%I #4 Nov 19 2017 10:43:12
%S 1,3,6,11,19,30,47,70,106,153,226,321,468,659,954,1338,1929,2698,3881,
%T 5420,7787,10866,15601,21760,31231,43551,62494,87135,125022,174305,
%U 250080,348647,500198,697333,1000436,1394707,2000914,2789457,4001872,5578959,8003790
%N Solution of the complementary equation a(n) = 2*a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, 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.
%C The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.43..., 1.39... .
%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) = 3, a(2) = 2, b(0) = 2, b(1) = 4,
%e a(2) = 2*a(0) + b(1) = 6
%e Complement: (b(n)) = (2, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, ...)
%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t a[0] = 1; a[1] = 3; b[0] = 2;
%t a[n_] := a[n] = 2 a[n - 2] + b[n - 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}] (* A295066 *)
%t Table[b[n], {n, 0, 10}]
%Y Cf. A295053, A295067.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Nov 19 2017