A319324 -id:A319324

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A319325

a(n) is the number of row convex polyglasses (polyiamonds which need only touch at corners) with n cells.

+10
2

2, 10, 52, 276, 1470, 7838, 41798, 222902, 1188696, 6339088, 33805118, 180276062, 961376842, 5126833922, 27340398612, 145800977348, 777527983398, 4146404063814, 22111958704510, 117918733974142, 628837454333128, 3353466668484240, 17883379272566534, 95368550166928198

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..24.

Adam Gruber, Java program to count lattice animals.

FORMULA

G.f.: (2*z*(1+z)*(1-z)^3)/(1-7z+9z^2+z^3-9z^4+3z^5).

EXAMPLE

The only diglasses that are not row convex are "two mountains" and its 180-degree rotation. So a(2) = A319324(2) - 2 = 10.

MATHEMATICA

Rest@CoefficientList[Series[(2z(1+z)(1-z)^3)/(1-7z+9z^2+z^3-9z^4+3z^5), {z, 0, 20}], z]

CROSSREFS

Cf. A319324 (fixed polyglasses), A319326 (column convex polyglasses), A238823 (row convex polyiamonds).

KEYWORD

nonn

AUTHOR

David Bevan, Sep 27 2018

STATUS

approved

A319326

a(n) is the number of column convex polyglasses (polyiamonds which need only touch at corners) with n cells.

+10
2

2, 12, 84, 612, 4532, 33762, 252106, 1884120, 14084674, 105295512, 787178752

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A polyglass is column convex if the intersection of its interior with any vertical line through the centers of the cells is connected.

LINKS

Table of n, a(n) for n=1..11.

Adam Gruber, Java program to count lattice animals.

EXAMPLE

There are four triglasses that are not column convex. So a(2) = A319324(3) - 4 = 84.

CROSSREFS

Cf. A319324 (fixed polyglasses), A319325 (row convex polyglasses), A319323 (column convex polyiamonds).

KEYWORD

nonn,more

AUTHOR

David Bevan, Sep 27 2018

STATUS

approved

A340341

Number of polymings with n cells, distinguishing mirror images.

+10
0

1, 3, 16, 129, 1009, 8997, 80816, 746483, 6983847, 66146105, 632186200, 6089173570

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A polyming is a generalized polyiamond whose cells may be joined at corners as well as at edges. I introduced the term in 2010. In A319324 and elsewhere, David Bevan calls these shapes "polyglasses." In A239658, Abe Wits and Ragnar Groot Koerkamp call them simply "triangular polyplets."

LINKS

Table of n, a(n) for n=1..12.

G. L. Sicherman, Catalogue of Polymings

EXAMPLE

a(3)=16, because there are 11 two-sided 3-mings (identifying mirror images), and 5 of them are chiral. See the link above.

CROSSREFS

Cf. A239658, A319324.

KEYWORD

nonn,hard,more

AUTHOR

George Sicherman, Jan 04 2021

EXTENSIONS

a(11) and a(12) from Aaron N. Siegel, May 22 2022

STATUS

approved

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