A250120 -id:A250120 - cp's OEIS frontend

A250126 Coordination sequence of point of type 3.3.4.12 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

1, 4, 9, 9, 12, 19, 21, 28, 27, 31, 38, 40, 48, 44, 49, 56, 57, 67, 63, 69, 73, 75, 85, 80, 88, 92, 95, 102, 98, 106, 109, 114, 121, 118, 123, 127, 132, 138, 137, 142, 147, 149, 156, 155, 159, 166, 168, 176, 172, 177, 184, 185, 195, 191, 197, 201, 203, 213, 208

Offset: 0

N. J. A. Sloane, Nov 29 2014

nonn

This tiling appears as an example in Connelly et al. (2014), Fig. 6 (the heavy black lines in the figures here are an artifact from that figure).

For the definition of k-uniform tiling see Section 2.2 of Chapter 2 of Grünbaum and Shephard (1987).

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987.

Joseph Myers, Table of n, a(n) for n = 0..1000
Robert Connelly, Jeffrey D. Shen, Alexander D. Smith, Ball Packings with Periodic Constraints, arXiv:1301.0664 [math.MG], 2013.
Robert Connelly, Jeffrey D. Shen, Alexander D. Smith, Ball Packings with Periodic Constraints, Discrete Comput. Geom. 52 (2014), no. 4, 754--779. MR3279548.
Brian Galebach, Tiling 132 (in list of 4-uniform tilings).
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
N. J. A. Sloane, A portion of the 3-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}. The four black dots labeled P,Q,R,S show the four types of point. The present sequence is for a point of type Q.
N. J. A. Sloane, Shows layers a(0)-a(6)

Cf. A250123, A250124, A250125.

List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706(3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).

Empirical g.f.: -(2*x^16 +x^14 -2*x^12 -7*x^11 -10*x^10 -10*x^9 -14*x^8 -18*x^7 -17*x^6 -18*x^5 -12*x^4 -9*x^3 -9*x^2 -4*x -1) / ((x -1)^2*(x^4 +x^3 +x^2 +x +1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)). - Colin Barker, Dec 02 2014

Galebach link from Joseph Myers, Nov 30 2014

Extended by Joseph Myers, Dec 02 2014

A299910 Coordination sequence of point of type 3.3.4.3.4 in 3-uniform tiling #3.54 in the Galebach listing.

Original entry on oeis.org

1, 5, 10, 17, 23, 28, 33, 40, 45, 48, 55, 63, 68, 71, 78, 85, 88, 93, 101, 108, 111, 116, 123, 128, 133, 139, 146, 151, 156, 161, 166, 173, 179, 184, 189, 196, 201, 204, 211, 219, 224, 227, 234, 241, 244, 249, 257, 264, 267, 272, 279, 284, 289, 295, 302, 307

Offset: 0

N. J. A. Sloane, Mar 07 2018

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..2500
Brian Galebach, Tiling 3.54
Brian Galebach, Tiling 3.54 [Annotated figure showing the 3 kinds of points mentioned in A299909, A299910, A299911]
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Printable Paper Web Site, Printable 3.3.3.3.3.3,3.3.4.3.4 Tessellation Small [Shows this tiling]
Rémy Sigrist, PARI program for A299910

See A299909, A299911 for the other two kinds of points.

PARI
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\\ See Links section.
```

Conjectures from Chai Wah Wu, Jul 10 2025: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-7) - a(n-8) + a(n-9) - a(n-10) for n > 10.
G.f.: (x^10 + 4*x^9 + 6*x^8 + 11*x^7 + 11*x^6 + 12*x^5 + 11*x^4 + 11*x^3 + 6*x^2 + 4*x + 1)/(x^10 - x^9 + x^8 - x^7 - x^3 + x^2 - x + 1). (End)

More terms from Rémy Sigrist, May 10 2021

A299911 Coordination sequence of point of type 3.3.3.4.4 in 3-uniform tiling #3.54 in the Galebach listing.

Original entry on oeis.org

1, 5, 11, 16, 22, 29, 35, 38, 43, 51, 56, 60, 67, 75, 78, 81, 89, 96, 100, 105, 113, 118, 121, 127, 134, 140, 145, 151, 156, 161, 167, 172, 178, 185, 191, 194, 199, 207, 212, 216, 223, 231, 234, 237, 245, 252, 256, 261, 269, 274, 277, 283, 290, 296, 301, 307

Offset: 0

N. J. A. Sloane, Mar 07 2018

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..2500
Brian Galebach, Tiling 3.54
Brian Galebach, Tiling 3.54 [Annotated figure showing the 3 kinds of points mentioned in A299909, A299910, A299911]
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Printable Paper Web Site, Printable 3.3.3.3.3.3,3.3.4.3.4 Tessellation Small [Shows this tiling]
Rémy Sigrist, PARI program for A299911

See A299909, A299910 for the other two kinds of points.

PARI
```
\\ See Links section.
```

Conjectures from Chai Wah Wu, Jul 10 2025: (Start)
a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-7) - a(n-8) + a(n-9) - a(n-10) for n > 10.
G.f.: (x^2 + x + 1)*(x^8 + 3*x^7 + 3*x^6 + 3*x^5 + 6*x^4 + 3*x^3 + 3*x^2 + 3*x + 1)/(x^10 - x^9 + x^8 - x^7 - x^3 + x^2 - x + 1). (End)

More terms from Rémy Sigrist, May 10 2021

A316316 Coordination sequence for tetravalent node in chamfered version of square grid.

Original entry on oeis.org

1, 4, 8, 8, 12, 20, 20, 20, 28, 32, 32, 36, 40, 44, 48, 48, 52, 60, 60, 60, 68, 72, 72, 76, 80, 84, 88, 88, 92, 100, 100, 100, 108, 112, 112, 116, 120, 124, 128, 128, 132, 140, 140, 140, 148, 152, 152, 156, 160, 164, 168, 168, 172, 180, 180, 180, 188, 192, 192

Offset: 0

N. J. A. Sloane, Jun 29 2018

Rémy Sigrist, Table of n, a(n) for n = 0..5000
Michel Deza and Mikhail Shtogrin, Isometric embedding of mosaics into cubic lattices, Discrete mathematics 244.1-3 (2002): 43-53. See Fig. 2.
Michel Deza and Mikhail Shtogrin, Isometric embedding of mosaics into cubic lattices, Discrete mathematics 244.1-3 (2002): 43-53. [Annotated scan of page 52 only]
Michel Deza and Mikhail Shtogrin, Enlargement of figure from previous link
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 on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.
Rémy Sigrist, PARI program for A316316
Rémy Sigrist, Illustration of first terms
N. J. A. Sloane, Initial terms of coordination sequence for tetravalent node
N. J. A. Sloane, Trunks and branches structure of tetravalent node (First part of proof that a(n+12)=a(n)+40).
N. J. A. Sloane, Calculation of coordination sequence (Second part of proof that a(n+12)=a(n)+40).
N. J. A. Sloane, "Basketweave" tiling by 3X1 rectangles which is equivalent (as far as the graph and coordination sequences are concerned) to this tiling
N. J. A. Sloane, An equivalent tiling seen on the sidewalk of East 70th St in New York City. As far as the graph and coordination sequences are concerned, this is the same as the chamfered square grid. The trivalent vertices labeled b and c are equivalent to each other.
Index entries for linear recurrences with constant coefficients, signature (1,-1,2,-1,1,-1).

See A316317 for trivalent node.
See A250120 for links to thousands of other coordination sequences.
Cf. A316357 (partial sums).
Cf. A056594, A102283.

Join[{1}, LinearRecurrence[{1, -1, 2, -1, 1, -1}, {4, 8, 8, 12, 20, 20}, 100]] (* Jean-François Alcover, Dec 13 2018 *)

PARI
```
See Links section.
```

Apparently, a(n + 12) = a(n) + 40 for any n > 0. - Rémy Sigrist, Jun 30 2018
From N. J. A. Sloane, Jun 30 2018: This conjecture is true.
Theorem: a(n + 12) = a(n) + 40 for any n > 0.
The proof uses the Coloring Book Method described in the Goodman-Strauss - Sloane article. For details see the two links.
From Colin Barker, Dec 13 2018: (Start)
G.f.: (1 + 3*x + 5*x^2 + 2*x^3 + 5*x^4 + 3*x^5 + x^6) / ((1 - x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-1) - a(n-2) + 2*a(n-3) - a(n-4) + a(n-5) - a(n-6) for n>6.
(End)
a(n) = (2/9)*(15*n + 9*A056594(n-1) - 6*A102283(n)) for n > 0. - Stefano Spezia, Jun 12 2021

More terms from Rémy Sigrist, Jun 30 2018

A316317 Coordination sequence for trivalent node in chamfered version of square grid.

Original entry on oeis.org

1, 3, 6, 11, 14, 15, 20, 25, 26, 29, 34, 37, 40, 43, 46, 51, 54, 55, 60, 65, 66, 69, 74, 77, 80, 83, 86, 91, 94, 95, 100, 105, 106, 109, 114, 117, 120, 123, 126, 131, 134, 135, 140, 145, 146, 149, 154, 157, 160, 163, 166, 171, 174, 175, 180, 185, 186, 189, 194

Offset: 0

N. J. A. Sloane, Jun 29 2018

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..5000
Michel Deza and Mikhail Shtogrin, Isometric embedding of mosaics into cubic lattices, Discrete mathematics 244.1-3 (2002): 43-53. See Fig. 2.
Michel Deza and Mikhail Shtogrin, Isometric embedding of mosaics into cubic lattices, Discrete mathematics 244.1-3 (2002): 43-53. [Annotated scan of page 52 only]
Rémy Sigrist, PARI program for A316317
Rémy Sigrist, Illustration of first terms
N. J. A. Sloane, Initial terms of coordination sequence for trivalent node
N. J. A. Sloane, "Basketweave" tiling by 3X1 rectangles which is equivalent (as far as the graph and coordination sequences are concerned) to this tiling

See A316316 for tetravalent node.
See A250120 for links to thousands of other coordination sequences.
Cf. A316358 (partial sums).
Cf. A056594, A102283.

PARI
```
See Links section.
```

Apparently, a(n + 12) = a(n) + 40 for any n > 0. - Rémy Sigrist, Jun 30 2018
This can surely be proved by the Coloring Book Method, although I have not worked out the details. See A316316 for the corresponding proof for a tetravalent node. - N. J. A. Sloane, Jun 30 2018
G.f. (assuming above conjecture): (1+x)^2*(1+3*x^2+x^4)/((1-x)^2*(1+x+x^2)*(1+x^2)). - Robert Israel, Jul 01 2018
a(n) = (30*n - 9*A056594(n-1) + 6*A102283(n))/9 for n > 0. - Conjectured by Stefano Spezia, Jun 12 2021

More terms from Rémy Sigrist, Jun 30 2018

A299909 Coordination sequence of node of type 3^6 in 3-uniform tiling #3.54 in the Galebach listing.

Original entry on oeis.org

1, 6, 12, 18, 24, 24, 30, 42, 48, 48, 54, 66, 66, 66, 78, 90, 90, 90, 102, 108, 108, 114, 126, 132, 132, 138, 144, 150, 156, 162, 168, 174, 180, 180, 186, 198, 204, 204, 210, 222, 222, 222, 234, 246, 246, 246, 258, 264, 264, 270, 282, 288, 288, 294, 300, 306

Offset: 0

N. J. A. Sloane, Mar 07 2018

This tiling has three kinds of nodes. So far the other two types (A299910, A299911) have nor been analyzed.

Colin Barker, Table of n, a(n) for n = 0..1000
Brian Galebach, Tiling 3.54
Brian Galebach, Tiling 3.54 [Annotated figure showing the 3 kinds of points mentioned in A299909, A299910, A299911]
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
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.
Printable Paper Web Site, Printable 3.3.3.3.3.3,3.3.4.3.4 Tessellation Small [Shows this tiling]
N. J. A. Sloane, Labeling of nodes in one sector used to determine the coordination sequence
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,0,0,0,1,-1,1,-1).

See A299910, A299911 for the other two kinds of nodes.

Join[{1}, LinearRecurrence[{1, -1, 1, 0, 0, 0, 1, -1, 1, -1}, {6, 12, 18, 24, 24, 30, 42, 48, 48, 54}, 60]] (* Jean-François Alcover, Jan 09 2019 *)

Vec((x^10+5*x^9+7*x^8+11*x^7+12*x^6+6*x^5+12*x^4+11*x^3+7*x^2+5*x+1) / ((1-x)^2*(1+x^2)*(x^6+x^5+x^4+x^3+x^2+x+1)) + O(x^60)) \\ Colin Barker, Mar 11 2018

G.f.: (x^10+5*x^9+7*x^8+11*x^7+12*x^6+6*x^5+12*x^4+11*x^3+7*x^2+5*x+1) / ((1-x)^2*(1+x^2)*(x^6+x^5+x^4+x^3+x^2+x+1)).
The denominator can also be written as (1-x)*(1+x^2)*(1-x^7).
Recurrence: (-n^2-5*n)*a(n)-n*a(n+1)+
(-n^2-6*n)*a(n+2)-2*n*a(n+3)-2*n*a(n+4)-2*n*a(n+5)-
2*n*a(n+6)+(n^2+3*n)*a(n+7)-n*a(n+8)+(n^2+4*n)*a(n+9) = 0,
with a(0) = 1, a(1) = 6, a(2) = 12, a(3) = 18, a(4) = 24, a(5) = 24, a(6) = 30, a(7) = 42, a(8) = 48, a(9) = 48.
a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-7) - a(n-8) + a(n-9) - a(n-10) for n>10. - Colin Barker, Mar 11 2018
Details of the calculation of the generations function. (Start)
The following lines are written in Maple notation, but should be intelligible as plain text. The colors refer to the labeling of one sector shown in the link.
This analysis did not directly use the "trunks and branches" method described in the Goodman-Strauss & Sloane paper, but was influenced by it.
# The generating function for one of the six sectors:
G:=1+2*x+2*x^2+2*x^3; # green sausages
QG:=G/((1-x^4)*(1-x^7)); # the lattice of green sausages
R:=2+2*x+2*x^2+x^3; # red sausages
QR:=R*(1/(1-x^3))*(x^4/(1-x^4)-x^7/(1-x^7)); # lattice of red sausages
XA:=-x^2/(1-x); # correction for "X-axis"
# red vertical lines of type a
RVLa := x^2/((1-x)*(1-x^4))+x^5*(1/(1-x^3))*(1/(1-x^4)-1/(1-x^7));
# red vertical lines of type b
RVLb:= x^3/((1-x^4)*(1-x^7)) + x^7/((1-x^3)*(1-x^4)) - x^10/((1-x^3)*(1-x^7));
# red vertical lines of type c (twigs to right of vertical sausages)
RVLc:= x^4/((1-x^4)*(1-x^7)) + x^8/((1-x^3)*(1-x^4)) - x^11/((1-x^3)*(1-x^7));
# Total for one sector
T:=QG+QR+XA+RVLa+RVLb+RVLc;
# Grand total, after correcting for overcounting where sectors meet:
U:=6*T-5-6*x;
series(U,x,30);
# After simplification, grand total is:
(x^10+5*x^9+7*x^8+11*x^7+12*x^6+6*x^5+12*x^4+11*x^3+7*x^2+5*x+1) / ((1-x)^2*(1+x^2)*(x^6+x^5+x^4+x^3+x^2+x+1));
(End) (These details added by N. J. A. Sloane, Apr 10 2018)

A350963 Coordination sequence for the XXOXX tiling with respect to a tile of type R.

Original entry on oeis.org

1, 9, 29, 42, 63, 75, 97, 106, 131, 139, 165, 170, 199, 203, 233, 234, 267, 267, 301, 298, 335, 331, 369, 362, 403, 395, 437, 426, 471, 459, 505, 490, 539, 523, 573, 554, 607, 587, 641, 618, 675, 651, 709, 682, 743, 715, 777, 746, 811, 779, 845, 810, 879, 843

Offset: 0

N. J. A. Sloane, Feb 25 2022

nonn

The tile consists of five squares in a row with the center square missing. The tiling of the plane studied here is that shown in Fig. 2 of Gruslys et al., and in the other illustrations below. (An email from Neil Bickford suggests that this may be the unique tiling of the plane using this tile.)
There are three orbits on tiles, which we denote by R, G, and B, corresponding to the colors in the illustrations. The tiles are drawn as two dominoes linked by a thin bridge. The symmetry group appears to be p2.
For the coordination sequence we regard two tiles as adjacent if they share a (long or short) edge. The two halves of a tile are marked with its generation number. The base (R) tile is marked with two 0's and is enclosed in a black border. The 9 generation 1 tiles are marked with two 1's and two black stars.
Each R tile touches 4 R's, 4 G's, and 1 B; each G tile touches 4 R's, 1 G, and 3 B's; each B tile touches 1 R, 3 G's, and 4 B's.

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 on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.
Vytautas Gruslys, Imre Leader, and Ta Sheng Tan, Tiling with arbitrary tiles, arXiv:1505.03697 [math.CO], 2015-2016. A small piece of this tiling is shown in Fig. 2.
Rémy Sigrist, Illustration of initial terms
Rémy Sigrist, Illustration of the configuration for n = 0..100 with cyclic colors
Rémy Sigrist, PARI program
N. J. A. Sloane, (Top) A large portion of the tiling. (Bottom) The start of the coordination sequence with respect to a tile of type R.
N. J. A. Sloane, A large portion of the tiling in higher resolution
Index entries for coordination sequences

Cf. A351966 (type G), A351967 (type B).

See A250120 for other examples of coordination sequences.

PARI
```
See Links section.
```

Conjectured g.f.: -(2*t^7-6*t^6-24*t^5-33*t^4-33*t^3-28*t^2-9*t-1)/((1-t^2)*(1-t^4)). Given the decomposition of this structure into eight sectors (see Sigrist's illustration of the first 100 generations), it should be possible to establish this g.f. and those of the other two coordination sequences by using the coloring book method. - N. J. A. Sloane, Feb 26 2022

More terms from Rémy Sigrist, Feb 26 2022

A315431 Coordination sequence Gal.6.669.6 where Gal.u.t.v denotes the coordination sequence for a vertex of type v in tiling number t in the Galebach list of u-uniform tilings.

Original entry on oeis.org

1, 6, 11, 15, 22, 28, 33, 39, 44, 52, 56, 61, 64, 74, 78, 83, 88, 98, 100, 105, 110, 120, 120, 129, 132, 142, 142, 153, 154, 164, 166, 175, 174, 186, 188, 197, 196, 212, 210, 219, 218, 234, 230, 241, 242, 256, 252, 265, 264, 278

Offset: 0

Brian Galebach and N. J. A. Sloane, Jun 18 2018

nonn

Note that there may be other vertices in the Galebach list of u-uniform tilings with u <= 6 that have this same coordination sequence. See the Galebach link for the complete list of A-numbers for all these tilings.

Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers

From Chai Wah Wu, Dec 20 2019: (Start)
a(n) = a(n-4) - a(n-5) + a(n-7) + a(n-9) - a(n-11) + a(n-12) - a(n-16) for n > 22 (conjectured).
G.f.: (-x^22 - x^17 + x^16 + 8*x^15 + 11*x^14 + 16*x^13 + 21*x^12 + 23*x^11 + 30*x^10 + 34*x^9 + 31*x^8 + 34*x^7 + 28*x^6 + 23*x^5 + 21*x^4 + 15*x^3 + 11*x^2 + 6*x + 1)/(x^16 - x^12 + x^11 - x^9 - x^7 + x^5 - x^4 + 1) (conjectured). (End)

A315440 Coordination sequence Gal.6.491.5 where Gal.u.t.v denotes the coordination sequence for a vertex of type v in tiling number t in the Galebach list of u-uniform tilings.

Original entry on oeis.org

1, 6, 11, 16, 19, 26, 32, 36, 41, 46, 47, 56, 64, 68, 75, 76, 79, 86, 92, 98, 107, 106, 109, 118, 122, 130, 137, 140, 141, 146, 152, 160, 165, 170, 175, 178, 184, 190, 197, 200, 203, 210, 216, 218, 227, 232, 233, 242, 248, 252

Offset: 0

Brian Galebach and N. J. A. Sloane, Jun 18 2018

nonn

Note that there may be other vertices in the Galebach list of u-uniform tilings with u <= 6 that have this same coordination sequence. See the Galebach link for the complete list of A-numbers for all these tilings.

Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers

A315442 Coordination sequence Gal.6.428.5 where Gal.u.t.v denotes the coordination sequence for a vertex of type v in tiling number t in the Galebach list of u-uniform tilings.

Original entry on oeis.org

1, 6, 11, 16, 20, 22, 30, 33, 40, 46, 54, 55, 62, 64, 66, 74, 75, 86, 92, 100, 99, 110, 108, 110, 120, 119, 130, 136, 146, 143, 156, 152, 156, 166, 163, 176, 182, 190, 185, 202, 196, 200, 212, 209, 222, 228, 236, 229, 246, 238

Offset: 0

Brian Galebach and N. J. A. Sloane, Jun 18 2018

nonn

Note that there may be other vertices in the Galebach list of u-uniform tilings with u <= 6 that have this same coordination sequence. See the Galebach link for the complete list of A-numbers for all these tilings.

Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers

cp's OEIS Frontend

A250126 Coordination sequence of point of type 3.3.4.12 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

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A299910 Coordination sequence of point of type 3.3.4.3.4 in 3-uniform tiling #3.54 in the Galebach listing.

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A299911 Coordination sequence of point of type 3.3.3.4.4 in 3-uniform tiling #3.54 in the Galebach listing.

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A316316 Coordination sequence for tetravalent node in chamfered version of square grid.

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A316317 Coordination sequence for trivalent node in chamfered version of square grid.

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A299909 Coordination sequence of node of type 3^6 in 3-uniform tiling #3.54 in the Galebach listing.

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Mathematica

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A350963 Coordination sequence for the XXOXX tiling with respect to a tile of type R.

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A315431 Coordination sequence Gal.6.669.6 where Gal.u.t.v denotes the coordination sequence for a vertex of type v in tiling number t in the Galebach list of u-uniform tilings.

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