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

Author

N. J. A. Sloane, Nov 19 2019

Keywords

• nonn
• tabl
• easy

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").

Examples

			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,
...
		

Links

• 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

Crossrefs

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

Cf. A008574, A329501-A329517.

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).