A301714 -id:A301714 - cp's OEIS frontend

A301298 Expansion of (1 + 4x + 4x^2 + 4x^3 + x^4)/((1 - x)(1 - x^3)).

1, 5, 9, 14, 19, 23, 28, 33, 37, 42, 47, 51, 56, 61, 65, 70, 75, 79, 84, 89, 93, 98, 103, 107, 112, 117, 121, 126, 131, 135, 140, 145, 149, 154, 159, 163, 168, 173, 177, 182, 187, 191, 196, 201, 205, 210, 215, 219, 224, 229, 233, 238, 243, 247, 252, 257, 261

Offset: 0

N. J. A. Sloane, Mar 24 2018

Coordination sequence for pentavalent node in the "krl" 2-D tiling (or net). (This is easily established using the "coloring book" method - see the Goodman-Strauss & Sloane link.)

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 Table 2.2.1, page 66, 3rd row, second tiling.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Brian Galebach, Collection of n-Uniform Tilings. See Number 4 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.
Reticular Chemistry Structure Resource (RCSR), The krl 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. A298024.

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.

I:=[1,5,9,14,19]; [n le 5 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..80]]; // Vincenzo Librandi, Mar 26 2018

[n eq 0 select 1 else 5*n-Floor((n+1)/3): n in [0..60]]; // Bruno Berselli, Mar 26 2018

CoefficientList[Series[(x^4 + 4 x^3 + 4 x^2 + 4 x + 1) / ((1 -x) (1 - x^3)), {x, 0, 60}], x] (* Vincenzo Librandi, Mar 26 2018 *)
LinearRecurrence[{1,0,1,-1},{1,5,9,14,19},60] (* Harvey P. Dale, Dec 30 2024 *)

lista(nn) = {x='x+O('x^nn); Vec((x^4+4*x^3+4*x^2+4*x+1)/((1-x)*(1-x^3)))} \\ Altug Alkan, Mar 26 2018

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

a(n) = 5*n - floor((n + 1)/3) for n>0, a(0)=1. - Bruno Berselli, Mar 26 2018

A301299 Coordination sequence for node of type V1 in "krq" 2-D tiling (or net).

Original entry on oeis.org

1, 4, 8, 13, 18, 22, 26, 29, 34, 40, 44, 48, 50, 55, 62, 66, 70, 71, 76, 84, 88, 92, 92, 97, 106, 110, 114, 113, 118, 128, 132, 136, 134, 139, 150, 154, 158, 155, 160, 172, 176, 180, 176, 181, 194, 198, 202, 197, 202, 216, 220, 224, 218, 223, 238, 242, 246, 239, 244, 260, 264, 268, 260, 265, 282

Offset: 0

N. J. A. Sloane, Mar 25 2018

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 Table 2.2.1, page 66, bottom row, 2nd tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 5 from the list of 20 2-uniform tilings.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
A. V. Maleev, A. A. Mokrova, and 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 krq 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 (0,0,0,0,2,0,0,0,0,-1).

Cf. A301301.

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[{0,0,0,0,2,0,0,0,0,-1},{1,4,8,13,18,22,26,29,34,40,44},100] (* Paolo Xausa, Nov 15 2023 *)

G.f.: -(-x^10-4*x^9-8*x^8-13*x^7-18*x^6-20*x^5-18*x^4-13*x^3-8*x^2-4*x-1)/(x^10-2*x^5+1). - N. J. A. Sloane, Mar 29 2018

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

A301301 Coordination sequence for node of type V2 in "krq" 2-D tiling (or net).

Original entry on oeis.org

1, 4, 8, 12, 16, 20, 25, 30, 34, 39, 43, 47, 53, 56, 60, 65, 68, 75, 78, 81, 87, 89, 97, 100, 102, 109, 110, 119, 122, 123, 131, 131, 141, 144, 144, 153, 152, 163, 166, 165, 175, 173, 185, 188, 186, 197, 194, 207, 210, 207, 219, 215, 229, 232, 228, 241, 236, 251, 254, 249, 263, 257, 273, 276, 270

Offset: 0

N. J. A. Sloane, Mar 25 2018

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 Table 2.2.1, page 66, bottom row, 2nd tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 5 from the list of 20 2-uniform tilings.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
A. V. Maleev, A. A. Mokrova, and 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 krq 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 (0,0,0,0,2,0,0,0,0,-1).

Cf. A301299.

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[{0,0,0,0,2,0,0,0,0,-1},{1,4,8,12,16,20,25,30,34,39,43,47,53,56,60,65,68,75},100] (* Paolo Xausa, Nov 15 2023 *)

G.f. = -(x^17+x^16+x^15+2*x^14-x^12-x^11-4*x^10-7*x^9-10*x^8-14*x^7-17*x^6-18*x^5-16*x^4-12*x^3-8*x^2-4*x-1)/(x^10-2*x^5+1). - N. J. A. Sloane, Mar 29 2018

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

A301670 Coordination sequence for node of type V1 in "krr" 2-D tiling (or net).

Original entry on oeis.org

1, 4, 8, 12, 16, 22, 26, 26, 36, 36, 44, 42, 54, 50, 64, 56, 72, 66, 82, 70, 92, 80, 100, 86, 110, 94, 120, 100, 128, 110, 138, 114, 148, 124, 156, 130, 166, 138, 176, 144, 184, 154, 194, 158, 204, 168, 212, 174, 222, 182, 232, 188, 240, 198, 250, 202, 260

Offset: 0

N. J. A. Sloane, Mar 25 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, bottom row, 1st tiling.

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 7 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 krr 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 A301670
Index entries for linear recurrences with constant coefficients, signature (-1,0,1,2,1,0,-1,-1).

Cf. A301672.
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.
Cf. A049347, A163805.

LinearRecurrence[{-1,0,1,2,1,0,-1,-1},{1,4,8,12,16,22,26,26,36,36},100] (* Paolo Xausa, Nov 15 2023 *)

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

Based on the b-file, the g.f. appears to be
(-2*x^9+x^8+5*x^7+16*x^6+21*x^5+22*x^4+19*x^3+12*x^2+5*x+1) / ((1+x)*(1-x^3)*(1-x^4)). - N. J. A. Sloane, Mar 25 2018
a(n) = (75*n + 9*(n - 4)*(-1)^n + 18*A163805(n+2) - 12*A049347(n+2))/18 for n >1. - Stefano Spezia, Jun 08 2024

More terms from Rémy Sigrist, Mar 25 2018

A301672 Coordination sequence for node of type V2 in "krr" 2-D tiling (or net).

Original entry on oeis.org

1, 4, 8, 13, 17, 20, 25, 30, 33, 37, 42, 46, 50, 54, 58, 63, 67, 70, 75, 80, 83, 87, 92, 96, 100, 104, 108, 113, 117, 120, 125, 130, 133, 137, 142, 146, 150, 154, 158, 163, 167, 170, 175, 180, 183, 187, 192, 196, 200, 204, 208, 213, 217, 220, 225, 230, 233

Offset: 0

N. J. A. Sloane, Mar 25 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, bottom row, 1st tiling.

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 7 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 krr 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 A301672
Index entries for linear recurrences with constant coefficients, signature (1,-1,2,-1,1,-1).

Cf. A301670.
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.
Cf. A049347, A163805.

LinearRecurrence[{1,-1,2,-1,1,-1},{1,4,8,13,17,20,25},100] (* Paolo Xausa, Nov 15 2023 *)

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

Based on the b-file, the g.f. appears to be
(x^3+2*x^2+x+1)*(x^3+x^2+2*x+1) / ((1-x)*(1+x^2)*(1-x^3)). - N. J. A. Sloane, Mar 25 2018
a(n) = (75*n - 9*A163805(n+2) + 6*A049347(n+2))/18 for n > 0. - Stefano Spezia, Jun 08 2024

More terms from Rémy Sigrist, Mar 25 2018

A301674 Coordination sequence for node of type V1 in "krs" 2-D tiling (or net).

Original entry on oeis.org

1, 4, 8, 14, 16, 26, 22, 34, 36, 38, 44, 54, 46, 62, 64, 62, 72, 82, 70, 90, 92, 86, 100, 110, 94, 118, 120, 110, 128, 138, 118, 146, 148, 134, 156, 166, 142, 174, 176, 158, 184, 194, 166, 202, 204, 182, 212, 222, 190, 230, 232, 206, 240, 250, 214, 258, 260

Offset: 0

N. J. A. Sloane, Mar 25 2018

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 Table 2.2.1, page 67, bottom row, 2nd tiling.

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 6 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 krs 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 A301674
Index entries for linear recurrences with constant coefficients, signature (-1,0,2,2,0,-1,-1).

Cf. A301676.

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[{-1,0,2,2,0,-1,-1},{1,4,8,14,16,26,22,34,36},100] (* Paolo Xausa, Nov 15 2023 *)

PARI
```
See Links section.
```

(a) G.f. = -(2*x^8-x^7-5*x^6-18*x^5-20*x^4-20*x^3-12*x^2-5*x-1)/((x+1)*(x-1)^2*(x^2+x+1)^2). (b) Satisfies the recurrence {( - 2*n^5 - 13*n^4 - 22*n^3 + 7*n^2 + 30*n)*a(n) + ( - 2*n^5 - 13*n^4 - 25*n^3 + n^2 + 39*n)*a(n + 1) + ( - 6*n^2 + 6*n)*a(n + 2) + (2*n^5 + 7*n^4 + 7*n^3 - 7*n^2 - 9*n)*a(n + 3) + (2*n^5 + 7*n^4 + 4*n^3 - 7*n^2 - 6*n)*a(n + 4) = 0, a(0) = 1, a(1) = 4, a(2) = 8, a(3) = 14, a(4) = 16, a(5) = 26}. - N. J. A. Sloane, Mar 28 2018

Equivalent conjecture: 9*a(n) = 40*n -18*(-1)^n -6*(-1)^n*A076118(n+1) +6*A049347(n) -4*A049347(n-1). - R. J. Mathar, Apr 01 2018

More terms from Rémy Sigrist, Mar 28 2018

A301676 Coordination sequence for node of type V2 in "krs" 2-D tiling (or net).

Original entry on oeis.org

1, 4, 8, 13, 18, 22, 27, 31, 35, 41, 44, 48, 55, 57, 61, 69, 70, 74, 83, 83, 87, 97, 96, 100, 111, 109, 113, 125, 122, 126, 139, 135, 139, 153, 148, 152, 167, 161, 165, 181, 174, 178, 195, 187, 191, 209, 200, 204, 223, 213, 217, 237, 226, 230, 251, 239, 243

Offset: 0

N. J. A. Sloane, Mar 25 2018

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 Table 2.2.1, page 67, bottom row, 2nd tiling.

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 6 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 krs 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 A301676
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A301674.

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},{1,4,8,13,18,22,27,31,35},100] (* Paolo Xausa, Nov 15 2023 *)

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

(a) G.f. = -(x^8+x^7-2*x^6-6*x^5-10*x^4-11*x^3-8*x^2-4*x-1)/(x^6-2*x^3+1). (b) Satisfies the recurrence {(-n^5+6*n^4-7*n^3-21*n^2+53*n-30)*a(n)+(3*n^3-24*n^2+51*n-30)*a(n+1)+(6*n^3-33*n^2+57*n-30)*a(n+2)+(n^5-9*n^4+25*n^3-12*n^2-35*n+30)*a(n+3) = 0, a(0) = 1, a(1) = 4, a(2) = 8, a(3) = 13, a(4) = 18, a(5) = 22}. - N. J. A. Sloane, Mar 28 2018

More terms from Rémy Sigrist, Mar 28 2018

A301678 Coordination sequence for node of type V1 in "krn" 2-D tiling (or net).

Original entry on oeis.org

1, 5, 10, 14, 20, 26, 31, 36, 41, 46, 52, 58, 60, 65, 74, 79, 80, 86, 96, 98, 99, 108, 117, 118, 120, 130, 136, 137, 142, 151, 156, 158, 164, 170, 175, 180, 185, 190, 196, 202, 204, 209, 218, 223, 224, 230, 240, 242, 243, 252, 261, 262, 264, 274, 280, 281, 286, 295, 300, 302, 308, 314, 319, 324, 329

Offset: 0

N. J. A. Sloane, Mar 25 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, 3rd row, 2nd tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 9 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 krn 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,-1,1,0,0,0,1,-1,1,-1).

Cf. A301680.

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,-1,1,0,0,0,1,-1,1,-1},{1,5,10,14,20,26,31,36,41,46,52,58,60},100] (* Paolo Xausa, Nov 15 2023 *)

G.f.: -(2*x^12-3*x^10-4*x^9-6*x^8-10*x^7-11*x^6-10*x^5-11*x^4-8*x^3-6*x^2-4*x-1)/(x^10-x^9+x^8-x^7-x^3+x^2-x+1). - N. J. A. Sloane, Mar 29 2018

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

A301680 Coordination sequence for node of type V2 in "krn" 2-D tiling (or net).

Original entry on oeis.org

1, 4, 9, 15, 20, 26, 32, 36, 40, 46, 52, 56, 62, 68, 72, 76, 82, 88, 92, 98, 104, 108, 112, 118, 124, 128, 134, 140, 144, 148, 154, 160, 164, 170, 176, 180, 184, 190, 196, 200, 206, 212, 216, 220, 226, 232, 236, 242, 248, 252, 256, 262, 268, 272, 278, 284, 288, 292, 298, 304, 308, 314, 320, 324, 328

Offset: 0

N. J. A. Sloane, Mar 25 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, 3rd row, 2nd tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 9 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 krn 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 (1,0,0,0,0,0,1,-1).

Cf. A301678.

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[{1,0,0,0,0,0,1,-1},{1,4,9,15,20,26,32,36,40,46,52,56},100] (* Paolo Xausa, Nov 15 2023 *)

G.f.: -(x^11-x^9-x^8-3*x^7-6*x^6-6*x^5-5*x^4-6*x^3-5*x^2-3*x-1)/(x^8-x^7-x+1). - N. J. A. Sloane, Mar 29 2018

Equivalent conjecture: 7*a(n) = 36*n-2*b(n) for n>3, where b(n>=1) = 4, 1, -2, 2, -1, -4, 0 (continued 7-periodic). - R. J. Mathar, Mar 30 2018

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

A301682 Coordination sequence for node of type V1 in "krg" 2-D tiling (or net).

Original entry on oeis.org

1, 6, 6, 18, 18, 18, 36, 30, 30, 54, 42, 42, 72, 54, 54, 90, 66, 66, 108, 78, 78, 126, 90, 90, 144, 102, 102, 162, 114, 114, 180, 126, 126, 198, 138, 138, 216, 150, 150, 234, 162, 162, 252, 174, 174, 270, 186, 186, 288, 198, 198, 306, 210, 210, 324, 222, 222, 342, 234, 234, 360, 246, 246, 378, 258

Offset: 0

N. J. A. Sloane, Mar 25 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, 2nd row, 2nd tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 10 from the list of 20 2-uniform tilings.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Reticular Chemistry Structure Resource (RCSR), The krg 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. A301684.
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.
Cf. A049347, A099837.

LinearRecurrence[{0,0,2,0,0,-1},{1,6,6,18,18,18,36},100] (* Paolo Xausa, Nov 15 2023 *)

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

a(n) = 2*(7*n + n*A099837(n+3) + 3*A049347(n+2))/3 for n > 0. - Stefano Spezia, Jun 08 2024

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

cp's OEIS Frontend

A301298 Expansion of (1 + 4*x + 4*x^2 + 4*x^3 + x^4)/((1 - x)*(1 - x^3)).

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A301299 Coordination sequence for node of type V1 in "krq" 2-D tiling (or net).

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A301301 Coordination sequence for node of type V2 in "krq" 2-D tiling (or net).

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A301670 Coordination sequence for node of type V1 in "krr" 2-D tiling (or net).

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A301672 Coordination sequence for node of type V2 in "krr" 2-D tiling (or net).

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A301674 Coordination sequence for node of type V1 in "krs" 2-D tiling (or net).

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A301676 Coordination sequence for node of type V2 in "krs" 2-D tiling (or net).

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A301298 Expansion of (1 + 4x + 4x^2 + 4x^3 + x^4)/((1 - x)(1 - x^3)).