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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 (list; graph; refs; listen; history; text; internal format)
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: 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

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Last modified August 11 23:45 EDT 2020. Contains 336434 sequences. (Running on oeis4.)