A008574 -id:A008574 - cp's OEIS frontend

A301716 Coordination sequence for node of type V1 in "kre" 2-D tiling (or net).

1, 6, 12, 18, 18, 30, 36, 36, 48, 48, 54, 66, 66, 72, 78, 84, 90, 96, 102, 102, 114, 120, 120, 132, 132, 138, 150, 150, 156, 162, 168, 174, 180, 186, 186, 198, 204, 204, 216, 216, 222, 234, 234, 240, 246, 252, 258, 264, 270, 270, 282, 288, 288, 300, 300, 306

Offset: 0

N. J. A. Sloane, Mar 26 2018

Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 30 2023

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, 1st row, 3rd tiling.

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 18 from the list of 20 2-uniform tilings.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Reticular Chemistry Structure Resource (RCSR), The kre tiling (or net)
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
Rémy Sigrist, Illustration of first terms
Rémy Sigrist, PARI program for A301716
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,1,0,0,-1).

Cf. A301718.

Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.

LinearRecurrence[{0,0,1,0,1,0,0,-1},{1,6,12,18,18,30,36,36,48},100] (* Paolo Xausa, Nov 16 2023 *)

PARI
```
See Links section.
```

G.f.: (x^8+6*x^7+12*x^6+17*x^5+12*x^4+17*x^3+12*x^2+6*x+1) / ((1-x^3)*(1-x^5)). - N. J. A. Sloane, Mar 28 2018

More terms from Rémy Sigrist, Mar 28 2018

A301718 Coordination sequence for node of type V2 in "kre" 2-D tiling (or net).

Original entry on oeis.org

1, 5, 11, 17, 23, 28, 33, 39, 45, 51, 56, 61, 67, 73, 79, 84, 89, 95, 101, 107, 112, 117, 123, 129, 135, 140, 145, 151, 157, 163, 168, 173, 179, 185, 191, 196, 201, 207, 213, 219, 224, 229, 235, 241, 247, 252, 257, 263, 269, 275, 280, 285, 291, 297, 303, 308

Offset: 0

N. J. A. Sloane, Mar 26 2018

Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 30 2023

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, 1st row, 3rd tiling.

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 18 from the list of 20 2-uniform tilings.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Reticular Chemistry Structure Resource (RCSR), The kre tiling (or net)
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
Rémy Sigrist, Illustration of first terms
Rémy Sigrist, PARI program for A301718
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A301716.

Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.

LinearRecurrence[{1,0,0,0,1,-1},{1,5,11,17,23,28,33},100] (* Paolo Xausa, Nov 16 2023 *)

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

G.f.: (x^4+2*x^3+x^2+2*x+1)*(x+1)^2 / ((1-x)*(1-x^5)). - N. J. A. Sloane, Mar 28 2018

Conjecture: a(n) ~ 28*n/5. - Stefano Spezia, Mar 29 2023

More terms from Rémy Sigrist, Mar 28 2018

A301720 Coordination sequence for node of type V1 in "krb" 2-D tiling (or net).

Original entry on oeis.org

1, 6, 9, 18, 21, 24, 36, 36, 39, 54, 51, 54, 72, 66, 69, 90, 81, 84, 108, 96, 99, 126, 111, 114, 144, 126, 129, 162, 141, 144, 180, 156, 159, 198, 171, 174, 216, 186, 189, 234, 201, 204, 252, 216, 219, 270, 231, 234, 288, 246, 249, 306, 261, 264, 324, 276, 279, 342, 291, 294, 360, 306, 309, 378, 321

Offset: 0

N. J. A. Sloane, Mar 26 2018

Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 30 2023

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 66, 1st row, 2nd tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 19 from the list of 20 2-uniform tilings.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Reticular Chemistry Structure Resource (RCSR), The krb tiling (or net)
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A301722.

Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.

LinearRecurrence[{0,0,2,0,0,-1},{1,6,9,18,21,24,36},100] (* Paolo Xausa, Nov 16 2023 *)

G.f.: -(-x^6-6*x^5-9*x^4-16*x^3-9*x^2-6*x-1)/(x^6-2*x^3+1). - N. J. A. Sloane, Mar 29 2018

a(11)-a(100) from Davide M. Proserpio, Mar 28 2018

A301722 Coordination sequence for node of type V2 in "krb" 2-D tiling (or net).

Original entry on oeis.org

1, 5, 10, 15, 22, 27, 31, 38, 43, 47, 54, 59, 63, 70, 75, 79, 86, 91, 95, 102, 107, 111, 118, 123, 127, 134, 139, 143, 150, 155, 159, 166, 171, 175, 182, 187, 191, 198, 203, 207, 214, 219, 223, 230, 235, 239, 246, 251, 255, 262, 267, 271, 278, 283, 287, 294, 299, 303, 310, 315, 319, 326, 331, 335

Offset: 0

N. J. A. Sloane, Mar 26 2018

Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 30 2023

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 66, 1st row, 2nd tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 19 from the list of 20 2-uniform tilings.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Reticular Chemistry Structure Resource (RCSR), The krb tiling (or net)
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A301720.

Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.

LinearRecurrence[{1,0,1,-1},{1,5,10,15,22,27,31},100] (* Paolo Xausa, Nov 16 2023 *)

G.f.: (-x^6+3*x^4+4*x^3+5*x^2+4*x-1)/(x^4-x^3-x+1). - Ray Chandler, Aug 30 2023

a(11)-a(100) from Davide M. Proserpio, Mar 28 2018

A301724 Coordination sequence for node of type V1 in "kra" 2-D tiling (or net).

Original entry on oeis.org

1, 6, 10, 16, 23, 27, 31, 38, 44, 48, 54, 60, 64, 70, 77, 81, 85, 92, 98, 102, 108, 114, 118, 124, 131, 135, 139, 146, 152, 156, 162, 168, 172, 178, 185, 189, 193, 200, 206, 210, 216, 222, 226, 232, 239, 243, 247, 254, 260, 264, 270, 276, 280, 286, 293, 297, 301, 308, 314, 318, 324, 330, 334, 340

Offset: 0

N. J. A. Sloane, Mar 26 2018

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 66, 1st row, 1st tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 20 from the list of 20 2-uniform tilings.
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.
A. V. Maleev, A. A. Mokrova, A. V. Shutov, Coordination sequences of the 2-uniform graphs (Russian), Algebra, number theory and discrete geometry: modern problems and application of past problems (2019), Proceedings of the XVI International Conference in honor of the 80th birthday of Professor Michel Deza, 262-266.
Reticular Chemistry Structure Resource (RCSR), The kra tiling (or net)
Anton Shutov and Andrey Maleev, Coordination sequences and layer-by-layer growth of periodic structures, Zeitschrift für Kristallographie - Crystalline Materials, Volume 234, Issue 5, Pages 291-299 (2018).
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-2,2,-1).

Cf. A301726.

Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.

CoefficientList[Series[(x^10+4x^9+6x^7+x^6+3x^5+x^4+6x^3+4x+1)/((x^4+x^3+x^2+x+1)(x^4-x^3+x^2-x+1)(x-1)^2),{x,0,100}],x] (* Harvey P. Dale, Aug 08 2021 *)

G.f.: (x^10+4*x^9+6*x^7+x^6+3*x^5+x^4+6*x^3+4*x+1)/((x^4+x^3+x^2+x+1)*(x^4-x^3+x^2-x+1)*(x-1)^2). - N. J. A. Sloane, Mar 29 2018

a(11)-a(100) from Davide M. Proserpio, Mar 28 2018

A301726 Coordination sequence for node of type V2 in "kra" 2-D tiling (or net).

Original entry on oeis.org

1, 5, 11, 16, 21, 27, 33, 38, 43, 49, 54, 59, 65, 70, 75, 81, 87, 92, 97, 103, 108, 113, 119, 124, 129, 135, 141, 146, 151, 157, 162, 167, 173, 178, 183, 189, 195, 200, 205, 211, 216, 221, 227, 232, 237, 243, 249, 254, 259, 265, 270, 275, 281, 286, 291, 297, 303, 308, 313, 319, 324, 329, 335, 340

Offset: 0

N. J. A. Sloane, Mar 26 2018

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 66, 1st row, 1st tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 20 from the list of 20 2-uniform tilings.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Reticular Chemistry Structure Resource (RCSR), The kra tiling (or net)
A. V. Maleev, A. A. Mokrova, A. V. Shutov, Coordination sequences of the 2-uniform graphs (Russian), Algebra, number theory and discrete geometry: modern problems and application of past problems (2019), Proceedings of the XVI International Conference in honor of the 80th birthday of Professor Michel Deza, 262-266.
Anton Shutov and Andrey Maleev, Coordination sequences and layer-by-layer growth of periodic structures, Zeitschrift für Kristallographie - Crystalline Materials, Volume 234, Issue 5, Pages 291-299 (2018).
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
Index entries for linear recurrences with constant coefficients, signature (2, -2, 2, -2, 2, -2, 2, -2, 2, -1).

Cf. A301724.

Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.

CoefficientList[Series[(x^2+x+1)(x^8+2x^7+3x^4+2x+1)/((x^4+x^3+x^2+x+1)(x^4-x^3+x^2-x+1)(x-1)^2),{x,0,110}],x] (* Harvey P. Dale, Sep 25 2020 *)

G.f. = (x^2+x+1)*(x^8+2*x^7+3*x^4+2*x+1)/((x^4+x^3+x^2+x+1)*(x^4-x^3+x^2-x+1)*(x-1)^2). - N. J. A. Sloane, Mar 29 2018

a(11)-a(100) from Davide M. Proserpio, Mar 28 2018

A265035 Coordination sequence of 2-uniform tiling {3.4.6.4, 4.6.12} with respect to a point of type 4.6.12.

Original entry on oeis.org

1, 3, 6, 9, 11, 14, 17, 21, 25, 28, 30, 32, 35, 39, 43, 46, 48, 50, 53, 57, 61, 64, 66, 68, 71, 75, 79, 82, 84, 86, 89, 93, 97, 100, 102, 104, 107, 111, 115, 118, 120, 122, 125, 129, 133, 136, 138, 140, 143, 147, 151, 154, 156, 158, 161, 165, 169, 172, 174, 176

Offset: 0

N. J. A. Sloane, Dec 12 2015

Joseph Myers (Dec 14 2015) reports that "My program for coordination sequences requires describing the tiling structure under translation, listing all edges in the form: (class1, 0, 0) has an edge to (class2, x, y). The present tiling has 18 orbits of vertices under translation and 30 orbits of edges under translation (each of which is described in both directions). So in principle it could generate the other 19 2-uniform tilings, but without a cross check with hand-computed terms there's a risk of e.g. missing some edges, and a fair amount of work producing all the descriptions of translation classes of edges."

Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 31 2023

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See page 67, 4th row, 3rd tiling.
Otto Krötenheerdt, Die homogenen Mosaike n-ter Ordnung in der euklidischen Ebene, I, II, III, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math-Natur. Reihe, 18 (1969), 273-290; 19 (1970), 19-38 and 97-122. [Includes classification of 2-uniform tilings]
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166.

Joseph Myers, Table of n, a(n) for n = 0..20000
Miguel Carlos Fernández-Cabo, Artisan Procedures to Generate Uniform Tilings, International Mathematical Forum, Vol. 9, 2014, no. 23, 1109-1130. [Background information]
Brian Galebach, Collection of n-Uniform Tilings. See Number 1 from the list of 20 2-uniform tilings.
Brian Galebach, The tiling {3.4.6.4, 4.6.12}, Number 1 from list of 20 2-uniform tilings. (From the 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 arXiv:1803.08530.
Reticular Chemistry Structure Resource (RCSR), The krt tiling (or net)
N. J. A. Sloane, Illustration of initial terms of A265035 (point of type 4.6.12)
N. J. A. Sloane, Illustration of initial terms of A265036 (point of type 3.4.6.4)
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 (3,-4,3,-1).

See A265036 for the other type of point.
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).
Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.

LinearRecurrence[{3,-4,3,-1},{1,3,6,9,11,14,17,21,25},100] (* Paolo Xausa, Nov 15 2023 *)

Based on the b-file, the g.f. appears to be (1+x^2+2*x^5-2*x^6+2*x^7-x^8)/(1-3*x+4*x^2-3*x^3+x^4). This matches the first 1000 terms, so is probably correct. - N. J. A. Sloane, Dec 14 2015

Conjectured g.f. is equivalent to a(n) = 3*n - A010892(n+1) for n >= 5. - R. J. Mathar, Oct 09 2020

Extended by Joseph Myers, Dec 13 2015

b-file extended by Joseph Myers, Dec 18 2015

A008579 Coordination sequence for planar net 3.6.3.6. Spherical growth function for a certain reflection group in plane.

Original entry on oeis.org

1, 4, 8, 14, 18, 22, 28, 30, 38, 38, 48, 46, 58, 54, 68, 62, 78, 70, 88, 78, 98, 86, 108, 94, 118, 102, 128, 110, 138, 118, 148, 126, 158, 134, 168, 142, 178, 150, 188, 158, 198, 166, 208, 174, 218, 182, 228, 190, 238, 198, 248, 206, 258, 214, 268, 222, 278

Offset: 0

N. J. A. Sloane

Interesting because coefficients never become monotonic.
Also the coordination sequence for a planar net made of densely packed circles. - Yuriy Sibirmovsky, Sep 11 2016
Described by J.-G. Eon (2014) as the coordination sequence of the Kagome net. - N. J. A. Sloane, Jan 03 2018

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 161 (but beware errors).

T. D. Noe, Table of n, a(n) for n = 0..1000
Pierre de La Harpe, and P. I. Grigorchuk, Local convexity of the growth function of finitely generated groups and question 5.2 in the Kourovka Notebook, Algebra and Logic 37.6 (1998): 353-356.
Jean-Guillaume Eon, Algebraic determination of generating functions for coordination sequences in crystal structures, Acta Cryst. A58 (2002), 47-53. See p. 51.
Jean-Guillaume Eon, Topological density of nets: a direct calculation, Acta Crystallographica Section A (Foundations of Crystallography), A60 (2014), 7-18; DOI: 10.1107/S0108767303022037. See Section 5.
Jean-Guillaume Eon, Symmetry and Topology: The 11 Uninodal Planar Nets Revisited, Symmetry, 10 (2018), 13 pages, doi:10.3390/sym10020035. See Section 4.
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, kgm
Yuriy Sibirmovsky, Illustration of initial terms with densely packed circles.
N. J. A. Sloane, Illustration of initial terms
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 (0,2,0,-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. A017137, A017365.

a008579 0 = 1
a008579 1 = 4
a008579 n = (10 - 2*m) * n' + 8*m - 2 where (n',m) = divMod n 2
a008579_list = 1 : 4 : concatMap (\x -> map (* 2) [5*x-1,4*x+3]) [1..]
-- Reinhard Zumkeller, Nov 12 2012

f := n->if n mod 2 = 0 then 10*(n/2)-2 else 8*(n-1)/2+6 fi;

a[n_?EvenQ] := 10*n/2-2; a[n_?OddQ] := 8*(n-1)/2+6; a[0] = 1; a[1] = 4; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Nov 18 2011, after Maple *)
CoefficientList[Series[(1+2x)(1+2x+2x^2+2x^3-x^4)/(1-x^2)^2,{x,0,50}],x] (* or *) LinearRecurrence[{0,2,0,-1},{1,4,8,14,18,22},50] (* Harvey P. Dale, Sep 05 2018 *)

G.f.: (1 + 2*x)*(1 + 2*x + 2*x^2 + 2*x^3 - x^4)/(1 - x^2)^2.
From R. J. Mathar, Nov 26 2014: (Start)
a(2n) = A017365(n), n > 0.
a(2n+1) = A017137(n), n > 0. (End)
From Stefano Spezia, Aug 07 2022: (Start)
a(n) = (9 + (-1)^n)*n/2 - 2*(-1)^n for n > 1.
E.g.f.: 3 - 2*x + (4*x - 2)*cosh(x) + (5*x + 2)*sinh(x). (End)

A250122 Coordination sequence for planar net 3.12.12.

Original entry on oeis.org

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

cp's OEIS Frontend

A301716 Coordination sequence for node of type V1 in "kre" 2-D tiling (or net).

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A301718 Coordination sequence for node of type V2 in "kre" 2-D tiling (or net).

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A301720 Coordination sequence for node of type V1 in "krb" 2-D tiling (or net).

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A301722 Coordination sequence for node of type V2 in "krb" 2-D tiling (or net).

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A301724 Coordination sequence for node of type V1 in "kra" 2-D tiling (or net).

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A301726 Coordination sequence for node of type V2 in "kra" 2-D tiling (or net).

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A265035 Coordination sequence of 2-uniform tiling {3.4.6.4, 4.6.12} with respect to a point of type 4.6.12.

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