,487,510,537,
%U A294863 561,589,614,643,669,699,726,757,786,818,848,881,912,946,979,1014
%N A294863 Solution of the complementary equation a(n) = a(n-2) + b(n-2) + 3, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A294863 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 A294863 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 A294863 a(0) = 1, a(1) = 2, b(0) = 3
%e A294863 b(1) = 4 (least "new number")
%e A294863 a(2) = a(0) + b(0) + 3 = 7
%e A294863 Complement: (b(n)) = (3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 16, ...)
%t A294863 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294863 a[0] = 1; a[1] = 2; b[0] = 3;
%t A294863 a[n_] := a[n] = a[n - 2] + b[n - 2] + 3;
%t A294863 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294863 Table[a[n], {n, 0, 18}]  (* A294863 *)
%t A294863 Table[b[n], {n, 0, 10}]
%Y A294863 Cf. A294860, A294864.
%K A294863 nonn,easy
%O A294863 0,2
%A A294863 _Clark Kimberling_, Nov 16 2017