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A329508
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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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8
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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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.
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FORMULA
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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).
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EXAMPLE
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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
...
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PROG
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(Magma)
c := 4; \\ set c
R<x> := RationalFunctionField(Integers());
FG3<R, S, T> := FreeGroup(3);
Q3 := quo<FG3| R^2, S^2, T^2, R*S*T = T*S*R, (R*T)^c >;
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)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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