A295070 - OEIS

%I #4 Nov 19 2017 10:43:39

%S 1,2,8,11,19,24,35,43,57,68,84,97,115,130,150,168,191,211,236,259,287,

%T 312,342,369,401,430,464,495,531,565,604,640,681,719,762,802,848,891,

%U 939,984,1034,1081,1133,1182,1236,1287,1343,1396,1454,1510,1571,1629

%N Solution of the complementary equation a(n) = a(n-2) + b(n-1) + b(n-2), 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.  Conjecture:  a(n)/a(n-1) -> 1.

%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) = 2, b(0) = 3

%e a(2) = a(0) + b(1) + b(0) = 8

%e Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, ... )

%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] = 2 a[n - 2] + b[n - 1] + 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}]  (* A295070 *)

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

%Y Cf. A295053.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Nov 19 2017