A295621 Solution of the complementary equation a(n) = a(n-1) + 3*a(n-2) -2*a(n-3) - 2*a(n-4) + b(n-3), where a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4, b(0) = 5, b(1) = 6, b(2) = 7, b(3) = 8, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 3, 4, 13, 22, 55, 96, 201, 346, 659, 1117, 2015, 3372, 5882, 9752, 16643, 27411, 46093, 75559, 125754, 205448, 339432, 553177, 909097, 1478897, 2421000, 3933174, 6420218, 10419979, 16972319, 27525507, 44762106, 72554068, 117844772, 190931789, 309833797

Offset: 0

Author

Clark Kimberling, Nov 25 2017

Keywords

• nonn
• easy

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.

Examples

			a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4, b(0) = 5, b(1) = 6, b(2) = 7, b(3) = 8, so that
b(4) = 9 (least "new number")
a(4) = a(3) + 3*a(2) -2*a(1) - 2*a(0) + b(1) = 13
Complement: (b(n)) = (5, 6, 7, 8, 9, 10, 11, 12, 14, 15, ...)
		

Links

• Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Crossrefs

Cf. A001622, A000045, A295619, A295620.

Programs

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; a[2] = 3; a[3] = 4;
b[0] = 5; b[1] = 6; b[2] = 7; b[3] = 8;
a[n_] := a[n] = a[n - 1] + 3*a[n - 2] - 2*a[n - 3] - 2 a[n - 4] + b[n - 3];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
z = 36;  Table[a[n], {n, 0, z}]   (* A295621 *)
Table[b[n], {n, 0, 20}]  (*complement *)

Extensions

Typo in the definition corrected by Georg Fischer, Jun 08 2022