A250120 -id:A250120 - cp's OEIS frontend

A315462 Coordination sequence Gal.6.339.6 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.

1, 6, 11, 16, 21, 25, 30, 35, 39, 44, 49, 54, 60, 66, 71, 76, 81, 85, 90, 95, 99, 104, 109, 114, 120, 126, 131, 136, 141, 145, 150, 155, 159, 164, 169, 174, 180, 186, 191, 196, 201, 205, 210, 215, 219, 224, 229, 234, 240, 246

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) = 2*a(n-1) - a(n-2) - a(n-4) + 2*a(n-5) - a(n-6) - a(n-8) + 2*a(n-9) - a(n-10) for n > 10 (conjectured).
G.f.: (x^10 + 4*x^9 + x^6 + 3*x^5 + x^4 + 4*x + 1)/(x^8*(x - 1)^2 + x^4*(x - 1)^2 + (x - 1)^2) (conjectured). (End)

A315464 Coordination sequence Gal.6.346.5 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, 21, 25, 31, 35, 40, 45, 50, 56, 62, 67, 72, 77, 81, 87, 91, 96, 101, 106, 112, 118, 123, 128, 133, 137, 143, 147, 152, 157, 162, 168, 174, 179, 184, 189, 193, 199, 203, 208, 213, 218, 224, 230, 235, 240, 245, 249

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-1) + a(n-11) - a(n-12) for n > 12 (conjectured).
G.f.: (x^12 + 5*x^11 + 5*x^10 + 5*x^9 + 5*x^8 + 4*x^7 + 6*x^6 + 4*x^5 + 5*x^4 + 5*x^3 + 5*x^2 + 5*x + 1)/(x^12 - x^11 - x + 1) (conjectured). (End)

A316319 Coordination sequence for a trivalent node in a chamfered version of the 3^6 triangular tiling of the plane.

Original entry on oeis.org

1, 3, 7, 14, 25, 38, 51, 63, 75, 87, 99, 111, 123, 135, 147, 159, 171, 183, 195, 207, 219, 231, 243, 255, 267, 279, 291, 303, 315, 327, 339, 351, 363, 375, 387, 399, 411, 423, 435, 447, 459, 471, 483, 495, 507, 519, 531, 543, 555, 567, 579, 591, 603, 615, 627, 639

Offset: 0

Rémy Sigrist and N. J. A. Sloane, Jul 01 2018

Let E denote the lattice of Eisenstein integers u + v*w in the plane, with each point joined to its six neighbors. Here u and v are ordinary integers and w = (-1+sqrt(-3))/2 is a complex cube root of unity. Let theta = w - w^2 = sqrt(-3). Then theta*E is a sublattice of E of index 3 (Conway-Sloane, Fig. 7.2). The tiling considered in this sequence is obtained by replacing each node in theta*E by a small hexagon.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., 1993. See Fig. 7.2, page 199.

Colin Barker, Table of n, a(n) for n = 0..1000
Rémy Sigrist, Illustration of initial terms
N. J. A. Sloane, The graph of the tiling. (The red dots indicate the nodes of the sublattice theta*E.)
Index entries for linear recurrences with constant coefficients, signature (2,-1).

See A316320 for hexavalent node.
See A250120 for links to thousands of other coordination sequences.
Cf. A017557.

Vec((1 + x + x^2)*(1 + x^2 + 2*x^3 + x^4 - x^5) / (1 - x)^2 + O(x^50)) \\ Colin Barker, Mar 11 2020

a(n) = 12*n-21 = A017557(n-2) for n > 5.
From Colin Barker, Mar 11 2020: (Start)
G.f.: (1 + x + x^2)*(1 + x^2 + 2*x^3 + x^4 - x^5) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>7.
(End)

Terms a(16) and beyond from Andrey Zabolotskiy, Sep 30 2019

A316320 Coordination sequence for a hexavalent node in a chamfered version of the 3^6 triangular tiling of the plane.

Original entry on oeis.org

1, 6, 15, 27, 39, 51, 63, 75, 87, 99, 111, 123, 135, 147, 159, 171, 183, 195, 207, 219, 231, 243, 255, 267, 279, 291, 303, 315, 327, 339, 351, 363, 375, 387, 399, 411, 423, 435, 447, 459, 471, 483, 495, 507, 519, 531, 543, 555, 567, 579, 591, 603, 615, 627, 639

Offset: 0

Rémy Sigrist and N. J. A. Sloane, Jul 01 2018

Let E denote the lattice of Eisenstein integers u + v*w in the plane, with each point joined to its six neighbors. Here u and v are ordinary integers and w = (-1+sqrt(-3))/2 is a complex cube root of unity. Let theta = w - w^2 = sqrt(-3). Then theta*E is a sublattice of E of index 3 (Conway-Sloane, Fig. 7.2). The tiling considered in this sequence is obtained by replacing each node in theta*E by a small hexagon.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., 1993. See Fig. 7.2, page 199.

Colin Barker, Table of n, a(n) for n = 0..1000
Rémy Sigrist, Illustration of initial terms
N. J. A. Sloane, The graph of the tiling. (The red dots indicate the nodes of the sublattice theta*E.)
Index entries for linear recurrences with constant coefficients, signature (2,-1).

See A316319 for trivalent node.
See A250120 for links to thousands of other coordination sequences.
Cf. A017557, A008486.

Vec((1 + 3*x)*(1 + x + x^2) / (1 - x)^2 + O(x^50)) \\ Colin Barker, Mar 11 2020

a(n) = 12*n-9 = A017557(n-1) for n > 1.
From Colin Barker, Mar 11 2020: (Start)
G.f.: (1 + 3*x)*(1 + x + x^2) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>3.
(End)

Terms a(15) and beyond from Andrey Zabolotskiy, Sep 30 2019

A342285 Coordination sequence with respect to the central vertex of a dodecagon-based tiling of the plane by copies of a certain Goldberg quadrilateral tile.

Original entry on oeis.org

1, 6, 6, 18, 30, 36, 42, 54, 60, 66, 72, 78, 84, 90, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 330, 336, 342, 348, 354, 360

Offset: 0

N. J. A. Sloane, Mar 23 2021

nonn

There are many ways to tile the plane with the Goldberg tile; this is a particularly symmetric one.

In Cye Waldman's drawing, six copies of the gray sector are placed at the degree-4 vertices of the decagon, and 6 copies of a similar sector at the degree-6 vertices of the decagon.

Goldberg, M. (1955). “Central Tessellations,” Scripta Mathematica, 21, pp. 253-260. See Fig.7b.
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987; Fig. 1.3.6(a), page 30.

Rémy Sigrist, Table of n, a(n) for n = 0..2500
Rémy Sigrist, Illustration of initial terms
Rémy Sigrist, PARI program for A342285
N. J. A. Sloane, Illustration of initial terms (For example, the a(3) = 18 vertices at 3 steps from the center are colored blue; the a(4) = 30 vertices at 4 steps from the center are green with a black circle.)
Cye Waldman, The Goldberg Tile
Cye Waldman, Central portion of tiling.
Cye Waldman and others, Numerous postings to the Google Groups Tiling Mailing List about tilings with this Goldberg quadrilateral tile, March 2021.

Cf. A138591, A250120.

PARI
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See Links section.
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Apparently, a(n) = 6*A138591(n-1) for n > 1. - Rémy Sigrist, Mar 30 2021

More terms from Rémy Sigrist, Mar 29 2021

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

A250121 Crystal ball sequence for planar net 3.3.3.3.6.

Original entry on oeis.org

1, 6, 15, 30, 49, 73, 102, 135, 174, 217, 265, 318, 375, 438, 505, 577, 654, 735, 822, 913, 1009, 1110, 1215, 1326, 1441, 1561, 1686, 1815, 1950, 2089, 2233, 2382, 2535, 2694, 2857, 3025, 3198, 3375, 3558, 3745, 3937, 4134

Offset: 0

Bradley Klee and N. J. A. Sloane, Nov 23 2014

The g.f. was proven; cf. the comment in A250120. - Georg Fischer, Jul 19 2020

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Partial sums of A250120.

CoefficientList[Series[(x^2+x+1)*(x^4+3*x^3+3*x+1)/((x^4+x^3+x^2+x+1)*(1-x)^3),{x,0,41}],x] (* Georg Fischer, Jul 19 2020 *)
LinearRecurrence[{2,-1,0,0,1,-2,1},{1,6,15,30,49,73,102},50] (* Harvey P. Dale, Apr 30 2022 *)

G.f.: (x^2+x+1)*(x^4+3*x^3+3*x+1)/((x^4+x^3+x^2+x+1)*(1-x)^3).

A310387 Coordination sequence Gal.6.588.6 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, 4, 10, 14, 18, 26, 30, 34, 38, 42, 54, 50, 58, 62, 74, 70, 78, 78, 98, 86, 98, 98, 118, 106, 118, 114, 142, 122, 138, 134, 162, 142, 158, 150, 186, 158, 178, 170, 206, 178, 198, 186, 230, 194, 218, 206, 250, 214, 238, 222

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

A311019 Coordination sequence Gal.6.492.2 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, 4, 8, 11, 12, 16, 24, 29, 34, 30, 32, 43, 50, 52, 58, 53, 56, 58, 70, 81, 82, 78, 76, 83, 96, 96, 104, 105, 98, 108, 114, 123, 132, 118, 122, 131, 138, 150, 150, 147, 148, 142, 162, 171, 176, 176, 164, 173, 184, 184

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

A315211 Coordination sequence Gal.3.20.3 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, 10, 14, 20, 26, 30, 34, 40, 46, 50, 54, 60, 66, 70, 74, 80, 86, 90, 94, 100, 106, 110, 114, 120, 126, 130, 134, 140, 146, 150, 154, 160, 166, 170, 174, 180, 186, 190, 194, 200, 206, 210, 214, 220, 226, 230, 234, 240, 246

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

A315462 Coordination sequence Gal.6.339.6 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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A315464 Coordination sequence Gal.6.346.5 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.

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A316319 Coordination sequence for a trivalent node in a chamfered version of the 3^6 triangular tiling of the plane.

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A316320 Coordination sequence for a hexavalent node in a chamfered version of the 3^6 triangular tiling of the plane.

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PARI

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A342285 Coordination sequence with respect to the central vertex of a dodecagon-based tiling of the plane by copies of a certain Goldberg quadrilateral tile.

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PARI

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

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Crossrefs

Extensions

A250121 Crystal ball sequence for planar net 3.3.3.3.6.

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A310387 Coordination sequence Gal.6.588.6 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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A311019 Coordination sequence Gal.6.492.2 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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