%I #4 Nov 19 2017 19:05:35
%S 1,3,9,25,64,159,389,945,2289,5534,13369,32285,77953,188206,454381,
%T 1096985,2648369,6393742,15435873,37265509,89966913,217199358,
%U 524365653,1265930690
%N Solution of the complementary equation a(n) = 2*a(n1) + a(n2) + b(n2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, 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), 113.
%F a(n+1)/a(n) > 1 + sqrt(2).
%e a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
%e a(2) =2*a(1) + a(0) + b(0) = 9
%e Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, ...)
%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t a[n_] := a[n] = 2 a[ n  1] + a[n  2] + b[n  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}] (* A295142 *)
%t Table[b[n], {n, 0, 10}]
%Y Cf. A295053, A295141, A295143, A295144.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Nov 19 2017
