A329501 -id:A329501 - cp's OEIS frontend

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

N. J. A. Sloane, Nov 19 2019

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").

			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,
...

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

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

Cf. A008574, A329501-A329517.

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).

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.

Original entry on oeis.org

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

N. J. A. Sloane, Nov 22 2019

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.

			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
...

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

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

Cf. A008486, A329501-A329517.

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)

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).

A329515 Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width 2*n+1 hexagons cut from the hexagonal grid by cuts perpendicular to grid lines.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 3, 6, 2, 1, 3, 6, 7, 2, 1, 3, 6, 9, 6, 2, 1, 3, 6, 9, 12, 6, 2, 1, 3, 6, 9, 12, 12, 6, 2, 1, 3, 6, 9, 12, 15, 10, 6, 2, 1, 3, 6, 9, 12, 15, 18, 10, 6, 2, 1, 3, 6, 9, 12, 15, 18, 17, 10, 6, 2, 1, 3, 6, 9, 12, 15, 18, 21, 14, 10, 6, 2

Offset: 0

N. J. A. Sloane, Nov 25 2019

The width of the strip is a little harder to define here. In the illustration for n=2, the strip is five hexagons wide if measured along hexagons that touch edge-to-edge. A path joining two vertices to be identified when the cylinder is formed has length 4n+2 edges (10 edges in the illustration for n=2).
Because the width 2*n+1 is odd, one edge of the cut has to be displaced sideways (by half the width of a hexagon) in order for the two edges to mesh properly.  The arrows in the figures indicate pairs of points which will coalesce.
For the case when the strip is 2*n hexagons wide see A329512.
For the case when the cuts are parallel to the grid lines, see A329508.
See A329501 and A329504 for coordination sequences for cylinders formed by rolling up the square grid.

			Array begins:
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
1, 3, 6, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, ...
1, 3, 6, 9, 12, 12, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, ...
1, 3, 6, 9, 12, 15, 18, 17, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 22, 18, 18, 18, 18, 18, 18, 18, 18, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 27, 22, 22, 22, 22, 22, 22, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 32, 26, 26, 26, 26, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 37, 30, 30, 30, 30, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 42, 34, 34, 34, 34, 34, 34, 34, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 47, 38, 38, 38, 38, 38, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 52, 42, 42, 42, ...
...
The initial antidiagonals are:
1,
1, 2,
1, 3, 2,
1, 3, 6, 2,
1, 3, 6, 7, 2,
1, 3, 6, 9, 6, 2,
1, 3, 6, 9, 12, 6, 2,
1, 3, 6, 9, 12, 12, 6, 2,
1, 3, 6, 9, 12, 15, 10, 6, 2,
1, 3, 6, 9, 12, 15, 18, 10, 6, 2,
1, 3, 6, 9, 12, 15, 18, 17, 10, 6, 2,
1, 3, 6, 9, 12, 15, 18, 21, 14, 10, 6, 2,
1, 3, 6, 9, 12, 15, 18, 21, 24, 14, 10, 6, 2,
...

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 row n = 0, showing vertices of cylinder of width (or circumference) 1 labeled with distance from base point 0. The cylinder is formed by identifying the black lines. Arrows indicate two points which will coalesce.
N. J. A. Sloane, Illustration for row n = 1, showing vertices of cylinder of width (or circumference) 3 labeled with distance from base point 0. The cylinder is formed by identifying the black lines. Arrows indicate two points which will coalesce.
N. J. A. Sloane, Illustration for row n = 2, showing vertices of cylinder of width (or circumference) 5 labeled with distance from base point 0. The cylinder is formed by identifying the black lines. Arrows indicate two points which will coalesce.
Index entries for coordination sequences

Rows 0,1,2,3 are A040000, A329516, A329517, A329771.

Cf. A008486, A329501-A329514.

n := 2; \\ set n
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, 12, 12, 10, 10, 10, 10, 10, 10, ... (row n)
f3; // G(n)
// (2*x^6 - 3*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1)

The g.f.s for the rows could be found using the "trunks and branches" method (see Goodman-Strauss and Sloane), as was done in A329508. This step has not yet been carried out, so the following g.f. is at present only conjectural.
The g.f. G(n) for row n (n>=0) is (strongly) conjectured to be
(1/(1-x))*(1 + 2*x + 3*x^2*(1-x^(2*n-1))/(1-x) - (n-2)*x^(2*n+1) - n*x^(2*n+2)).
The values of G(0) through G(6) (certified by MAGMA) are:
(1 + x)/(1 - x),
(x^4 - x^3 - 3*x^2 - 2*x - 1)/(x - 1),
(2*x^6 - 3*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1),
(3*x^8 + x^7 - 3*x^6 - 3*x^5 - 3*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1),
(4*x^10 + 2*x^9 - 3*x^8 - 3*x^7 - 3*x^6 - 3*x^5 - 3*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1),
(5*x^12 + 3*x^11 - 3*x^10 - 3*x^9 - 3*x^8 - 3*x^7 - 3*x^6 - 3*x^5 - 3*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1),
(6*x^14 + 4*x^13 - 3*x^12 - 3*x^11 - 3*x^10 - 3*x^9 - 3*x^8 - 3*x^7 - 3*x^6 - 3*x^5 - 3*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1).
Note that row n is equal to 4*n+2 once the (2*n+2)-nd term has been reached.
The g.f.s for the rows can also be obtained 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).

A329512 Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width 2*n hexagons cut from the hexagonal grid by cuts perpendicular to grid lines.

Original entry on oeis.org

1, 1, 3, 1, 3, 4, 1, 3, 6, 4, 1, 3, 6, 9, 4, 1, 3, 6, 9, 9, 4, 1, 3, 6, 9, 12, 8, 4, 1, 3, 6, 9, 12, 15, 8, 4, 1, 3, 6, 9, 12, 15, 14, 8, 4, 1, 3, 6, 9, 12, 15, 18, 12, 8, 4, 1, 3, 6, 9, 12, 15, 18, 21, 12, 8, 4

Offset: 1

N. J. A. Sloane, Nov 24 2019

The width of the strip is a little harder to define here. In the illustration for n=2, the strip is four hexagons wide if measured along hexagons that touch edge-to-edge. A path joining two vertices to be identified when the cylinder is formed has length 4n edges (8 edges in the illustration for n=2).
For the case when the strip is 2*n+1 hexagons wide see A329515.
For the case when the cuts are parallel to the grid lines, see A329508.
See A329501 and A329504 for coordination sequences for cylinders formed by rolling up the square grid.

			Array begins:
1, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...
1, 3, 6, 9, 9, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, ...
1, 3, 6, 9, 12, 15, 14, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, ...
1, 3, 6, 9, 12, 15, 18, 21, 19, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 24, 20, 20, 20, 20, 20, 20, 20, 20, 20, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 29, 24, 24, 24, 24, 24, 24, 24, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 34, 28, 28, 28, 28, 28, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 39, 32, 32, 32, ...
1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 44, 36, 36, ...
...
The initial antidiagonals are:
1,
1, 3,
1, 3, 4,
1, 3, 6, 4,
1, 3, 6, 9, 4,
1, 3, 6, 9, 9, 4,
1, 3, 6, 9, 12, 8, 4,
1, 3, 6, 9, 12, 15, 8, 4,
1, 3, 6, 9, 12, 15, 14, 8, 4,
1, 3, 6, 9, 12, 15, 18, 12, 8, 4,
...

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 row n = 1, showing vertices of cylinder of width (or circumference) 2 labeled with distance from base point 0. The cylinder is formed by identifying the black lines. Arrows indicate two points which will coalesce.
N. J. A. Sloane, Illustration for row n = 2, showing vertices of cylinder of width (or circumference) 4 labeled with distance from base point 0. The cylinder is formed by identifying the black lines. Arrows indicate two points which will coalesce.
Index entries for coordination sequences

Rows 1,2,3 are A113311, A329513, A329514.

Cf. A008486, A329501-A329517.

n := 2; \\ set n
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, 12, 15, 14, 12, 12, 12, 12, 12, 12, 12, 12, ... (row n)
f3; // G(n)
// (x^3-2*x^2-1)*(1+x)^2/(x-1)

The g.f.s for the rows could be found using the "trunks and branches" method (see Goodman-Strauss and Sloane), as was done in A329508. This step has not yet been carried out, so the following g.f. is at present only conjectural.
The g.f. G(n) for row n (n>=1) is (strongly) conjectured to be
(1/(1-x))*(1 + 2*x + 3*x^2*(1-x^(2*n-2))/(1-x) - (n-2)*x^(2*n) - (n-1)*x^(2*n+1)).
The values of G(1) through G(6) (certified by MAGMA) are:
(1+x)^2/(1-x),
(x^3-2*x^2-1)*(1+x)^2/(x-1),
(2*x^5-3*x^4+x^3-2*x^2-1)*(1+x)^2/(x-1),
(3*x^7-4*x^6+2*x^5-3*x^4+x^3-2*x^2-1)*(1+x)^2/(x-1),
(4*x^9-5*x^8+3*x^7-4*x^6+2*x^5-3*x^4+x^3-2*x^2-1)*(1+x)^2/(x-1),
(5*x^11-6*x^10+4*x^9-5*x^8+3*x^7-4*x^6+2*x^5-3*x^4+x^3-2*x^2-1)*(1+x)^2/(x-1).
Note that row n is equal to 4*n once the (2*n+1)-st term has been reached.
The g.f.s for the rows can also be obtained 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).

A329502 Expansion of (1+x)(1+2x)/(1-x).

Original entry on oeis.org

1, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

N. J. A. Sloane, Nov 19 2019

Decimal expansion of 11/75. - Elmo R. Oliveira, Aug 09 2024

Index entries for linear recurrences with constant coefficients, signature (1).

Row 3 of array in A329501.

RealDigits[11/75,10,120][[1]] (* or *) PadRight[{1,4},120,{6}] (* Harvey P. Dale, Jul 16 2025 *)

Vec((1 + x)*(1 + 2*x) / (1 - x) + O(x^80)) \\ Colin Barker, Nov 19 2019

a(n) = 6 for n >= 2.

E.g.f.: 6*exp(x) - 2*x - 5. - Elmo R. Oliveira, Aug 09 2024

cp's OEIS Frontend

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.

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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.

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A329515 Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width 2*n+1 hexagons cut from the hexagonal grid by cuts perpendicular to grid lines.

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A329512 Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width 2*n hexagons cut from the hexagonal grid by cuts perpendicular to grid lines.

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A329502 Expansion of (1+x)*(1+2*x)/(1-x).

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A329503 Expansion of g.f. (1+x)*(1+2*x+2*x^2)/(1-x).

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A329502 Expansion of (1+x)(1+2x)/(1-x).

A329503 Expansion of g.f. (1+x)(1+2x+2*x^2)/(1-x).