A329504 Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width n squares cut from the square grid by cuts at 45 degrees to grid lines.

1, 1, 2, 1, 4, 2, 1, 4, 5, 2, 1, 4, 8, 4, 2, 1, 4, 8, 8, 4, 2, 1, 4, 8, 12, 6, 4, 2, 1, 4, 8, 12, 11, 6, 4, 2, 1, 4, 8, 12, 16, 8, 6, 4, 2, 1, 4, 8, 12, 16, 14, 8, 6, 4, 2, 1, 4, 8, 12, 16, 20, 10, 8, 6, 4, 2, 1, 4, 8, 12, 16, 20, 17, 10, 8, 6, 4, 2

Offset

1,3

Comments

By the "width" of the strip is meant the number of squares in a corner-to-corner ring around the cylinder.

For the case when the cuts are parallel to grid lines, see A329501.

See A329508 ... for coordination sequences for cylinders formed by rolling up the hexagonal grid ("carbon nanotubes").

Links

Table of n, a(n) for n=1..78.

N. J. A. Sloane, Illustration for rows 1 through 5, showing vertices of cylinder labeled with distance from base point (c = n is the width (or circumference)). The cylinders are formed by identifying the black lines.

Index entries for coordination sequences

Formula

Let theta = (1+x)/(1-x). The g.f. for the coordination sequence for row n is theta*(1+2x+2x^2+...+2x^(n-1)-(n-1)*x^n).

Example

Array begins:
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
1, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...
1, 4, 8, 8, 6, 6, 6, 6, 6, 6, 6, 6, ...
1, 4, 8, 12, 11, 8, 8, 8, 8, 8, 8, 8, ...
1, 4, 8, 12, 16, 14, 10, 10, 10, 10, 10, 10, ...
1, 4, 8, 12, 16, 20, 17, 12, 12, 12, 12, 12, ...
1, 4, 8, 12, 16, 20, 24, 20, 14, 14, 14, 14, ...
1, 4, 8, 12, 16, 20, 24, 28, 23, 16, 16, 16, ...
1, 4, 8, 12, 16, 20, 24, 28, 32, 26, 18, 18, ...
1, 4, 8, 12, 16, 20, 24, 28, 32, 36, 29, 20, ...
...
The initial antidiagonals are:
1,
1,2,
1,4,2,
1,4,5,2,
1,4,8,4,2,
1,4,8,8,4,2,
1,4,8,12,6,4,2,
1,4,8,12,11,6,4,2,
1,4,8,12,16,8,6,4,2,
...

Crossrefs

Rows 2,3,4 are A329505, A329506, A329507.

Cf. A008574, A329501-A329517.

Sequence in context: A083653 A278290 A135152 * A147542 A347069 A365901

Adjacent sequences: A329501 A329502 A329503 * A329505 A329506 A329507

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Nov 19 2019

Status

approved