A008579 -id:A008579 - cp's OEIS frontend

A250122 Coordination sequence for planar net 3.12.12.

1, 3, 4, 6, 8, 12, 14, 15, 18, 21, 22, 24, 28, 30, 30, 33, 38, 39, 38, 42, 48, 48, 46, 51, 58, 57, 54, 60, 68, 66, 62, 69, 78, 75, 70, 78, 88, 84, 78, 87, 98, 93, 86, 96, 108, 102, 94, 105, 118, 111, 102, 114, 128, 120, 110, 123, 138, 129

Offset: 0

Darrah Chavey, Nov 23 2014

Also, growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^6 = 1 >. See Magma program in A298805. - N. J. A. Sloane, Feb 06 2018

Maurizio Paolini, Table of n, a(n) for n = 0..1021
Agnes Azzolino, Regular and Semi-Regular Tessellation Paper, 2011.
Agnes Azzolino, Illustration of 3.12.12 tiling [From previous link]
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 on arXiv, arXiv:1803.08530 [math.CO], 2018-2019. See section 10 The 3.12^2 tiling.
Rostislav Grigorchuk and Cosmas Kravaris, On the growth of the wallpaper groups, arXiv:2012.13661 [math.GR], 2020. See section 4.6 p. 23.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Tom Karzes, Tiling Coordination Sequences
Maurizio Paolini, C program for A250122
Reticular Chemistry Structure Resource, hca
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).

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).

Cf. A298805.

Join[{1, 3, 4}, LinearRecurrence[{2, -3, 4, -3, 2, -1}, {6, 8, 12, 14, 15, 18}, 100]] (* Jean-François Alcover, Aug 05 2018 *)

From Joseph Myers, Nov 28 2014: (Start)
Empirically,
a(4n) = 10n - 2 except for a(0) = 1
a(4n+1) = 9n + 3
a(4n+2) = 8n + 6 except for a(2) = 4
a(4n+3) = 9n + 6. (End)
If these are correct, the sequence has g.f.
-(-1 - x - x^2 - 3*x^3 + x^4 - 5*x^5 + 3*x^6 - 4*x^7 + 2*x^8)/((x - 1)^2*(x^2 + 1)^2). - N. J. A. Sloane, Nov 28 2014
All the above conjectures are true. - N. J. A. Sloane, Dec 31 2015
E.g.f.: (9*x*cosh(x) - 4*(2*cos(x) + x^2 - 3) + 9*x*sinh(x) - (x - 3)*sin(x))/4. - Stefano Spezia, Jan 05 2023

a(8) onwards from Maurizio Paolini and Joseph Myers (independently), Nov 28 2014

A072154 Coordination sequence for the planar net 4.6.12.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 15, 17, 19, 21, 24, 27, 29, 31, 33, 36, 39, 41, 43, 45, 48, 51, 53, 55, 57, 60, 63, 65, 67, 69, 72, 75, 77, 79, 81, 84, 87, 89, 91, 93, 96, 99, 101, 103, 105, 108, 111, 113, 115, 117, 120, 123, 125, 127, 129, 132, 135, 137

Offset: 0

N. J. A. Sloane, Jun 28 2002

There is only one type of node in this structure: each node meets a square, a hexagon and a 12-gon.
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.
Also, coordination sequence for the aluminophosphate AlPO_4-5 structure.

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; translation in J. Math. Sci. (N.Y.) 129 (2005), no. 3, 3922-3926 [MR2023041]. See Table 1, line "p6m" (but beware typos).

Sean A. Irvine, Table of n, a(n) for n = 0..999
Joerg Arndt, The 4.6.12 planar net
Agnes Azzolino, Regular and Semi-Regular Tessellation Paper, 2011.
Agnes Azzolino, Larger illustration of 4.6.12 planar net [From previous link]
M. E. Davis, Ordered porous materials for emerging applications, Nature, 417 (Jun 20 2002), 813-821 (gives structure).
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 on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.
Rostislav Grigorchuk and Cosmas Kravaris, On the growth of the wallpaper groups, arXiv:2012.13661 [math.GR], 2020. See section 4.7 p. 23.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Sean A. Irvine, Java implementation with explicit counting
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource, fxt
N. J. A. Sloane, AlPO_4-5 structure, after Davis
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
N. J. A. Sloane, The subgraph H used in the proof of the formulas
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A072149, A072150, A072151, A072152, A072153.
For partial sums see A265078.
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).
See also A301730.

Join[{1}, LinearRecurrence[{1, 0, 0, 0, 1, -1}, {3, 5, 7, 9, 12, 15}, 100]] (* Jean-François Alcover, Dec 13 2018 *)

Empirical g.f.: (x+1)^2*(x^2-x+1)*(x^2+x+1)/((x-1)^2*(x^4+x^3+x^2+x+1)). - Colin Barker, Nov 18 2012
This empirical g.f. can also be written as (1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + x^6)/(1 - x - x^5 + x^6). - N. J. A. Sloane, Dec 20 2015
Theorem: For n >= 7, a(n) = a(n-1) + a(n-5) - a(n-6), and a(5k) = 12k (k > 0), a(5k+m) = 12k + 2m + 1 (k >= 0, 1 <= m < 5). This also implies the conjectured g.f.'s. - N. J. A. Sloane, conjectured Dec 20 2015, proved Jan 20 2018.
Notes on the proof, from N. J. A. Sloane, Jan 20 2018 (Start)
The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.
The figure is divided into 6 sectors by the blue trunks. In the interior of each sector, working outwards from the base point P at the origin, there are successively 1,2,3,4,... (red) 12-gons. All the 12-gons (both red and blue) have a unique closest point to P.
If the closest point in a 12-gon is at distance d from P, then the contributions of the 12 points of the 12-gon to a(d), a(d+1), ..., a(d+6) are 1,2,2,2,2,2,1, respectively.
The rest of the proof is now a matter of simple counting.
The blue 12-gons (along the trunks) are especially easy to count, because there is a unique blue 12-gon at shortest distance d from P for d = 1,2,3,4,...
(End)
a(n) = 2*(6*n + sqrt(1 + 2/sqrt(5))*sin(2*n*Pi/5) + sqrt(1 - 2/sqrt(5))*sin(4*n*Pi/5))/5 for n > 0. - Stefano Spezia, Jan 05 2023

More terms from Sean A. Irvine, Sep 29 2011

Thanks to Darrah Chavey for pointing out that this is the planar net 4.6.12. - N. J. A. Sloane, Nov 24 2014

A298808 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^4 = 1 >.

Original entry on oeis.org

1, 3, 4, 6, 6, 3, 1

Offset: 0

John Cannon and N. J. A. Sloane, Feb 06 2018

This group is finite, so the growth series is a polynomial.

Coordination sequence for truncated cube (see Karzes link). The coordination sequence for the cuboctahedron is 1,4,6,1, which is too short to have its own entry. - N. J. A. Sloane, Nov 20 2019

Tom Karzes, Polyhedron Coordination Sequences
Index entries for coordination sequences

Cf. A008579, A298802, A298805.

Magma
```
// See Magma program in A298805.
```

A008706 Coordination sequence for 3.3.3.4.4 planar net.

Original entry on oeis.org

1, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260, 265, 270, 275

Offset: 0

N. J. A. Sloane

Also the Engel expansion of exp^(1/5); cf. A006784 for the Engel expansion definition. - Benoit Cloitre, Mar 03 2002

			G.f. = 1 + 5*x + 10*x^2 + 15*x^3 + 20*x^4 + 25*x^5 + 30*x^6 + 35*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
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.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource, cem
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A006784, A048476 (binomial Transf.)
Essentially the same as A008587.
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).
First differences of A005891.

[0^n+5*n: n in [0..50] ]; // Vincenzo Librandi, Aug 21 2011

Join[{1}, LinearRecurrence[{2, -1}, {5, 10}, 100]] (* Jean-François Alcover, Dec 13 2018 *)

a(n)=0^n+5*n \\ Charles R Greathouse IV, Mar 19 2015

From Paul Barry, Jul 21 2003: (Start)
G.f.: (1 + 3*x + x^2)/(1 - x)^2.
a(n) = 0^n + 5n. (End)
G.f.: A(x) + 1, where A(x) is the g.f. of A008587. - Gennady Eremin, Feb 21 2021
E.g.f.: 1 + 5*x*exp(x). - Stefano Spezia, Jan 05 2023

A296368 Coordination sequence for the Cairo or dual-3.3.4.3.4 tiling with respect to a trivalent point.

Original entry on oeis.org

1, 3, 8, 12, 15, 20, 25, 28, 31, 36, 41, 44, 47, 52, 57, 60, 63, 68, 73, 76, 79, 84, 89, 92, 95, 100, 105, 108, 111, 116, 121, 124, 127, 132, 137, 140, 143, 148, 153, 156, 159, 164, 169, 172, 175, 180, 185, 188, 191, 196, 201, 204, 207, 212, 217, 220, 223, 228

Offset: 0

N. J. A. Sloane, Dec 21 2017

There are two types of point in this tiling. This is the coordination sequence with respect to a point of degree 3.

The coordination sequence with respect to a point of degree 4 (see second illustration) is simply 1, 4, 8, 12, 16, 20, ..., the same as the coordination sequence for the 4.4.4.4 square grid (A008574). See the CGS-NJAS link for the proof.

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Fig. 9.1.3, drawing P_5-24, page 480.
Herbert C. Moore, U.S. Patents 928,320 and 928,321, Patented July 20 1909. [Shows Cairo tiling.]

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Chaim Goodman-Strauss and N. J. A. Sloane, A portion of the Cairo tiling
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.
Chung, Ping Ngai, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, and Elena Wikner. Isoperimetric Pentagonal Tilings, Notices of the AMS 59, no. 5 (2012), pp. 632-640. See Fig. 1 (left).
Tom Karzes, Tiling Coordination Sequences
Frank Morgan, Optimal Pentagonal Tilings, Video, May 2021 [Has much to say about the Cairo tiling]
Reticular Chemistry Structure Resource (RCSR), The mcm tiling (or net)
Rémy Sigrist, PARI program for A296368
N. J. A. Sloane, Illustration of initial terms (for a trivalent point)
N. J. A. Sloane, Illustration of initial terms of coordination sequence 1,4,8,12,... for a tetravalent point
N. J. A. Sloane, A tiling by rectangles which has the same graph and coordination sequences as the Cairo tiling (Seen on the streets of Piscataway, New Jersey, USA)
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)
Index entries for linear recurrences with constant coefficients, signature (2, -2, 2, -1).

For partial sums see A296909.
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).
List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

Join[{1, 3, 8}, LinearRecurrence[{2, -2, 2, -1}, {12, 15, 20, 25}, 100]] (* Jean-François Alcover, Aug 05 2018 *)

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

The simplest formula is: a(0)=1, a(1)=2, a(2)=8, and thereafter a(n) = 4n if n is odd, 4n - 1 if n == 0 (mod 4), and 4n+1 if n == 2 (mod 4). (See the CGS-NJAS link for proof. - N. J. A. Sloane, May 10 2018)
a(n + 4) = a(n) + 16 for any n >= 3. - Rémy Sigrist, Dec 23 2017 (See the CGS-NJAS link for a proof. - N. J. A. Sloane, Dec 30 2017)
G.f.: -(x^6-x^5-2*x^4-4*x^2-x-1)/((x^2+1)*(x-1)^2).
From Colin Barker, Dec 23 2017: (Start)
a(n) = (8*n - (-i)^n - i^n) / 2 for n>2, where i=sqrt(-1).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>6.
(End)

Terms a(8)-a(20) and RCSR link from Davide M. Proserpio, Dec 22 2017

More terms from Rémy Sigrist, Dec 23 2017

A298026 Coordination sequence of Dual(3.6.3.6) tiling with respect to a hexavalent node.

Original entry on oeis.org

1, 6, 6, 18, 12, 30, 18, 42, 24, 54, 30, 66, 36, 78, 42, 90, 48, 102, 54, 114, 60, 126, 66, 138, 72, 150, 78, 162, 84, 174, 90, 186, 96, 198, 102, 210, 108, 222, 114, 234, 120, 246, 126, 258, 132, 270, 138, 282, 144, 294, 150, 306, 156, 318, 162, 330, 168, 342, 174, 354, 180, 366, 186, 378, 192, 390

Offset: 0

N. J. A. Sloane, Jan 21 2018

Also known as the kgd net.

This is one of the Laves tilings.

Robert Israel, Table of n, a(n) for n = 0..10000
Reticular Chemistry Structure Resource (RCSR), The kgd tiling (or net)
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008579, A298027 (partial sums), A298028 (trivalent point).

List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

f6:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 3*n else 6*n; fi; end;
[seq(f6(n),n=0..80)];

Join[{1}, LinearRecurrence[{0, 2, 0, -1}, {6, 6, 18, 12}, 80]] (* Jean-François Alcover, Mar 23 2020 *)

Vec((1 + 6*x + 4*x^2 + 6*x^3 + x^4) / ((1 - x)^2*(1 + x)^2) + O(x^60)) \\ Colin Barker, Jan 22 2018

a(0)=1; a(2*k)=6*k, a(2*k+1)=12*k+6.
G.f.: 1 + 6*x*(1+x+x^2)/(1-x^2)^2. - Robert Israel, Jan 21 2018
From Colin Barker, Jan 22 2018: (Start)
a(n) = 3*n for n>0 and even.
a(n) = 6*n for n odd.
a(n) = 2*a(n-2) - a(n-4) for n>4.
(End)
a(n) = 6*A026741(n), n>0. - R. J. Mathar, Jan 29 2018

A298028 Coordination sequence of Dual(3.6.3.6) tiling with respect to a trivalent node.

Original entry on oeis.org

1, 3, 12, 9, 24, 15, 36, 21, 48, 27, 60, 33, 72, 39, 84, 45, 96, 51, 108, 57, 120, 63, 132, 69, 144, 75, 156, 81, 168, 87, 180, 93, 192, 99, 204, 105, 216, 111, 228, 117, 240, 123, 252, 129, 264, 135, 276, 141, 288, 147, 300, 153, 312, 159, 324, 165, 336, 171, 348, 177, 360, 183, 372, 189, 384, 195

Offset: 0

N. J. A. Sloane, Jan 21 2018

Also known as the kgd net.

This is one of the Laves tilings.

Robert Israel, Table of n, a(n) for n = 0..10000
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource (RCSR), The kgd tiling (or net)
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008579, A135556 (partial sums), A298026 (trivalent point).
If the initial 1 is changed to 0 we get A165988 (but we need both sequences, just as we have both A008574 and A008586).
List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

f3:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 6*n else 3*n; fi; end;
[seq(f3(n),n=0..80)];

Join[{1}, LinearRecurrence[{0, 2, 0, -1}, {3, 12, 9, 24}, 80]] (* Jean-François Alcover, Mar 23 2020 *)

a(0)=1; a(2*k) = 12*k, a(2*k+1) = 6*k+3.
G.f.: 1 + 3*x*(x^2+4*x+1)/(1-x^2)^2. - Robert Israel, Jan 21 2018
a(n) = 3*A022998(n), n>0. - R. J. Mathar, Jan 29 2018

A298802 Growth series for group with presentation < S, T : S^4 = T^4 = (S*T)^4 = 1 >.

Original entry on oeis.org

1, 4, 10, 24, 56, 128, 294, 676, 1552, 3564, 8186, 18800, 43176, 99160, 227734, 523020, 1201184, 2758676, 6335658, 14550664, 33417496, 76747632, 176260934, 404806196, 929690160, 2135154556, 4903660570, 11261895264, 25864409480, 59400985544, 136422101046, 313311125788, 719559813184

Offset: 0

John Cannon and N. J. A. Sloane, Feb 04 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,2,-1).

Cf. A008579.

R := RationalFunctionField(Integers());
PSR25 := PowerSeriesRing(Integers():Precision := 25);
FG := FreeGroup(2);
TG := quo;
f, A :=IsAutomaticGroup(TG);
gf := GrowthFunction(A);
R!gf;
Coefficients(PSR25!gf);

LinearRecurrence[{2,0,2,-1},{1,4,10,24,56},40] (* Harvey P. Dale, Jan 02 2020 *)

Vec((1 + x)^2*(1 + x^2) / (1 - 2*x - 2*x^3 + x^4) + O(x^40)) \\ Colin Barker, Feb 04 2018

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

a(n) = 2*a(n-1) + 2*a(n-3) - a(n-4) for n>4. - Colin Barker, Feb 04 2018

A298805 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^7 = 1 >.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 16, 22, 24, 34, 40, 56, 62, 83, 98, 133, 152, 202, 236, 322, 368, 496, 570, 776, 892, 1202, 1384, 1871, 2158, 2915, 3352, 4534, 5218, 7060, 8120, 10976, 12636, 17084, 19664, 26580, 30592, 41367, 47604, 64365, 74072, 100152, 115264, 155836, 179352, 242488, 279076, 377324, 434246, 587126

Offset: 0

John Cannon and N. J. A. Sloane, Feb 04 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,0,0,1,2,1,0,1,0,1,2,1,0,0,-1,-1).

Cf. A008579, A298802.

// To get the growth function for the group with presentation
// < S, T | S^a = T^b = (S*I)^c = 1 >
a:=2; b:=3; c:=7;
R := RationalFunctionField(Integers());
PSR := PowerSeriesRing(Integers():Precision := 100);
FG := FreeGroup(2);
TG := quo;
f, A :=IsAutomaticGroup(TG);
gf := GrowthFunction(A);
R!gf;
Coefficients(PSR!gf);

LinearRecurrence[{-1,0,0,1,2,1,0,1,0,1,2,1,0,0,-1,-1},{1,3,4,6,8,12,16,22,24,34,40,56,62,83,98,133,152,202,236},60] (* Harvey P. Dale, Jun 15 2021 *)

Vec((1 + 4*x + 7*x^2 + 10*x^3 + 13*x^4 + 15*x^5 + 17*x^6 + 21*x^7 + 21*x^8 + 21*x^9 + 21*x^10 + 19*x^11 + 15*x^12 + 12*x^13 + 9*x^14 + 6*x^15 + 3*x^16 - 2*x^17 - 2*x^18) / ((1 + x + x^2 + x^3 + x^4)*(1 - x^2 - x^4 + x^6 - x^8 - x^10 + x^12)) + O(x^60)) \\ Colin Barker, Feb 06 2018

G.f.: (-2*x^18 - 2*x^17 + 3*x^16 + 6*x^15 + 9*x^14 + 12*x^13 + 15*x^12 + 19*x^11 + 21*x^10 + 21*x^9 + 21*x^8 + 21*x^7 + 17*x^6 + 15*x^5 + 13*x^4 + 10*x^3 + 7*x^2 + 4*x + 1)/(x^16 + x^15 - x^12 - 2*x^11 - x^10 - x^8 - x^6 - 2*x^5 - x^4 + x + 1).
The denominator can be factored: G.f. also = -(2*x^18 + 2*x^17 - 3*x^16 - 6*x^15 - 9*x^14 - 12*x^13 - 15*x^12 - 19*x^11 - 21*x^10 - 21*x^9 - 21*x^8 - 21*x^7 - 17*x^6 - 15*x^5 - 13*x^4 - 10*x^3 - 7*x^2 - 4*x - 1) / ((x^4 + x^3 + x^2 + x + 1) * (x^12 - x^10 - x^8 + x^6 - x^4 - x^2 + 1)).
a(n) = -a(n-1) + a(n-4) + 2*a(n-5) + a(n-6) + a(n-8) + a(n-10) + 2*a(n-11) + a(n-12) - a(n-15) - a(n-16) for n>18. - Colin Barker, Feb 06 2018

A068600 Number of n-uniform tilings having n different arrangements of polygons about their vertices.

Original entry on oeis.org

11, 20, 39, 33, 15, 10, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Brian Galebach, Mar 28 2002

nonn

Sequence gives the number of edge-to-edge regular-polygon tilings having n topologically distinct vertex types, with each vertex type having a different arrangement of surrounding polygons. Does not allow for tilings with two or more vertex types having the same arrangement of surrounding polygons, even when those vertices are topologically distinct. There are no 8- or higher-uniform tilings having the equivalent number of distinct polygon arrangements.

There are eleven 1-uniform tilings (also called the "Archimedean" tessellations) which comprise the three regular tessellations (all triangles, squares, or hexagons) plus the eight semiregular tessellations. (See A250120. - N. J. A. Sloane, Nov 29 2014)

This sequence was originally calculated by Otto Krotenheerdt.
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, page 69.
Krotenheerdt, Otto. "Die homogenen Mosaike n-ter Ordnung in der euklidischen Ebene," Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-natur. Reihe, 18(1969), 273-290; 19 (1970)19-38 and 97-122.

Steven Dutch, Uniform Tilings
Brian L. Galebach, n-Uniform Tilings
Ng Lay Ling, Honours Project - Tilings and Patterns.

Cf. A068599.

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).

cp's OEIS Frontend

A250122 Coordination sequence for planar net 3.12.12.

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A072154 Coordination sequence for the planar net 4.6.12.

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A298808 Growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^4 = 1 >.

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Magma

A008706 Coordination sequence for 3.3.3.4.4 planar net.

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A296368 Coordination sequence for the Cairo or dual-3.3.4.3.4 tiling with respect to a trivalent point.

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A298026 Coordination sequence of Dual(3.6.3.6) tiling with respect to a hexavalent node.

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A298028 Coordination sequence of Dual(3.6.3.6) tiling with respect to a trivalent node.

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