A072154 -id:A072154 - cp's OEIS frontend

A250123 Coordination sequence of point of type 3.3.4.3.4 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

1, 5, 8, 8, 11, 17, 25, 27, 24, 30, 38, 46, 47, 44, 46, 50, 64, 68, 65, 66, 70, 80, 80, 83, 87, 91, 100, 100, 99, 99, 109, 121, 121, 119, 119, 125, 133, 139, 140, 140, 145, 153, 155, 152, 158, 166, 174, 175, 172, 174, 178, 192, 196, 193, 194, 198, 208, 208, 211

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 P.
N. J. A. Sloane, Shows layers a(0)-a(6)

Cf. A250124, A250125, A250126.

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.: -(x+1)*(x^15 +3*x^14 -4*x^11 -6*x^10 -7*x^9 -4*x^8 -7*x^7 -11*x^6 -9*x^5 -7*x^4 -4*x^3 -4*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

A250124 Coordination sequence of point of type 3.3.12.4 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

Original entry on oeis.org

1, 4, 7, 10, 15, 16, 21, 29, 28, 34, 33, 40, 48, 45, 53, 51, 59, 65, 64, 72, 68, 78, 83, 83, 89, 87, 97, 100, 102, 107, 106, 114, 119, 121, 124, 125, 132, 138, 138, 143, 144, 149, 157, 156, 162, 161, 168, 176, 173, 181, 179, 187, 193, 192, 200, 196, 206, 211, 211

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 R.
N. J. A. Sloane, Shows layers a(0)-a(6)

Cf. A250123, A250125, A250126.

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.: -(3*x^14 -4*x^12 -4*x^11 -7*x^10 -12*x^9 -14*x^8 -21*x^7 -17*x^6 -15*x^5 -15*x^4 -10*x^3 -7*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

A250125 Coordination sequence of point of type 3.4.3.12 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

Original entry on oeis.org

1, 4, 6, 11, 13, 15, 23, 23, 33, 30, 33, 42, 41, 54, 46, 54, 58, 58, 73, 64, 75, 74, 79, 89, 81, 94, 92, 100, 105, 102, 110, 109, 119, 123, 123, 126, 130, 135, 140, 142, 144, 151, 151, 161, 158, 161, 170, 169, 182, 174, 182, 186, 186, 201, 192, 203, 202, 207, 217

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 S.
N. J. A. Sloane, Shows layers a(0)-a(6)

Cf. A250123, A250124, A250126.

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.: -(x^17 +x^16 +x^15 +x^14 -2*x^13 -4*x^12 -6*x^11 -7*x^10 -11*x^9 -18*x^8 -16*x^7 -19*x^6 -14*x^5 -13*x^4 -11*x^3 -6*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

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}.

Original entry on oeis.org

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

A072151 Coordination sequence for AlPO_4-11 structure with respect to node (Z) where decagon, hexagon and square meet and is not adjacent to node of type (X).

Original entry on oeis.org

1, 3, 5, 7, 11, 14, 15, 18, 22, 24, 26, 28, 31, 36, 37, 37, 43, 47, 46, 49, 53, 55, 59, 60, 61, 68, 70, 68, 74, 78, 77, 82, 85, 85, 91, 93, 92, 99, 101, 99, 107, 110, 107, 114, 118, 116, 122, 124, 123, 132, 133, 129, 139, 143, 138, 145, 149, 147, 155, 156

Offset: 0

N. J. A. Sloane, Jun 28 2002

There are three types of nodes in this structure.

The coordination sequence with respect to a particular node gives the number of nodes that can be reached from that node in n steps along edges.

Joseph Myers, Table of n, a(n) for n = 0..1000
M. E. Davis, Ordered porous materials for emerging applications, Nature, 417 (Jun 20 2002), 813-821 (gives structure).
R. J. Mathar, Illustration of counts and adjacencies (PostScript)
N. J. A. Sloane, AlPO_4-11 structure, after Davis

Cf. A072149-A072154.

Empirical g.f.: 1 +x *(3 +5*x +10*x^2 +13*x^3 +16*x^4 +16*x^5 +16*x^6 +16*x^7 +16*x^8 +13*x^9 +10*x^10 +5*x^11 +3*x^12) / ( (1+x^2) *(x^6+x^3+1) *(x-1)^2 *(1+x+x^2)^2 ). - R. J. Mathar, Sep 30 2011

More terms from R. J. Mathar, Mar 29 2007

Extended by Joseph Myers, Sep 29 2011

A072152 Coordination sequence for VPl-5 structure with respect to node (X) where 18-gon and two squares meet.

Original entry on oeis.org

1, 3, 4, 6, 8, 8, 10, 14, 16, 17, 20, 23, 22, 22, 28, 32, 30, 31, 36, 37, 36, 40, 46, 46, 44, 47, 50, 51, 54, 58, 60, 60, 60, 61, 64, 69, 72, 72, 74, 76, 74, 75, 82, 87, 86, 86, 90, 90, 88, 93, 100, 101, 100, 102, 104, 104, 106, 111, 114, 115, 116, 116, 118, 122

Offset: 0

N. J. A. Sloane, Jun 28 2002

There are two types of nodes in this structure.

The coordination sequence with respect to a particular node gives the number of nodes that can be reached from that node in n steps along edges.

Joseph Myers, Table of n, a(n) for n = 0..1000
M. E. Davis, Ordered porous materials for emerging applications, Nature, 417 (Jun 20 2002), 813-821 (gives structure).
N. J. A. Sloane, VPl-5 structure, after Davis

Cf. A072149-A072154.

Empirical G.f.: -(x^17+x^16-x^15-2*x^14-2*x^13-2*x^12-4*x^11-5*x^10-5*x^9-5*x^8-7*x^7-6*x^6-4*x^5-4*x^4-4*x^3-2*x^2-2*x-1)/((x-1)^2*(x^2+1)*(x^4+1)*(x^6+x^5+x^4+x^3+x^2+x+1)). [Colin Barker, Nov 18 2012]

Extended by Joseph Myers, Sep 30 2011

A072153 Coordination sequence for VPl-5 structure with respect to node (Y) where 18-gon, hexagon and square meet.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 13, 16, 19, 20, 20, 22, 25, 27, 29, 32, 34, 34, 35, 38, 41, 43, 45, 47, 48, 49, 51, 54, 57, 59, 60, 61, 63, 65, 67, 70, 73, 74, 74, 76, 79, 81, 83, 86, 88, 88, 89, 92, 95, 97, 99, 101, 102, 103, 105, 108, 111, 113, 114, 115, 117, 119, 121, 124

Offset: 0

N. J. A. Sloane, Jun 28 2002

There are two types of nodes in this structure.

The coordination sequence with respect to a particular node gives the number of nodes that can be reached from that node in n steps along edges.

Joseph Myers, Table of n, a(n) for n = 0..1000
M. E. Davis, Ordered porous materials for emerging applications, Nature, 417 (Jun 20 2002), 813-821 (gives structure).
N. J. A. Sloane, VPl-5 structure, after Davis

Cf. A072149, A072150, A072151, A072152, A072154.

Empirical G.f.: (x^2+x+1)^2*(x^6+x^3+1)/((x-1)^2*(x^2+1)*(x^6+x^5+x^4+x^3+x^2+x+1)). [Colin Barker, Nov 18 2012]

Extended by Joseph Myers, Sep 30 2011

A242941 a(n) is the number of convex uniform tessellations in dimension n.

Original entry on oeis.org

1, 11, 28, 143

Offset: 1

Felix Fröhlich, May 27 2014

Terms for n > 4 have not been determined so far. Alfredo Andreini in 1905 gave a value of 25 for a(3), later found to be incorrect. The value 28 for a(3) was given by Norman Johnson in 1991 and later in 1994 independently by Branko Grünbaum. The value for a(4) was given by George Olshevsky in 2006.
Deza and Shtogrin (2000) agree that the value of a(3) is 28, although the authors do not provide a proof. - Felix Fröhlich, Nov 29 2014
From Felix Fröhlich, Feb 03 2019: (Start)
The 11 convex uniform tilings are all illustrated in Kepler, 1619. For an argument that exactly 11 such tilings exist, see Grünbaum, Shephard, 1977.
In dimension 2, the definition of "uniform polytope" usually seems to be equivalent to the regular polygons in order to exclude polygons that alternate two different edge-lengths. Applying this principle retroactively to dimension 1 (as done, as I assume, by Coxeter, see Coxeter, 1973, p. 129) yields a(1) = 1. (End)

H. S. M. Coxeter, Regular Polytopes, Third Edition, Dover Publications, 1973, ISBN 9780486614809.
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, Vol. 4, No. 2 (1994), 49-56.
N. W. Johnson, Uniform Polytopes, [To appear, cf. Weiss, Stehle, 2017].

A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle correspondenti reto correlative [On the regular and semiregular nets of polyhedra and on the corresponding correlative nets], Mem. Società Italiana della Scienze, Ser.3, 14 (1905), 75-129.
M. Deza and M. Shtogrin, Uniform Partitions of 3-space, their Relatives and Embedding, European Journal of Combinatorics, Vol. 21, No. 6 (2000), 807-814.
B. Grünbaum and G. C. Shephard, Tilings by Regular Polygons, Mathematics Magazine, Vol. 50, No. 5 (1977), 227-247.
J. Kepler, Harmonices Mundi [The Harmony of the World] (1619).
G. Olshevsky, Uniform Panoploid Tetracombs (2006)
A. I. Weiss and E. M. Stehle, Norman W. Johnson (12 November 1930 to 13 July 2017), The Art of Discrete and Applied Mathematics, Vol. 1, No. 1 (2018).
Wikipedia, Convex uniform honeycomb
Wikipedia, List of convex uniform tilings

Cf. A068599.
List of coordination sequences for the 11 uniform 2D tilings: 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).
List of coordination sequences for the 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e:  A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.

Edited by N. J. A. Sloane, Feb 15 2018

Edited by Felix Fröhlich, Feb 03-10 2019

A301730 Expansion of (x^8-x^7+x^6+5x^5+4x^4+3x^3+5x^2+5*x+1)/(x^6-x^5-x+1).

Original entry on oeis.org

1, 6, 11, 14, 18, 24, 30, 34, 38, 42, 48, 54, 58, 62, 66, 72, 78, 82, 86, 90, 96, 102, 106, 110, 114, 120, 126, 130, 134, 138, 144, 150, 154, 158, 162, 168, 174, 178, 182, 186, 192, 198, 202, 206, 210, 216, 222, 226, 230, 234, 240, 246, 250, 254, 258, 264

Offset: 0

N. J. A. Sloane, Mar 30 2018

Growth series for group with presentation < X, Y, Z | X^2 = Y^2, X^2 = Z^2, X^2 = (Y*Z)^3, X^2 = (Z*X)^2, X^2 = (X*Y)^6 >. Probably Shutov intended to add "X^2 = Id" to the presentation, which would have produced the sequence A072154.

A. V. Shutov, On the number of words of a given length in plane crystallographic groups (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 302 (2003), Anal. Teor. Chisel i Teor. Funkts. 19, 188--197, 203. See Table 1, line "p6m".
A. V. Shutov, On the number of words of a given length in plane crystallographic groups (English translation), J. Math. Sci. (N.Y.) 129 (2005), no. 3, 3922-3926 [MR2023041]. See Table 1, line "p6m".
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A072154.

R := RationalFunctionField(Integers());
FG3 := FreeGroup(3);
Q3 := quo;
G3 := AutomaticGroup(Q3);
f3 := GrowthFunction(G3);
R!f3;
PSR := PowerSeriesRing(Integers():Precision := 60);
Coefficients(PSR!f3);

From Bruno Berselli, Apr 09 2018: (Start)
G.f.: (x + 1)*(x^7 - 2*x^6 + 3*x^5 + 2*x^4 + 2*x^3 + x^2 + 4*x + 1)/((x - 1)^2*(x^4 + x^3 + x^2 + x + 1)).
a(5*k)   = 24*k with k>0, a(0)=1;
a(5*k+1) = 24*k + 6;
a(5*k+2) = 24*k + 10 with k>0, a(2)=11;
a(5*k+3) = 24*k + 14;
a(5*k+4) = 24*k + 18. (End)

cp's OEIS Frontend

A250123 Coordination sequence of point of type 3.3.4.3.4 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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A250124 Coordination sequence of point of type 3.3.12.4 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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A250125 Coordination sequence of point of type 3.4.3.12 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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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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A072151 Coordination sequence for AlPO_4-11 structure with respect to node (Z) where decagon, hexagon and square meet and is not adjacent to node of type (X).

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A072152 Coordination sequence for VPl-5 structure with respect to node (X) where 18-gon and two squares meet.

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A072153 Coordination sequence for VPl-5 structure with respect to node (Y) where 18-gon, hexagon and square meet.

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A242941 a(n) is the number of convex uniform tessellations in dimension n.

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A301730 Expansion of (x^8-x^7+x^6+5*x^5+4*x^4+3*x^3+5*x^2+5*x+1)/(x^6-x^5-x+1).

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A301730 Expansion of (x^8-x^7+x^6+5x^5+4x^4+3x^3+5x^2+5*x+1)/(x^6-x^5-x+1).