A301291 -id:A301291 - cp's OEIS frontend

A301686 Coordination sequence for node of type V1 in "krh" 2-D tiling (or net).

1, 5, 7, 14, 17, 19, 28, 29, 31, 42, 41, 43, 56, 53, 55, 70, 65, 67, 84, 77, 79, 98, 89, 91, 112, 101, 103, 126, 113, 115, 140, 125, 127, 154, 137, 139, 168, 149, 151, 182, 161, 163, 196, 173, 175, 210, 185, 187, 224, 197, 199, 238, 209, 211, 252, 221, 223

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, 2nd row, 1st tiling.

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 12 from the list of 20 2-uniform tilings.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
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 A301686
Reticular Chemistry Structure Resource (RCSR), The krh tiling (or net)
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A301688.
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},{1,5,7,14,17,19,28},100] (* Paolo Xausa, Nov 16 2023 *)

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

a(n) = 2*(19*n + n*A099837(n+3) + 9*A049347(n+2)/2)/9 for n > 0. - Stefano Spezia, Jun 08 2024

More terms from Rémy Sigrist, Mar 26 2018

A301688 Coordination sequence for node of type V2 in "krh" 2-D tiling (or net).

Original entry on oeis.org

1, 4, 9, 12, 17, 22, 24, 30, 35, 36, 43, 48, 48, 56, 61, 60, 69, 74, 72, 82, 87, 84, 95, 100, 96, 108, 113, 108, 121, 126, 120, 134, 139, 132, 147, 152, 144, 160, 165, 156, 173, 178, 168, 186, 191, 180, 199, 204, 192, 212, 217, 204, 225, 230, 216, 238, 243

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, 2nd row, 1st tiling.

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 12 from the list of 20 2-uniform tilings.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
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 A301688
Reticular Chemistry Structure Resource (RCSR), The krh tiling (or net)
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A301686.
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},{1,4,9,12,17,22,24},100] (* Paolo Xausa, Nov 16 2023 *)

PARI
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\\ See Links section.
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G.f.: -(-x^6-4*x^5-9*x^4-10*x^3-9*x^2-4*x-1)/(x^6-2*x^3+1). - N. J. A. Sloane, Mar 28 2018

a(n) = 2*(19*n - n*A099837(n+3)/2 - 3*A049347(n+2)/2)/9 for n > 0. - Stefano Spezia, Jun 08 2024

More terms from Rémy Sigrist, Mar 26 2018

A301690 Coordination sequence for node of type V1 in "krf" 2-D tiling (or net).

Original entry on oeis.org

1, 6, 6, 9, 12, 18, 24, 21, 24, 30, 36, 42, 36, 39, 48, 54, 60, 51, 54, 66, 72, 78, 66, 69, 84, 90, 96, 81, 84, 102, 108, 114, 96, 99, 120, 126, 132, 111, 114, 138, 144, 150, 126, 129, 156, 162, 168, 141, 144, 174, 180, 186, 156, 159, 192, 198, 204, 171, 174, 210, 216, 222, 186, 189, 228, 234, 240

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, 1st tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 13 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 krf 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. A301692.

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},{1,6,6,9,12,18,24,21,24,30,36},100] (* Paolo Xausa, Nov 16 2023 *)

G.f.: -(-x^10-6*x^9-6*x^8-9*x^7-12*x^6-16*x^5-12*x^4-9*x^3-6*x^2-6*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

A301692 Coordination sequence for node of type V2 in "krf" 2-D tiling (or net).

Original entry on oeis.org

1, 4, 7, 9, 12, 16, 21, 25, 26, 29, 34, 38, 41, 42, 46, 52, 55, 57, 58, 63, 70, 72, 73, 74, 80, 88, 89, 89, 90, 97, 106, 106, 105, 106, 114, 124, 123, 121, 122, 131, 142, 140, 137, 138, 148, 160, 157, 153, 154, 165, 178, 174, 169, 170, 182, 196, 191, 185, 186, 199, 214, 208, 201, 202, 216, 232, 225

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, 1st tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 13 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 krf 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. A301690.

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},{1,4,7,9,12,16,21,25,26,29,34,38,41,42},100] (* Paolo Xausa, Nov 16 2023 *)

G.f.: -(x^13+2*x^12-3*x^10-5*x^9-8*x^8-11*x^7-13*x^6-14*x^5-12*x^4-9*x^3-7*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

A301694 Expansion of (1 + 5x + 4x^2 + 5x^3 + x^4)/((1 - x)(1 - x^3)).

Original entry on oeis.org

1, 6, 10, 16, 22, 26, 32, 38, 42, 48, 54, 58, 64, 70, 74, 80, 86, 90, 96, 102, 106, 112, 118, 122, 128, 134, 138, 144, 150, 154, 160, 166, 170, 176, 182, 186, 192, 198, 202, 208, 214, 218, 224, 230, 234, 240, 246, 250, 256, 262, 266, 272, 278, 282, 288, 294

Offset: 0

N. J. A. Sloane, Mar 25 2018

Appears to be coordination sequence for node of type V1 in "krd" 2-D tiling (or net). This should be easy to prove by the coloring book method (see link).

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, 1st row, 2nd tiling.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Brian Galebach, Collection of n-Uniform Tilings. See Number 14 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 krd 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. A219529.

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:=[1,6,10,16,22]; [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 6*n-2*Floor((n+1)/3): n in [0..60]]; // Bruno Berselli, Mar 26 2018

CoefficientList[Series[(x^4 + 5 x^3 + 4 x^2 + 5 x + 1) / ((1 - x) (1 - x^3)), {x, 0, 80}], x] (* Vincenzo Librandi, Mar 26 2018 *)

lista(nn) = {x='x+O('x^nn); Vec((x^4+5*x^3+4*x^2+5*x+1)/((1-x)*(1-x^3)))} \\ Altug Alkan, Mar 26 2018

G.f.: (1 + 5*x + 4*x^2 + 5*x^3 + x^4)/((1 - x)*(1 - x^3)).

a(n) = 6*n - 2*floor((n + 1)/3) for n>0, a(0)=1. - Bruno Berselli, Mar 26 2018

A301697 Coordination sequence for node of type V2 in "krj" 2-D tiling (or net).

Original entry on oeis.org

1, 5, 10, 16, 22, 27, 32, 37, 42, 48, 54, 59, 64, 69, 74, 80, 86, 91, 96, 101, 106, 112, 118, 123, 128, 133, 138, 144, 150, 155, 160, 165, 170, 176, 182, 187, 192, 197, 202, 208, 214, 219, 224, 229, 234, 240, 246, 251, 256, 261, 266, 272, 278, 283, 288, 293, 298, 304, 310, 315, 320, 325, 330, 336

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, 2nd row, 3rd tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 17 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 krj 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 (2,-2,2,-2,2,-1).

Cf. A219529.

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[{2,-2,2,-2,2,-1},{1,5,10,16,22,27,32},100] (* Paolo Xausa, Nov 16 2023 *)

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

Equivalent conjecture: 6*a(n) = 32*n -3*A010892(n-1) + A049347(n-1) for n>0. - R. J. Mathar, Nov 26 2018

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

A301708 Coordination sequence for node of type V1 in "krc" 2-D tiling (or net).

Original entry on oeis.org

1, 6, 11, 16, 22, 28, 33, 38, 44, 50, 55, 60, 66, 72, 77, 82, 88, 94, 99, 104, 110, 116, 121, 126, 132, 138, 143, 148, 154, 160, 165, 170, 176, 182, 187, 192, 198, 204, 209, 214, 220, 226, 231, 236, 242, 248, 253, 258, 264, 270, 275, 280, 286, 292, 297, 302

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, 1st tiling.

Rémy Sigrist, Table of n, a(n) for n = 0..1000 (first 100 terms from Davide M. Proserpio)
Brian Galebach, Collection of n-Uniform Tilings. See Number 15 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 krc 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 A301708.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A301710.

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[{2,-2,2,-1},{1,6,11,16,22},100] (* Paolo Xausa, Nov 14 2023 *)

PARI
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\\ See Links section.
```

G.f. = (x^4+4*x^3+x^2+4*x+1)/((x^2+1)*(x-1)^2); for n>0, a(2*t)=11*t, a(4*t+1)=22*t+6, a(4*t+3)=22*t+16. These should be easy to prove by the coloring book method (see link).
Conjecture: a(n) = (i*((-i)^n - i^n) + 22*n) / 4 where i=sqrt(-1). - Colin Barker, Apr 07 2018
E.g.f.: (2 + 11*exp(x)*x + sin(x))/2. - Stefano Spezia, Jun 08 2024

More terms from Davide M. Proserpio, Mar 28 2018

A301710 Coordination sequence for node of type V2 in "krc" 2-D tiling (or net).

Original entry on oeis.org

1, 5, 11, 17, 22, 27, 33, 39, 44, 49, 55, 61, 66, 71, 77, 83, 88, 93, 99, 105, 110, 115, 121, 127, 132, 137, 143, 149, 154, 159, 165, 171, 176, 181, 187, 193, 198, 203, 209, 215, 220, 225, 231, 237, 242, 247, 253, 259, 264, 269, 275, 281, 286, 291, 297, 303

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, 1st tiling.

Rémy Sigrist, Table of n, a(n) for n = 0..1000 (first 100 terms from Davide M. Proserpio)
Brian Galebach, Collection of n-Uniform Tilings. See Number 15 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 krc 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 A301710.
Index entries for linear recurrences with constant coefficients, signature (2, -2, 2, -1).

Cf. A301708.

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[{2,-2,2,-1},{1,5,11,17,22},100] (* Paolo Xausa, Nov 14 2023 *)

PARI
```
See Links section.
```

G.f.: (x^4+3*x^3+3*x^2+3*x+1)/((x^2+1)*(x-1)^2); for n>0, a(2*t)=11*t, a(4*t+1)=22*t+5, a(4*t+3)=22*t+17. These should be easy to prove by the coloring book method (see link).

a(n) = ((-i)^(1+n) + i^(1+n) + 22*n) / 4 for n>0, where i=sqrt(-1) (conjectured). - Colin Barker, Apr 07 2018

More terms from Davide M. Proserpio, Mar 28 2018

A301712 Coordination sequence for node of type V1 in "usm" 2-D tiling (or net).

Original entry on oeis.org

1, 5, 10, 16, 22, 27, 33, 38, 43, 49, 53, 59, 65, 70, 77, 81, 86, 92, 96, 103, 108, 113, 120, 124, 130, 135, 139, 146, 151, 157, 163, 167, 173, 178, 183, 189, 194, 200, 206, 211, 216, 221, 226, 232, 238, 243, 249, 254, 259, 265, 269, 275, 281, 286, 293, 297, 302, 308, 312, 319, 324, 329, 336, 340

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, 2nd row, 2nd tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 16 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 usm 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,-1,2,-1,0,0,1,-1).

Cf. A301714.

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,-1,2,-1,0,0,1,-1},{1,5,10,16,22,27,33,38,43,49,53},100] (* Paolo Xausa, Nov 16 2023 *)

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

Equivalent conjecture: 5*a(n) = 27*n -b(n) -5*A014017(n-2) for n>0, where b(n) = 2,-1,1,-2,0 (5-periodic) for n>=1. - R. J. Mathar, Mar 30 2018

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

A301714 Coordination sequence for node of type V2 in "usm" 2-D tiling (or net).

Original entry on oeis.org

1, 5, 12, 16, 21, 28, 31, 38, 44, 47, 56, 59, 64, 72, 73, 82, 87, 90, 100, 101, 108, 115, 116, 126, 129, 134, 143, 144, 152, 157, 160, 169, 172, 178, 185, 188, 195, 200, 204, 211, 216, 221, 228, 232, 237, 244, 247, 254, 260, 263, 272, 275, 280, 288, 289, 298, 303, 306, 316, 317, 324, 331, 332, 342

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, 2nd row, 2nd tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 16 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 usm 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,-1,2,-1,0,0,1,-1).

Cf. A301712.

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,-1,2,-1,0,0,1,-1},{1,5,12,16,21,28,31,38,44,47,56},100] (* Paolo Xausa, Nov 16 2023 *)

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

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

cp's OEIS Frontend

A301686 Coordination sequence for node of type V1 in "krh" 2-D tiling (or net).

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A301688 Coordination sequence for node of type V2 in "krh" 2-D tiling (or net).

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A301690 Coordination sequence for node of type V1 in "krf" 2-D tiling (or net).

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A301692 Coordination sequence for node of type V2 in "krf" 2-D tiling (or net).

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A301694 Expansion of (1 + 5*x + 4*x^2 + 5*x^3 + x^4)/((1 - x)*(1 - x^3)).

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A301697 Coordination sequence for node of type V2 in "krj" 2-D tiling (or net).

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A301708 Coordination sequence for node of type V1 in "krc" 2-D tiling (or net).

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A301694 Expansion of (1 + 5x + 4x^2 + 5x^3 + x^4)/((1 - x)(1 - x^3)).