A329501 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 parallel to grid lines.

1, 1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 4, 6, 4, 2, 1, 4, 7, 6, 4, 2, 1, 4, 8, 8, 6, 4, 2, 1, 4, 8, 10, 8, 6, 4, 2, 1, 4, 8, 11, 10, 8, 6, 4, 2, 1, 4, 8, 12, 12, 10, 8, 6, 4, 2, 1, 4, 8, 12, 14, 12, 10, 8, 6, 4, 2

Offset: 1

Author

N. J. A. Sloane, Nov 19 2019

Keywords

• nonn
• tabl
• easy

Comments

For the case when the cuts are at 45 degrees to the grid lines, see A329504.

See A329508, A329512, and A329515 for coordination sequences for cylinders formed by rolling up the hexagonal grid ("carbon nanotubes").

The g.f.s for the rows can easily be found using the "trunks and branches" method (see Goodman-Strauss and Sloane). In the illustration for n=5, there are two trunks (blue) and ten branches (red).

Examples

			Array begins:
  1, 2, 2,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
  1, 3, 4,  4,  4,  4,  4,  4,  4,  4,  4,  4, ...
  1, 4, 6,  6,  6,  6,  6,  6,  6,  6,  6,  6, ...
  1, 4, 7,  8,  8,  8,  8,  8,  8,  8,  8,  8, ...
  1, 4, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, ...
  1, 4, 8, 11, 12, 12, 12, 12, 12, 12, 12, 12, ...
  1, 4, 8, 12, 14, 14, 14, 14, 14, 14, 14, 14, ...
  1, 4, 8, 12, 15, 16, 16, 16, 16, 16, 16, 16, ...
  1, 4, 8, 12, 16, 18, 18, 18, 18, 18, 18, 18, ...
  1, 4, 8, 12, 16, 19, 20, 20, 20, 20, 20, 20, ...
  ...
The initial antidiagonals are:
  1;
  1,  2;
  1,  3,  2;
  1,  4,  4,  2;
  1,  4,  6,  4,  2;
  1,  4,  7,  6,  4,  2;
  1,  4,  8,  8,  6,  4,  2;
  1,  4,  8, 10,  8,  6,  4,  2;
  1,  4,  8, 11, 10,  8,  6,  4,  2;
  1,  4,  8, 12, 12, 10,  8,  6,  4,  2;
  1,  4,  8, 12, 14, 12, 10,  8,  6,  4,  2;
  ...
		

Links

• Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
• 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 1,2,3,4,5 are A040000, A113311, A329502, A115291, A329503.

Cf. A008574, A329504-A329517.

Formula

Let theta = (1+x)/(1-x).

If n = 2*k, the g.f. for the coordination sequence for row n is theta*(1+2*x+2*x^2+...+2*x^(k-1)+x^k).

If n = 2*k+1, the g.f. for the coordination sequence for row n is theta*(1+2*x+2*x^2+...+2*x^k).