A329508 - OEIS

%I #52 Sep 08 2022 08:46:24

%S 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,

%T 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,

%U 4,2,1,3,6,9,12,15,17,13,8,6,4,2

%N 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.

%C This is the structure of carbon nanotubes.

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

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

%H Chaim Goodman-Strauss and N. J. A. Sloane, <a href="https://doi.org/10.1107/S2053273318014481">A Coloring Book Approach to Finding Coordination Sequences</a>, Acta Cryst. A75 (2019), 121-134, also <a href="http://NeilSloane.com/doc/Cairo_final.pdf">on NJAS's home page</a>. Also <a href="http://arxiv.org/abs/1803.08530">arXiv:1803.08530</a>.

%H N. J. A. Sloane, <a href="/A329508/a329508.pdf">Illustration for rows 1 through 3</a>, 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.

%H N. J. A. Sloane, <a href="/A329508/a329508_1.pdf">Illustration for row 4</a>, 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.

%H N. J. A. Sloane, <a href="/A329508/a329508_2.pdf">Illustration for row 5</a>, 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.

%H <a href="/index/Con#coordination_sequences">Index entries for coordination sequences</a>

%F 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.

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

%F (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)).

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

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

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

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

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

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

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

%F (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),

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

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

%F 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).

%e Array begins:

%e 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...

%e 1, 3, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, ...

%e 1, 3, 6, 8, 8, 7, 6, 6, 6, 6, 6, 6, ...

%e 1, 3, 6, 9, 11, 11, 10, 9, 8, 8, 8, 8, ...

%e 1, 3, 6, 9, 12, 14, 14, 13, 12, 11, 10, 10, ...

%e 1, 3, 6, 9, 12, 15, 17, 17, 16, 15, 14, 13, ...

%e 1, 3, 6, 9, 12, 15, 18, 20, 20, 19, 18, 17, ...

%e 1, 3, 6, 9, 12, 15, 18, 21, 23, 23, 22, 21, ...

%e 1, 3, 6, 9, 12, 15, 18, 21, 24, 26, 26, 25, ...

%e 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 29, 29, ...

%e The initial antidiagonals are:

%e 1

%e 1,2

%e 1,3,2

%e 1,3,5,2

%e 1,3,6,5,2

%e 1,3,6,8,4,2

%e 1,3,6,9,8,4,2

%e 1,3,6,9,11,7,4,2

%e 1,3,6,9,12,11,6,4,2

%e 1,3,6,9,12,14,10,6,4,2

%e ...

%o (Magma)

%o c := 4; \\ set c

%o R<x> := RationalFunctionField(Integers());

%o FG3<R, S, T> := FreeGroup(3);

%o Q3 := quo<FG3| R^2, S^2, T^2, R*S*T = T*S*R, (R*T)^c >;

%o H := AutomaticGroup(Q3);

%o f3 := GrowthFunction(H);

%o PSR := PowerSeriesRing(Integers():Precision := 60);

%o Coefficients(PSR!f3);

%o // 1, 3, 6, 9, 11, 11, 10, 9, 8, 8, 8, 8, 8, 8, 8, ... (row c)

%o f3;  // g.f. for row c

%o // (x^8 + x^7 + x^6 - 2*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1)

%o // = (1+x)*(x^3-x-1)*(x^2+1)^2/(x-1)

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

%Y Cf. A008486, A329501-A329517.

%K nonn,tabl,easy

%O 1,3

%A _N. J. A. Sloane_, Nov 22 2019