OFFSET
0,2
COMMENTS
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.
The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.52..., 1.96...
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4
a(2) =3*a(0) + b(0) = 6
Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 11, 12, 13, 14, 15, ...)
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3; b[1]=4;
a[n_] := a[n] = 3 a[n - 2] + b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295139 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 19 2017
STATUS
approved