A329508 Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width n hexagons cut from the hexagonal grid by cuts parallel to grid lines.

1, 1, 2, 1, 3, 2, 1, 3, 5, 2, 1, 3, 6, 5, 2, 1, 3, 6, 8, 4, 2, 1, 3, 6, 9, 8, 4, 2, 1, 3, 6, 9, 11, 7, 4, 2, 1, 3, 6, 9, 12, 11, 6, 4, 2, 1, 3, 6, 9, 12, 14, 10, 6, 4, 2, 1, 3, 6, 9, 12, 15, 14, 9, 6, 4, 2, 1, 3, 6, 9, 12, 15, 17, 13, 8, 6, 4, 2

Offset: 1

Author

N. J. A. Sloane, Nov 22 2019

Keywords

• nonn
• tabl
• easy

Comments

This is the structure of carbon nanotubes.

For the case when the cuts are perpendicular to the grid lines, see A329512 and A329515.

See A329501 and A329504 for coordination sequences for cylinders formed by rolling up the square grid.

Examples

			Array begins:
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
1, 3, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, ...
1, 3, 6, 8, 8, 7, 6, 6, 6, 6, 6, 6, ...
1, 3, 6, 9, 11, 11, 10, 9, 8, 8, 8, 8, ...
1, 3, 6, 9, 12, 14, 14, 13, 12, 11, 10, 10, ...
1, 3, 6, 9, 12, 15, 17, 17, 16, 15, 14, 13, ...
1, 3, 6, 9, 12, 15, 18, 20, 20, 19, 18, 17, ...
1, 3, 6, 9, 12, 15, 18, 21, 23, 23, 22, 21, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 26, 26, 25, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 29, 29, ...
The initial antidiagonals are:
1
1,2
1,3,2
1,3,5,2
1,3,6,5,2
1,3,6,8,4,2
1,3,6,9,8,4,2
1,3,6,9,11,7,4,2
1,3,6,9,12,11,6,4,2
1,3,6,9,12,14,10,6,4,2
...
		

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 3, showing vertices of cylinder labeled with distance from base point 0 (c = n is the width (or circumference)). The cylinders are formed by identifying the black lines.
• N. J. A. Sloane, Illustration for row 4, showing vertices of cylinder labeled with distance from base point 0 (c = n is the width (or circumference)). The cylinder is formed by identifying the black lines. The trunks are colored blue, the branches red, and the twigs green.
• N. J. A. Sloane, Illustration for row 5, showing vertices of cylinder labeled with distance from base point 0 (c = n is the width (or circumference)). The cylinder is formed by identifying the black lines. The trunks are colored blue, the branches red, and the twigs green.
• Index entries for coordination sequences

Crossrefs

Rows 1,2,3,4 are A040000, A329509, A329510, A329511.

Cf. A008486, A329501-A329517.

Programs

Magma

c := 4; \\ set c
R := RationalFunctionField(Integers());
FG3 := FreeGroup(3);
Q3 := quo;
H := AutomaticGroup(Q3);
f3 := GrowthFunction(H);
PSR := PowerSeriesRing(Integers():Precision := 60);
Coefficients(PSR!f3);
// 1, 3, 6, 9, 11, 11, 10, 9, 8, 8, 8, 8, 8, 8, 8, ... (row c)
f3;  // g.f. for row c
// (x^8 + x^7 + x^6 - 2*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1)
// = (1+x)*(x^3-x-1)*(x^2+1)^2/(x-1)

Formula

The g.f.s for the rows were found and proved using the "trunks and branches" method (see Goodman-Strauss and Sloane). In the illustrations for n=4 and n=5, the trunks are colored blue, the branches red, and the twigs green.

The g.f. G(c) for row c (c>=1) is

(1/(1-x))*(1 + 2*x + 3*x^2*(1-x^(c-2))/(1-x) + 2*x^c - x^(c+2)*(1-x^(c-1))/(1-x)).

The values of G(1) through G(8) are:

(1+x)/(1-x),

(1+x)*(x^3-x^2-x-1)/(x-1),

(1+x)*(x^2+x+1)*(x^3-x^2-1)/(x-1),

(1+x)*(x^3-x-1)*(x^2+1)^2/(x-1),

(1+x)*(x^4+x^3+x^2+x+1)*(x^5-x^4+x^3-x^2-1)/(x-1),

(1+x)*(x^2+x+1)*(x^2-x+1)*(x^7-x^2-x-1)/(x-1),

(1+x)*(x^6+x^5+x^4+x^3+x^2+x+1)*(x^7-x^6+x^5-x^4+x^3-x^2-1)/(x-1),

(1+x)*(x^7-x^5+x^3-x-1)*(x^4+1)*(x^2+1)^2/(x-1).

Note that row n is equal to 2*n once the 2*n-th term has been reached.

The g.f.s for the rows can also be calculated by regarding the 1-skeleton of the cylinder as the Cayley diagram for an appropriate group H, and computing the growth function for H (see the MAGMA code).