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A294862 Solution of the complementary equation a(n) = a(n-2) + b(n-2) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A294862 #4 Nov 18 2017 09:05:49
%S A294862 1,2,6,8,13,17,24,29,37,43,53,60,71,80,92,102,115,126,140,153,168,182,
%T A294862 198,214,231,248,266,284,303,322,343,363,385,406,429,452,476,500,525,
%U A294862 550,576,602,629,656,685,713,743,772,803,833,866,897,931,963,998
%N A294862 Solution of the complementary equation a(n) = a(n-2) + b(n-2) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A294862 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 A294862 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 A294862 a(0) = 1, a(1) = 2, b(0) = 3
%e A294862 b(1) = 4 (least "new number")
%e A294862 a(2) = a(0) + b(0) + 2 = 6
%e A294862 Complement: (b(n)) = (3, 4, 5, 7, 9, 10, 11, 12, 14, 15, 16, ...)
%t A294862 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A294862 a[0] = 1; a[1] = 2; b[0] = 3;
%t A294862 a[n_] := a[n] = a[n - 2] + b[n - 2] + 2;
%t A294862 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A294862 Table[a[n], {n, 0, 18}]  (* A294862 *)
%t A294862 Table[b[n], {n, 0, 10}]
%Y A294862 Cf. A294860, A294863.
%K A294862 nonn,easy
%O A294862 0,2
%A A294862 _Clark Kimberling_, Nov 16 2017