A322038 Irregular triangle read by rows: for n >= 0, row n gives the coordination sequence for the tiling of a flat torus by a square grid with n points along each circuit.

1, 1, 1, 2, 1, 1, 4, 4, 1, 4, 6, 4, 1, 1, 4, 8, 8, 4, 1, 4, 8, 10, 8, 4, 1, 1, 4, 8, 12, 12, 8, 4, 1, 4, 8, 12, 14, 12, 8, 4, 1, 1, 4, 8, 12, 16, 16, 12, 8, 4, 1, 4, 8, 12, 16, 18, 16, 12, 8, 4, 1, 1, 4, 8, 12, 16, 20, 20, 16, 12, 8, 4

Offset

0,4

Comments

More precisely, this is the coordination sequence for the quotient graph Z^2 / (nZ X nZ). The graph has n^2 vertices.

There are obvious generalizations: for example, Z^2 / (mZ X nZ) where m and n are not necessarily equal.

Links

Table of n, a(n) for n=0..71.

N. J. A. Sloane, Illustration showing the tilings and coordination sequences for n = 4 and 5

Formula

Since the underlying graphs are finite, the coordination sequences are polynomial P_n(x).

For n even, P_n(x) = (1+x)^2*(Sum_{i=0..(n-2)/2} x^i)^2;

for n odd, P_n(x) = (1 + 2*Sum_{i=0..(n-1)/2} x^i)^2.

Example

The triangle begins:
1,
1,
1, 2, 1,
1, 4, 4,
1, 4, 6, 4, 1,
1, 4, 8, 8, 4,
1, 4, 8, 10, 8, 4, 1,
1, 4, 8, 12, 12, 8, 4,
1, 4, 8, 12, 14, 12, 8, 4, 1,
1, 4, 8, 12, 16, 16, 12, 8, 4,
1, 4, 8, 12, 16, 18, 16, 12, 8, 4, 1,
...

Crossrefs

The rows converge to A008574.

Sequence in context: A306614 A264336 A350012 * A123521 A322115 A294217

Adjacent sequences: A322035 A322036 A322037 * A322039 A322040 A322041

Keyword

nonn,tabf

Author

N. J. A. Sloane, Dec 01 2018

Status

approved