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%I #4 Nov 19 2017 10:43:55
%S 1,2,8,27,85,260,787,2369,7116,21358,64085,192267,576814,1730456,
%T 5191383,15574165,46722512,140167554
%N Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1) - 2, 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 A295053 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 a(2) =3*a(1) + b(1) - 2 = 8
%e Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, ... )
%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] = 3 a[n - 1] + b[n - 1] -2;
%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}] (* A295135 *)
%t Table[b[n], {n, 0, 10}]
%Y Cf. A295053.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Nov 19 2017
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