

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.


4



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
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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..(n2)/2} x^i)^2;
for n odd, P_n(x) = (1 + 2*Sum_{i=0..(n1)/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: A260625 A306614 A264336 * A123521 A322115 A294217
Adjacent sequences: A322035 A322036 A322037 * A322039 A322040 A322041


KEYWORD

nonn,tabf


AUTHOR

N. J. A. Sloane, Dec 01 2018


STATUS

approved



