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A295137 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1) - n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295137 #4 Nov 19 2017 19:04:49
%S A295137 1,2,8,26,80,242,729,2190,6573,19722,59169,177510,532533,1597602,
%T A295137 4792809,14378430,43135293,129405882
%N A295137 Solution of the complementary equation a(n) = 3*a(n-1) + b(n-1) - n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295137 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 A295137 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 A295137 a(0) = 1, a(1) = 2, b(0) = 3
%e A295137 a(2) =3*a(1) + b(1) - n = 8
%e A295137 Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, ... )
%t A295137 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295137 a[0] = 1; a[1] = 2; b[0] = 3;
%t A295137 a[n_] := a[n] = 3 a[n - 1] + b[n - 1] - n;
%t A295137 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295137 Table[a[n], {n, 0, 18}]  (* A295137 *)
%t A295137 Table[b[n], {n, 0, 10}]
%Y A295137 Cf. A295053.
%K A295137 nonn,easy
%O A295137 0,2
%A A295137 _Clark Kimberling_, Nov 19 2017