A296220 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 10, 13, 16, 19, 22, 25, 29, 34, 38, 43, 47, 52, 56, 61, 65, 70, 74, 79, 82, 86, 91, 94, 97, 101, 106, 109, 113, 118, 121, 124, 128, 133, 136, 140, 145, 148, 151, 155, 160, 163, 167, 172, 175, 178, 182, 187, 190, 194, 199, 202, 205, 209, 214, 217, 221

Offset: 0

Author

Clark Kimberling, Dec 08 2017

Keywords

• nonn
• easy

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.

P. Majer proved that a(n)/n -> 4, that (a(n) - 4*n) is unbounded, and that a( ) is not a linear recurrence sequence; see the Math Overflow link and A297964. - Clark Kimberling, Feb 10 2018

Examples

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4;
a(2) = a(0)*b(1) + a(1)*b(0) = 10.
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, ...).
		

Links

• Peter J. C. Moses, Table of n, a(n) for n = 0..9999
• Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
• Pietro Majer, Limit associated with complementary sequences Math Overflow.

Crossrefs

Cf. A296000.

Programs

Mathematica

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[0]*b[n - 1] + a[1]*b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 500}];  (* A296220 *)
Table[b[n], {n, 0, 20}]