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%I #4 Nov 18 2017 09:05:41
%S 1,2,5,7,12,16,22,27,34,41,49,57,67,76,87,97,109,121,134,147,161,176,
%T 191,207,223,240,257,276,294,314,333,354,374,397,418,442,464,489,512,
%U 538,563,590,616,644,671,700,728,759,788,820,850,883,914,948,980
%N Solution of the complementary equation a(n) = a(n-2) + b(n-2) + 1, 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), 1-13.
%e a(0) = 1, a(1) = 2, b(0) = 3
%e b(1) = 4 (least "new number")
%e a(2) = a(0) + b(0) + 1 = 5
%e Complement: (b(n)) = (3, 4, 6, 8, 9, 10, 11, 12, 14, 15, 17, ...)
%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] + 1;
%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}] (* A294861 *)
%t Table[b[n], {n, 0, 10}]
%Y Cf. A294860, A294862.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Nov 16 2017
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