cp's OEIS Frontend

A295070 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.

%I A295070 #4 Nov 19 2017 10:43:39
%S A295070 1,2,8,11,19,24,35,43,57,68,84,97,115,130,150,168,191,211,236,259,287,
%T A295070 312,342,369,401,430,464,495,531,565,604,640,681,719,762,802,848,891,
%U A295070 939,984,1034,1081,1133,1182,1236,1287,1343,1396,1454,1510,1571,1629
%N A295070 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 A295070 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 A295070 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 A295070 a(0) = 1, a(1) = 2, b(0) = 3
%e A295070 a(2) = a(0) + b(1) + b(0) = 8
%e A295070 Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, ... )
%t A295070 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295070 a[0] = 1; a[1] = 2; b[0] = 3;
%t A295070 a[n_] := a[n] = 2 a[n - 2] + b[n - 1] + b[ n - 2];
%t A295070 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295070 Table[a[n], {n, 0, 18}]  (* A295070 *)
%t A295070 Table[b[n], {n, 0, 10}]
%Y A295070 Cf. A295053.
%K A295070 nonn,easy
%O A295070 0,2
%A A295070 _Clark Kimberling_, Nov 19 2017