%I #4 Nov 18 2017 20:54:56

%S 1,4,10,23,51,108,223,454,917,1845,3702,7417,14848,29711,59438,118893,

%T 237804,475627,951274,1902569,3805160,7610344,15220713,30441452,

%U 60882931,121765890,243531809,487063648

%N Solution of the complementary equation a(n) = 2*a(n-1) + b(n-2), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 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) = 4, b(0) = 2, b(1) = 3

%e a(2) = 2*a(1) + b(0) = 10

%e Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, ...)

%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

%t a[0] = 2; a[1] = 5; b[0] = 1;

%t a[n_] := a[n] = 2 a[n - 1] + 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}] (* A295059 *)

%t Table[b[n], {n, 0, 10}]

%Y Cf. A295053.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Nov 18 2017