%I
%S 1,2,7,11,27,41,90,133,282,412,860,1251,2596,3770,7806,11329,23438,
%T 34008,70336,102047,211032,306166,633122,918526,1899395,2755608,
%U 5698216,8266856,17094681,24800602,51284078,74401842
%N Solution of the complementary equation a(n) = 3*a(n2) + b(n1), 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.
%C The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.45..., 2.06...
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 113.
%e a(0) = 1, a(1) = 2, b(0) = 3
%e a(2) =3*a(0) + b(1) = 7
%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  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}] (* A295138 *)
%t Table[b[n], {n, 0, 10}]
%Y Cf. A295053.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Nov 19 2017
