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

0,2

Comments

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

Links

Table of n, a(n) for n=0..36.

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

Example

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, ...)

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 *)

Crossrefs

Cf. A001622, A000045, A295619, A295620.

Sequence in context: A357954 A162222 A010346 * A295755 A089142 A123215

Adjacent sequences: A295618 A295619 A295620 * A295622 A295623 A295624

Keyword

nonn,easy

Author

Clark Kimberling, Nov 25 2017

Extensions

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

Status

approved