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A294864 Solution of the complementary equation a(n) = a(n-2) + b(n-2) + n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A294864 #4 Nov 18 2017 09:06:05
%S A294864 1,2,6,9,15,21,29,38,48,59,71,84,99,114,131,148,167,187,208,230,253,
%T A294864 277,302,328,356,384,414,444,476,508,542,576,613,649,688,726,767,807,
%U A294864 850,892,937,982,1029,1076,1125,1174,1225,1276,1329,1382,1437,1493,1550
%N A294864 Solution of the complementary equation a(n) = a(n-2) + b(n-2) + n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A294864 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 A294864 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 A294864 a(0) = 1, a(1) = 2, b(0) = 3
%e A294864 b(1) = 4 (least "new number")
%e A294864 a(2) = a(0) + b(0) + 2 = 6
%e A294864 Complement: (b(n)) = (3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 16, ...)
%t A294864 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294864 a[0] = 1; a[1] = 2; b[0] = 3;
%t A294864 a[n_] := a[n] = a[n - 2] + b[n - 2] + n;
%t A294864 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294864 Table[a[n], {n, 0, 18}]  (* A294864 *)
%t A294864 Table[b[n], {n, 0, 10}]
%Y A294864 Cf. A294860.
%K A294864 nonn,easy
%O A294864 0,2
%A A294864 _Clark Kimberling_, Nov 16 2017