A219529 -id:A219529 - cp's OEIS frontend

A301287 Coordination sequence for node of type 3.12.12 in "cph" 2-D tiling (or net).

1, 3, 6, 7, 8, 15, 18, 17, 20, 25, 28, 29, 30, 35, 40, 39, 40, 47, 50, 49, 52, 57, 60, 61, 62, 67, 72, 71, 72, 79, 82, 81, 84, 89, 92, 93, 94, 99, 104, 103, 104, 111, 114, 113, 116, 121, 124, 125, 126, 131, 136, 135, 136, 143, 146, 145, 148, 153, 156, 157, 158

Offset: 0

N. J. A. Sloane, Mar 23 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, first tiling.

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 2 from the list of 20 2-uniform tilings.
Brian Galebach, Enlarged illustration of tiling, suitable for coloring (taken from the web site in 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 cph 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 initial terms
Rémy Sigrist, PARI program for A301287
N. J. A. Sloane, Trunks and branches for determining coordination sequence, central view. Blue = trunks, red = branches, green = twigs.
N. J. A. Sloane, Trunks and branches, a different scan, truncated on right but otherwise shows quadrants I and IV in detail
N. J. A. Sloane, Trunks and branches in first quadrant, in full.
N. J. A. Sloane, Trunks and branches in fourth quadrant, in full.
N. J. A. Sloane, Details of proof (page 1)
N. J. A. Sloane, Details of proof (page 2)
Index entries for linear recurrences with constant coefficients, signature (1,-1,2,-1,1,-1).

Cf. A301289.

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.

Join[{1, 3, 6}, LinearRecurrence[{1, -1, 2, -1, 1, -1}, {7, 8, 15, 18, 17, 20}, 100]] (* Jean-François Alcover, Aug 05 2018 *)

PARI
```
See Links section.
```

G.f. = -(2*x^8-2*x^7-x^6-4*x^5-2*x^4-2*x^3-4*x^2-2*x-1) / ((x^2+1)*(x^2+x+1)*(x-1)^2). N. J. A. Sloane, Mar 28 2018 (This is now a theorem. - N. J. A. Sloane, Apr 05 2018)
Equivalent conjecture: 3*a(n) = 8*n+2*A057078(n+1)+3*A228826(n+2). - R. J. Mathar, Mar 31 2018 (This is now a theorem. - N. J. A. Sloane, Apr 05 2018)
Theorem: G.f. = (1+2*x+4*x^2+2*x^3+2*x^4+4*x^5+1*x^6+2*x^7-2*x^8) / ((1-x)*(1+x^2)*(1-x^3)).
Proof. This follows by applying the coloring book method described in the Goodman-Strauss & Sloane article. The trunks and branches structure is shown in the links, and the details of the proof (by calculating the generating function) are on the next two scanned pages. - N. J. A. Sloane, Apr 05 2018

More terms from Rémy Sigrist, Mar 27 2018

A301289 Coordination sequence for a tetravalent node of type 3.4.3.12 in "cph" 2-D tiling (or net).

Original entry on oeis.org

1, 4, 5, 6, 12, 14, 15, 18, 21, 26, 28, 26, 31, 38, 37, 38, 44, 46, 47, 50, 53, 58, 60, 58, 63, 70, 69, 70, 76, 78, 79, 82, 85, 90, 92, 90, 95, 102, 101, 102, 108, 110, 111, 114, 117, 122, 124, 122, 127, 134, 133, 134, 140, 142, 143, 146, 149, 154, 156, 154

Offset: 0

N. J. A. Sloane, Mar 23 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, first tiling.

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 2 from the list of 20 2-uniform tilings.
Brian Galebach, Enlarged illustration of tiling, suitable for coloring (taken from the web site in 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 cph 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 A301289
N. J. A. Sloane, Trunks and branches for determining coordination sequence, central view. Blue = trunks, red = branches, green = twigs.
N. J. A. Sloane, Trunks and branches, a different scan, truncated on right but otherwise shows quadrants I and IV in detail
N. J. A. Sloane, Page 1 of the proof of the theorem: counts for various classes of nodes [Page 1 will be added as soon as I can find a wide-bed scanner]
N. J. A. Sloane, Page 2 of the proof of the theorem: the corresponding generating functions
Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-4,4,-3,2,-1).

Cf. A301287.

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.

Join[{1, 4}, LinearRecurrence[{2, -3, 4, -4, 4, -3, 2, -1}, {5, 6, 12, 14, 15, 18, 21, 26}, 100]] (* Jean-François Alcover, Aug 05 2018 *)

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

Theorem: G.f. = (1+2*x+4*x^3+3*x^4+6*x^6-4*x^7+6*x^8-2*x^9) / ((1-x)^2*(1+x^2)*(1+x^2+x^4)).
The proof uses the coloring book method described in the Goodman-Strauss & Sloane article. The trunks and branches structure is shown in the first scan. (Not yet added.) The trunks are blue, the branches are red, and the twigs are green. There is mirror symmetry about the Y-axis, and quadrants I and II are essentially identical, as are quadrants III and IV. The counts of the various classes of nodes are given in the second scan, and the corresponding generating functions are in the third scan. Adding up the different terms gives the g.f. stated above. - N. J. A. Sloane, Apr 07 2018
E.g.f.: (6*(3 + (4*exp(x) - 3)*x + 3*sin(x)) - 9*cos(sqrt(3)*x/2)*cosh(x/2) + sqrt(3)*sin(sqrt(3)*x/2)*(8*cosh(x/2) - 5*sinh(x/2)))/9. - Stefano Spezia, Jun 08 2024

More terms from Rémy Sigrist, Mar 27 2018

A301291 Expansion of (x^4+3x^3+x^2+3x+1)/((x^2+1)*(x-1)^2).

Original entry on oeis.org

1, 5, 9, 13, 18, 23, 27, 31, 36, 41, 45, 49, 54, 59, 63, 67, 72, 77, 81, 85, 90, 95, 99, 103, 108, 113, 117, 121, 126, 131, 135, 139, 144, 149, 153, 157, 162, 167, 171, 175, 180, 185, 189, 193, 198, 203, 207, 211, 216, 221, 225, 229, 234, 239, 243, 247, 252

Offset: 0

N. J. A. Sloane, Mar 23 2018

Appears to be coordination sequence for node of type 3^3.4^2 in "krm" 2-D tiling (or net).
Also appears to be coordination sequence for pentavalent node in "krk" 2-D tiling (or net).
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, row 3, first tiling; also p. 66, row 3, first tiling.

Colin Barker, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Numbers 3 and 8 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 krm tiling (or net)
Reticular Chemistry Structure Resource (RCSR), The krk 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,-1).

Cf. A301293.

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.

f:=proc(n) if n=0 then 1
elif (n mod 2) = 0 then 9*n/2
elif (n mod 4) = 1 then 18*(n-1)/4+5
else 18*(n-3)/4+13; fi; end;
s1:=[seq(f(n),n=0..60)];

Join[{1}, LinearRecurrence[{2, -2, 2, -1}, {5, 9, 13, 18}, 60]] (* Jean-François Alcover, Jan 08 2019 *)

Vec((x^4+3*x^3+x^2+3*x+1)/((x^2+1)*(x-1)^2) + O(x^60)) \\ Colin Barker, Mar 23 2018

For explicit formula for a(n) see Maple code.
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n > 4. - Colin Barker, Mar 23 2018
E.g.f.: (2 + 9*x*exp(x) + sin(x))/2. - Stefano Spezia, Jan 31 2023

A301293 Expansion of (x^2+x+1)^2 / ((x^2+1)*(x-1)^2).

Original entry on oeis.org

1, 4, 9, 14, 18, 22, 27, 32, 36, 40, 45, 50, 54, 58, 63, 68, 72, 76, 81, 86, 90, 94, 99, 104, 108, 112, 117, 122, 126, 130, 135, 140, 144, 148, 153, 158, 162, 166, 171, 176, 180, 184, 189, 194, 198, 202, 207, 212, 216, 220, 225, 230, 234, 238, 243, 248, 252

Offset: 0

N. J. A. Sloane, Mar 23 2018

Appears to be coordination sequence for node of type 4^4 in "krm" 2-D tiling (or net).
Also appears to be coordination sequence for tetravalent node in "krk" 2-D tiling (or net).
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, row 3, first tiling; also p. 66, row 3, first tiling.

Colin Barker, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Numbers 3 and 8 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 krm tiling (or net)
Reticular Chemistry Structure Resource (RCSR), The krk 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,-1).

Cf. A301291.

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.

f:=proc(n) if n=0 then 1
elif (n mod 2) = 0 then 9*n/2
elif (n mod 4) = 1 then 18*(n-1)/4+4
else 18*(n-3)/4+14; fi; end;
s1:=[seq(f(n),n=0..60)];

Join[{1}, LinearRecurrence[{2, -2, 2, -1}, {4, 9, 14, 18}, 60]] (* Jean-François Alcover, Jan 08 2019 *)

Vec((x^2+x+1)^2 / ((x^2+1)*(x-1)^2) + O(x^60)) \\ Colin Barker, Mar 23 2018

For explicit formula for a(n) see Maple code.
a(n) = 9*n/2 + (1 - (-1)^n)*i^(n*(n + 1))/4 for n>0, a(0)=1 and i=sqrt(-1). Therefore, for even n>0 a(n) = 9*n/2, otherwise a(n) = 9*n/2 - (-1)^((n-1)/2)/2. - Bruno Berselli, Mar 23 2018
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4. - Colin Barker, Mar 23 2018

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

Original entry on oeis.org

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
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\\ 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

cp's OEIS Frontend

A301287 Coordination sequence for node of type 3.12.12 in "cph" 2-D tiling (or net).

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A301289 Coordination sequence for a tetravalent node of type 3.4.3.12 in "cph" 2-D tiling (or net).

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A301291 Expansion of (x^4+3*x^3+x^2+3*x+1)/((x^2+1)*(x-1)^2).

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A301293 Expansion of (x^2+x+1)^2 / ((x^2+1)*(x-1)^2).

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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).

A301291 Expansion of (x^4+3x^3+x^2+3x+1)/((x^2+1)*(x-1)^2).

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