A072154 -id:A072154 - cp's OEIS frontend

A299258 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.6.12 2D tiling (cf. A072154).

1, 5, 13, 25, 41, 62, 89, 121, 157, 197, 242, 293, 349, 409, 473, 542, 617, 697, 781, 869, 962, 1061, 1165, 1273, 1385, 1502, 1625, 1753, 1885, 2021, 2162, 2309, 2461, 2617, 2777, 2942, 3113, 3289, 3469, 3653, 3842, 4037, 4237, 4441, 4649, 4862, 5081, 5305, 5533, 5765, 6002, 6245, 6493, 6745

Offset: 0

N. J. A. Sloane, Feb 07 2018

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #23.

Colin Barker, Table of n, a(n) for n = 0..1000
Reticular Chemistry Structure Resource (RCSR), The fst tiling (or net)
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A072154.
Partial sums: A299264.
The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e:  A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.

LinearRecurrence[{2,-1,0,0,1,-2,1},{1,5,13,25,41,62,89,121},60] (* Harvey P. Dale, Mar 14 2023 *)

Vec((1 + x)^3*(1 - x + x^2)*(1 + x + x^2) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Feb 09 2018

G.f.: (x^2+x+1)*(x^2-x+1)*(x+1)^3 / ((x^4+x^3+x^2+x+1)*(1-x)^3).
a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) for n>7. - Colin Barker, Feb 09 2018
a(n) ~ 12*n^2/5. - Stefano Spezia, Jun 06 2024

A265078 Partial sums of A072154.

Original entry on oeis.org

1, 4, 9, 16, 25, 37, 52, 69, 88, 109, 133, 160, 189, 220, 253, 289, 328, 369, 412, 457, 505, 556, 609, 664, 721, 781, 844, 909, 976, 1045, 1117, 1192, 1269, 1348, 1429, 1513, 1600, 1689, 1780, 1873, 1969, 2068, 2169, 2272, 2377, 2485, 2596, 2709, 2824, 2941, 3061, 3184, 3309, 3436, 3565, 3697

Offset: 0

N. J. A. Sloane, Dec 29 2015

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

Cf. A072154.

I:=[1,4,9,16,25,37,52]; [n le 7 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-5)-2*Self(n-6)+Self(n-7): n in [1..60]]; // Vincenzo Librandi, Jan 01 2016

LinearRecurrence[{2, -1, 0, 0, 1, -2, 1}, {1, 4, 9, 16, 25, 37, 52}, 60] (* Vincenzo Librandi, Jan 01 2016 *)

Vec((1+x)^2*(1-x+x^2)*(1+x+x^2)/((1-x)^3*(1+x+x^2+x^3+x^4)) + O(x^70)) \\ Colin Barker, Jan 01 2016

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

a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7). - Vincenzo Librandi, Jan 01 2016

A250120 Coordination sequence for planar net 3.3.3.3.6 (also called the fsz net).

Original entry on oeis.org

1, 5, 9, 15, 19, 24, 29, 33, 39, 43, 48, 53, 57, 63, 67, 72, 77, 81, 87, 91, 96, 101, 105, 111, 115, 120, 125, 129, 135, 139, 144, 149, 153, 159, 163, 168, 173, 177, 183, 187, 192, 197, 201, 207, 211, 216, 221, 225, 231, 235

Offset: 0

N. J. A. Sloane, Nov 23 2014

There are eleven uniform (or Archimedean) tilings (or planar nets), with vertex symbols 3^6, 3^4.6, 3^3.4^2, 3^2.4.3.4, 4^4, 3.4.6.4, 3.6.3.6, 6^3, 3.12^2, 4.6.12, and 4.8^2. Grünbaum and Shephard (1987) is the best reference.
a(n) is the number of vertices at graph distance n from any fixed vertex.
The Mathematica notebook can compute 30 or 40 iterations, and colors them with period 5. You could also change out images if you want to. These graphs are better for analyzing 5-iteration chunks of the pattern. You can see that under iteration all fragments of the circumferences are preserved in shape and translated outwards a distance approximately sqrt(21) (relative to small triangle edge), the length of a long diagonal of larger rhombus unit cell. The conjectured recurrence should follow from an analysis of how new pieces occur in between the translated pieces. - Bradley Klee, Nov 26 2014

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Fig. 2.1.5, p. 63.
Marjorie Senechal, Quasicrystals and geometry, Cambridge University Press, Cambridge, 1995, Fig. 1.10, Section 1.3, pp. 13-16.

Stefano Spezia, Table of n, a(n) for n = 0..10000 (first 512 terms from Maurizio Paolini)
Darrah Chavey, Illustration of a(0)-a(12)
Jean-Guillaume Eon, Symmetry and Topology: The 11 Uninodal Planar Nets Revisited, Symmetry, 10 (2018), 13 pages, doi:10.3390/sym10020035. See Section 9.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers [updated version May 09 2020]
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.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Tom Karzes, Tiling Coordination Sequences
Bradley Klee, Illustration of a(0)-a(7).
Bradley Klee, Mathematica notebook for A250120
Maurizio Paolini, C program for A250120
Reticular Chemistry Structure Resource, fsz
N. J. A. Sloane, Initial hand-drawn illustration of a(0)-a(5)
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Grünbaum and Shephard (1977)]
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part 1, Part 2, Slides. (Mentions this sequence)
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-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).

For partial sums of the present sequence, see A250121.

/* Comments on the C program (see link) from Maurizio Paolini, Nov 23 2014: Basically what I do is deform the net onto the integral lattice, connect nodes aligned either horizontally, vertically or diagonally from northeast to southwest, marking as UNREACHABLE the nodes with coordinates (i, j) satisfying i + 2*j = 0 mod 7. Then the code computes the distance from each node to the central node of the grid. */

CoefficientList[Series[(x^2+x+1)(x^4+3x^3+3x+1)/((x^4+x^3+x^2+x+1)(x-1)^2), {x, 0, 80}], x] (* or *) LinearRecurrence[{1, 0, 0, 0, 1, -1}, {1, 5, 9, 15, 19, 24, 29}, 60] (* Harvey P. Dale, May 05 2018 *)

Based on the computations of Darrah Chavey, Bradley Klee, and Maurizio Paolini, there is a strong conjecture that the first differences of this sequence are 4, 4, 6, 4, 5, 5, 4, 6, 4, 5, 5, 4, 6, 4, 5, 5, ..., that is, 4 followed by (4,6,4,5,5)  repeated.
This would imply that the sequence satisfies the recurrence:
for n > 2, a(n) = a(n-1) + { n == 0,3 (mod 5), 4; n == 4 (mod 5), 6; n == 1,2 (mod 5), 5 }
(from Darrah Chavey)
and has generating function
(x^2+x+1)*(x^4+3*x^3+3*x+1)/((x^4+x^3+x^2+x+1)*(x-1)^2)
(from N. J. A. Sloane).
All the above conjectures are true - for proof see link to my article with Chaim Goodman-Strauss. - N. J. A. Sloane, Jan 14 2018; link added Mar 26 2018
a(n) ~ 24*n/5. - Stefano Spezia, May 08 2022
For n>0, a(n) = 2*(12*n + sqrt(1+2/sqrt(5))*sin(4*Pi*n/5) - sqrt(1-2/sqrt(5))*sin(2*Pi*n/5))/5. - Natalia L. Skirrow, Apr 13 2025

a(6)-a(10) from Bradley Klee, Nov 23 2014

a(11)-a(49) from Maurizio Paolini, Nov 23 2014

A008574 a(0) = 1, thereafter a(n) = 4n.

Original entry on oeis.org

1, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232

Offset: 0

N. J. A. Sloane; entry revised Aug 24 2014

Number of squares on the perimeter of an (n+1) X (n+1) board. - Jon Perry, Jul 27 2003
Coordination sequence for square lattice (or equivalently the planar net 4.4.4.4).
Apparently also the coordination sequence for the planar net 3.4.6.4. - Darrah Chavey, Nov 23 2014
From N. J. A. Sloane, Nov 26 2014: (Start)
I confirm that this is indeed the coordination sequence for the planar net 3.4.6.4. The points at graph distance n from a fixed point in this net essentially lie on a hexagon (see illustration in link).
If n = 3k, k >= 1, there are 2k + 1 nodes on each edge of the hexagon. This counts the corners of the hexagon twice, so the number of points in the shell is 6(2k + 1) - 6 = 4n. If n = 3k + 1, the numbers of points on the six edges of the hexagon are 2k + 2 (4 times) and 2k + 1 (twice), for a total of 12k + 10 - 6 = 4n. If n = 3k + 2 the numbers are 2k + 2 (4 times) and 2k + 3 twice, and again we get 4n points.
The illustration shows shells 0 through 12, as well as the hexagons formed by shells 9 (green, 36 points), 10 (black, 40 points), 11 (red, 44 points), and 12 (blue, 48 points).
It is clear from the net that this period-3 structure continues forever, and establishes the theorem.
In contrast, for the 4.4.4.4 planar net, the successive shells are diamonds instead of hexagons, and again the n-th shell (n > 0) contains 4n points.
Of course the two nets are very different, since 4.4.4.4 has the symmetry of the square, while 3.4.6.4 has only mirror symmetry (with respect to a point), and has the symmetry of a regular hexagon with respect to the center of any of the 12-gons. (End)
Also the coordination sequence for a 6.6.6.6 point in the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6}, see A265045, A265046. - N. J. A. Sloane, Dec 27 2015
Also the coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_4].
Susceptibility series H_1 for 2-dimensional Ising model (divided by 2).
Also the Engel expansion of exp^(1/4); cf. A006784 for the Engel expansion definition. - Benoit Cloitre, Mar 03 2002
This sequence differs from A008586, multiples of 4, only in its initial term. - Alonso del Arte, Apr 14 2011
Number of 2 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00,0), (00;1) and (10;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2 and j1 < j2 and these elements are in same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004
Central terms of the triangle in A118013. - Reinhard Zumkeller, Apr 10 2006
Also the coordination sequence for the htb net. - N. J. A. Sloane, Mar 31 2018
This is almost certainly also the coordination sequence for Dual(3.3.4.3.4) with respect to a tetravalent node. - Tom Karzes, Apr 01 2020
Minimal number of segments (equivalently, corners) in a rook circuit of a 2n X 2n board (maximal number is A085622). - Ruediger Jehn, Jan 02 2021

			From _Omar E. Pol_, Aug 20 2011 (Start):
Illustration of initial terms as perimeters of squares (cf. Perry's comment above):
.                                         o o o o o o
.                             o o o o o   o         o
.                   o o o o   o       o   o         o
.           o o o   o     o   o       o   o         o
.     o o   o   o   o     o   o       o   o         o
. o   o o   o o o   o o o o   o o o o o   o o o o o o
.
. 1    4      8        12         16           20
(End)

T. D. Noe, Table of n, a(n) for n = 0..1000
Joerg Arndt, The 3.4.6.4 net
Matthias Beck and Serkan Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
Pierre de la Harpe, On the prehistory of growth of groups, arXiv:2106.02499 [math.GR], 2021.
Jean-Guillaume Eon, Symmetry and Topology: The 11 Uninodal Planar Nets Revisited, Symmetry, 10 (2018), 13 pages, doi:10.3390/sym10020035. See Section 7.
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 4.2 p. 20.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189.
D. Hansel et al., Analytical properties of the anisotropic cubic Ising model, J. Stat. Phys., 48 (1987), 69-80.
Tom Karzes, Tiling Coordination Sequences
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Reticular Chemistry Structure Resource, sql and htb
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
N. J. A. Sloane, Illustration of points in shells 0 through 12 of the 3.4.6.4 planar net (see Comments for discussion)
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001844 (partial sums), A008586, A054275, A054410, A054389, A054764.
Convolution square of A040000.
Row sums of A130323 and A131032.
List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579(3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529(3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).
List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.
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.
See also A265045, A265046.

a008574 0 = 1; a008574 n = 4 * n
a008574_list = 1 : [4, 8 ..]  -- Reinhard Zumkeller, Apr 16 2015

f[0] = 1; f[n_] := 4 n; Array[f, 59, 0] (* or *)
CoefficientList[ Series[(1 + x)^2/(1 - x)^2, {x, 0, 58}], x] (* Robert G. Wilson v, Jan 02 2011 *)
Join[{1},Range[4,232,4]] (* Harvey P. Dale, Aug 19 2011 *)
a[ n_] := 4 n + Boole[n == 0]; (* Michael Somos, Jan 07 2019 *)

{a(n) = 4*n + !n}; /* Michael Somos, Apr 16 2007 */

Binomial transform is A000337 (dropping the 0 there). - Paul Barry, Jul 21 2003
Euler transform of length 2 sequence [4, -2]. - Michael Somos, Apr 16 2007
G.f.: ((1 + x) / (1 - x))^2. E.g.f.: 1 + 4*x*exp(x). - Michael Somos, Apr 16 2007
a(-n) = -a(n) unless n = 0. - Michael Somos, Apr 16 2007
G.f.: exp(4*atanh(x)). - Jaume Oliver Lafont, Oct 20 2009
a(n) = a(n-1) + 4, n > 1. - Vincenzo Librandi, Dec 31 2010
a(n) = A005408(n-1) + A005408(n), n > 1. - Ivan N. Ianakiev, Jul 16 2012
a(n) = 4*n = A008586(n), n >= 1. - Tom Karzes, Apr 01 2020

A008486 Expansion of (1 + x + x^2)/(1 - x)^2.

Original entry on oeis.org

1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174, 177, 180, 183, 186

Offset: 0

N. J. A. Sloane

Also the Engel expansion of exp^(1/3); cf. A006784 for the Engel expansion definition. - Benoit Cloitre, Mar 03 2002
Coordination sequence for planar net 6^3 (the graphite net, or the graphene crystal) - that is, the number of atoms at graph distance n from any fixed atom. Also for the hcb or honeycomb net. - N. J. A. Sloane, Jan 06 2013, Mar 31 2018
Coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_3].
Conjecture: This is also the maximum number of edges possible in a planar simple graph with n+2 vertices. - Dmitry Kamenetsky, Jun 29 2008
The conjecture is correct. Proof: For n=0 the theorem holds, the maximum planar graph has n+2=2 vertices and 1 edge. Now suppose that we have a connected planar graph with at least 3 vertices. If it contains a face that is not a triangle, we can add an edge that divides this face into two without breaking its planarity. Hence all maximum planar graphs are triangulations. Euler's formula for planar graphs states that in any planar simple graph with V vertices, E edges and F faces we have V+F-E=2. If all faces are triangles, then F=2E/3, which gives us E=3V-6. Hence for n>0 each maximum planar simple graph with n+2 vertices has 3n edges. - Michal Forisek, Apr 23 2009
a(n) = sum of natural numbers m such that n - 1 <= m <= n + 1. Generalization: If a(n,k) = sum of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = (k + n)*(k + n + 1)/2 = A000217(k+n) for 0 <= n <= k, a(n,k) = a(n-1,k) +2k + 1 = ((k + n - 1)*(k + n)/2) + 2k + 1 = A000217(k+n-1) +2k +1 for n >= k + 1 (see e.g. A008486). - Jaroslav Krizek, Nov 18 2009
a(n) = partial sums of A158799(n). Partial sums of a(n) = A005448(n). - Jaroslav Krizek, Dec 06 2009
Integers n dividing a(n) = a(n-1) - a(n-2) with initial conditions a(0)=0, a(1)=1 (see A128834 with offset 0). - Thomas M. Bridge, Nov 03 2013
a(n) is conjectured to be the number of polygons added after n iterations of the polygon expansions (type A, B, C, D & E) shown in the Ngaokrajang link. The patterns are supposed to become the planar Archimedean net 3.3.3.3.3.3, 3.6.3.6, 3.12.12, 3.3.3.3.6 and 4.6.12 respectively when n - > infinity. - Kival Ngaokrajang, Dec 28 2014
Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I. - Ray Chandler, Nov 21 2016
Conjecture: let m = n + 2, p is the polyhedron formed by the convex hull of m points, q is the number of quadrilateral faces of p (see the Wikipedia link below), and f(m) = a(n) - q. Then f(m) would be the solution of the Thompson problem for all m in 3-space. - Sergey Pavlov, Feb 03 2017
Also, sequence defined by a(0)=1, a(1)=3, c(0)=2, c(1)=4; and thereafter a(n) = c(n-1) + c(n-2), and c consists of the numbers missing from a (see A001651). - Ivan Neretin, Mar 28 2017

			G.f. = 1 + 3*x + 6*x^2 + 9*x^3 + 12*x^4 + 15*x^5 + 18*x^6 + 21*x^7 + 24*x^8 + ...
From _Omar E. Pol_, Aug 20 2011: (Start)
Illustration of initial terms as triangles:
.                                              o
.                                 o           o o
.                      o         o o         o   o
.             o       o o       o   o       o     o
.      o     o o     o   o     o     o     o       o
. o   o o   o o o   o o o o   o o o o o   o o o o o o
.
. 1    3      6        9          12           15
(End)

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 158.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
David Applegate, The movie version
M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
Jean-Guillaume Eon, Algebraic determination of generating functions for coordination sequences in crystal structures, Acta Cryst. A58 (2002), 47-53.
Jean-Guillaume Eon, Symmetry and Topology: The 11 Uninodal Planar Nets Revisited, Symmetry, 10 (2018), 13 pages, doi:10.3390/sym10020035. See Section 5.
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004. (See Table 5.)
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 4.3 p. 20.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Tom Karzes, Tiling Coordination Sequences
Kival Ngaokrajang, Illustration of polygon expansions (planar net)
Reticular Chemistry Structure Resource, hcb
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
University of Manchester, Graphene
Wikipedia, Thomson problem
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Partial sums give A005448.
List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574(4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579(3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529(3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).
List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.
Cf. A000217, A001651, A005448, A006784, A008585, A022424, A113062, A128834, A139250, A158799.

a008486 0 = 1; a008486 n = 3 * n
a008486_list = 1 : [3, 6 ..]  -- Reinhard Zumkeller, Apr 17 2015

[0^n+3*n: n in [0..90] ]; // Vincenzo Librandi, Aug 21 2011

CoefficientList[Series[(1 + x + x^2) / (1 - x)^2, {x, 0, 80}], x] (* Vincenzo Librandi, Nov 23 2014 *)
a[ n_] := If[ n == 0, 1, 3 n]; (* Michael Somos, Apr 17 2015 *)

{a(n) = if( n==0, 1, 3 * n)}; /* Michael Somos, May 05 2015 */

a(0) = 1; a(n) = 3*n = A008585(n), n >= 1.
Euler transform of length 3 sequence [3, 0, -1]. - Michael Somos, Aug 04 2009
a(n) = a(n-1) + 3 for n >= 2. - Jaroslav Krizek, Nov 18 2009
a(n) = 0^n + 3*n. - Vincenzo Librandi, Aug 21 2011
a(n) = -a(-n) unless n = 0. - Michael Somos, May 05 2015
E.g.f.: 1 + 3*exp(x)*x. - Stefano Spezia, Aug 07 2022

A008458 Coordination sequence for hexagonal lattice.

Original entry on oeis.org

1, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 330, 336, 342, 348

Offset: 0

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice. It is also the planar net 3.3.3.3.3.3.
Coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_6].
Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 20 ).
Also the Engel expansion of exp^(1/6); cf. A006784 for the Engel expansion definition. - Benoit Cloitre, Mar 03 2002
Numbers k such that k+floor(k/2) | k*floor(k/2). - Wesley Ivan Hurt, Dec 01 2020

			From _Omar E. Pol_, Aug 20 2011: (Start)
Illustration of initial terms:
.                                             o o o o o
.                            o o o o         o         o
.               o o o       o       o       o           o
.      o o     o     o     o         o     o             o
. o   o   o   o       o   o           o   o               o
.      o o     o     o     o         o     o             o
. 1             o o o       o       o       o           o
.       6                    o o o o         o         o
.                 12                          o o o o o
.                               18
.                                                 24
(End)
G.f. = 1 + 6*x + 12*x^2 + 18*x^3 + 24*x^4 + 30*x^5 + 36*x^6 + 42*x^7 + 48*x^8 + 54*x^9 + ...

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
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 4.1 p. 19.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Tom Karzes, Tiling Coordination Sequences
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Reticular Chemistry Structure Resource, hxl
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))
William A. Stein, The modular forms database
Index entries for sequences related to A2 = hexagonal = triangular lattice
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index entries for coordination sequences

Essentially the same as A008588.
List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574(4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579(3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529(3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).
List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.
Cf. A032528. - Omar E. Pol, Aug 20 2011
Cf. A048477 (binomial Transf.)

[0^n+6*n: n in [0..60] ]; // Vincenzo Librandi, Aug 21 2011

Maple
```
1, seq(6*n, n=1..65);
```

Join[{1},6*Range[60]] (* Harvey P. Dale, Jul 21 2013 *)
a[ n_] := Boole[n == 0] + 6 n; (* Michael Somos, May 21 2015 *)

makelist(if n=0 then 1 else 6*n,n,0,65); /* Martin Ettl, Nov 12 2012 */

PARI
```
{a(n) = 6*n + (!n)};
```

[6*n+int(n==0) for n in range(66)] # G. C. Greubel, May 25 2023

G.f.: (1 + 4*x + x^2)/(1 - x)^2.
a(n) = A003215(n) - A003215(n-1), n > 0.
Equals binomial transform of [1, 5, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Jul 08 2008
G.f.: Hypergeometric2F1([3,-2], [1], -x/(1-x)). - Paul Barry, Sep 18 2008
a(n) = 0^n + 6*n. - Vincenzo Librandi, Aug 21 2011
n*a(1) + (n-1)*a(2) + (n-2)*a(3) + ... + 2*a(n-1) + a(n) = n^3. - Warren Breslow, Oct 28 2013
E.g.f.: 1 + 6*x*exp(x). - Stefano Spezia, Jun 26 2022

A219529 Coordination sequence for 3.3.4.3.4 Archimedean tiling.

Original entry on oeis.org

1, 5, 11, 16, 21, 27, 32, 37, 43, 48, 53, 59, 64, 69, 75, 80, 85, 91, 96, 101, 107, 112, 117, 123, 128, 133, 139, 144, 149, 155, 160, 165, 171, 176, 181, 187, 192, 197, 203, 208, 213, 219, 224, 229, 235, 240, 245, 251, 256, 261, 267, 272, 277, 283, 288, 293, 299

Offset: 0

Allan C. Wechsler, Nov 21 2012

a(n) is the number of vertices of the 3.3.4.3.4 tiling (which has three triangles and two squares, in the given cyclic order, meeting at each vertex) whose shortest path connecting them to a given origin vertex contains n edges.
This is the dual tiling to the Cairo tiling (cf. A296368). - N. J. A. Sloane, Nov 02 2018
First few terms provided by Allan C. Wechsler; Fred Lunnon and Fred Helenius gave the next few; Fred Lunnon suggested that the recurrence was a(n+3) = a(n) + 16 for n > 1. [This conjecture is true - see the CGS-NJAS link for a proof. - N. J. A. Sloane, Dec 31 2017]
Appears also to be coordination sequence for node of type V2 in "krd" 2-D tiling (or net). This should be easy to prove by the coloring book method (see link). - N. J. A. Sloane, Mar 25 2018
Appears also to be coordination sequence for node of type V1 in "krj" 2-D tiling (or net). This also should be easy to prove by the coloring book method (see link). - N. J. A. Sloane, Mar 26 2018
First differences of A301696. - Klaus Purath, May 23 2020

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, also 2nd row, third tiling.

Joseph Myers, Table of n, a(n) for n = 0..1000
Giedrius Alkauskas, Colouring tiles in an isohedral tiling: automaton, defects and grain boundaries, arXiv:2301.10975 [math.CO], 2023.
Brian Galebach, Collection of n-Uniform Tilings. See Numbers 14 and 17 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 on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.
Chaim Goodman-Strauss and N. J. A. Sloane, Trunks and branches coloring (taken from preceding reference)
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Tom Karzes, Tiling Coordination Sequences
Kival Ngaokrajang, Illustration of initial terms
Reticular Chemistry Structure Resource, tts
Reticular Chemistry Structure Resource (RCSR), The krd tiling (or net)
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.
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
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 (1,0,1,-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).
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. A296368, A301694, A301697.

-- Very slow, could certainly be accelerated.  SST stands for Snub Square Tiling.
setUnion [] l2 = l2
setUnion (a:rst) l2 = if (elem a l2) then doRest else (a:doRest)
  where doRest = setUnion rst l2
setDifference [] l2 = []
setDifference (a:rst) l2 = if (elem a l2) then doRest else (a:doRest)
  where doRest = setDifference rst l2
adjust k = (if (even k) then 1 else -1)
weirdAdjacent (x,y) = (x+(adjust y),y+(adjust x))
sstAdjacents (x,y) = [(x+1,y),(x-1,y),(x,y+1),(x,y-1),(weirdAdjacent (x,y))]
sstNeighbors core = foldl setUnion core (map sstAdjacents core)
sstGlob n core = if (n == 0) then core else (sstGlob (n-1) (sstNeighbors core))
sstHalo core = setDifference (sstNeighbors core) core
origin = [(0,0)]
a219529 n = length (sstHalo (sstGlob (n-1) origin))
-- Allan C. Wechsler, Nov 30 2012

A219529:= n -> `if`(n=0, 1, (16*n +1 - `mod`(n+1,3))/3);
seq(A219529(n), n = 0..60); # G. C. Greubel, May 27 2020

Join[{1}, LinearRecurrence[{1,0,1,-1}, {5,11,16,21}, 60]] (* Jean-François Alcover, Dec 13 2018 *)
Table[If[n==0, 1, (16*n +1 - Mod[n+1, 3])/3], {n, 0, 60}] (* G. C. Greubel, May 27 2020 *)
CoefficientList[Series[(x+1)^4/((x^2+x+1)(x-1)^2),{x,0,70}],x] (* Harvey P. Dale, Jul 03 2021 *)

[1]+[(16*n+1 -(n+1)%3)/3 for n in (1..60)] # G. C. Greubel, May 27 2020

Conjectured to be a(n) = floor((16n+1)/3) for n>0; a(0) = 1; this is a consequence of the suggested recurrence due to Lunnon (see comments). [This conjecture is true - see the CGS-NJAS link in A296368 for a proof. - N. J. A. Sloane, Dec 31 2017]
G.f.: (x+1)^4/((x^2+x+1)*(x-1)^2). - N. J. A. Sloane, Feb 07 2018
From G. C. Greubel, May 27 2020: (Start)
a(n) = (16*n - ChebyshevU(n-1, -1/2))/3 for n>0 with a(0)=1.
a(n) = (A008598(n) - A049347(n-1))/3 for n >0 with a(0)=1. (End)

Corrected attributions and epistemological status in Comments; provided slow Haskell code - Allan C. Wechsler, Nov 30 2012

Extended by Joseph Myers, Dec 04 2014

A008576 Coordination sequence for planar net 4.8.8.

Original entry on oeis.org

1, 3, 5, 8, 11, 13, 16, 19, 21, 24, 27, 29, 32, 35, 37, 40, 43, 45, 48, 51, 53, 56, 59, 61, 64, 67, 69, 72, 75, 77, 80, 83, 85, 88, 91, 93, 96, 99, 101, 104, 107, 109, 112, 115, 117, 120, 123, 125, 128, 131, 133

Offset: 0

N. J. A. Sloane

Also, growth series for the affine Coxeter (or Weyl) groups B_2. - N. J. A. Sloane, Jan 11 2016

N. Bourbaki, Groupes et Algèbres de Lie, Chap. 4, 5 and 6, Hermann, Paris, 1968. See Chap. VI, Section 4, Problem 10b, page 231, W_a(t).
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.

T. D. Noe, Table of n, a(n) for n = 0..1000
Agnes Azzolino, Regular and Semi-Regular Tessellation Paper, 2011
Agnes Azzolino, Illustration of 4.8.8 tiling [From previous link]
Jillian Cervantes and Pamela E. Harris, (t,r) Broadcast Domination Numbers and Densities of the Truncated Square Tiling Graph, arXiv:2408.13331 [math.CO], 2024. See p. 8.
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 4.5 p. 22.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Tom Karzes, Tiling Coordination Sequences
W. M. Meier and H. J. Moeck, Topology of 3-D 4-connected nets ..., J. Solid State Chem 27 1979 349-355, esp. p. 351.
Reticular Chemistry Structure Resource, fes
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 (1,0,1,-1).

Cf. A042965, A102283.
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).
For partial sums see A008577.
The growth series for the finite Coxeter (or Weyl) groups B_3 through B_12 are A161696-A161699, A161716, A161717, A161733, A161755, A161776, A161858. These are all rows of A128084. The growth series for the affine Coxeter (or Weyl) groups B_2 through B_12 are A008576, A008137, A267167-A267175.

if n mod 3 = 0 then 8*n/3 elif n mod 3 = 1 then 8*(n-1)/3+3 else 8*(n-2)/3+5 fi;

cspn[n_]:=Module[{c=Mod[n,3]},Which[c==0,(8n)/3,c==1,(8(n-1))/3+3,True,(8(n-2))/3+5]]; Join[{1},Array[cspn,50]] (* or *) Join[{1}, LinearRecurrence[ {1,0,1,-1},{3,5,8,11},50]] (* Harvey P. Dale, Nov 24 2011 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,1,0,1]^n*[1;3;5;8])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

G.f.: ((1+x)^2*(1+x^2))/((1-x)^2*(1+x+x^2)). - Ralf Stephan, Apr 24 2004
a(0)=1, a(1)=3, a(2)=5, a(3)=8, a(4)=11, a(n) = a(n-1) + a(n-3) - a(n-4). - Harvey P. Dale, Nov 24 2011
a(0)=1; thereafter a(3k)=8k, a(3k+1)=8k+3, a(3k+2)=8k+5. - N. J. A. Sloane, Dec 22 2015
The above g.f. and recurrence were originally empirical observations, but I now have a proof (details will be added later). This also justifies the Maple and Mma programs and the b-file. - N. J. A. Sloane, Dec 22 2015
Sum of alternate terms of A042965 (numbers not congruent to 2 mod 4), such that A042965(n) = A042965(n+1) + A042965(n-1). - Gary W. Adamson, Sep 12 2007
a(n) = (2/9)*(12*n + (3/2)*A102283(n)) for n > 0. - Stefano Spezia, Aug 07 2022

A265036 Coordination sequence of 2-uniform tiling {3.4.6.4, 4.6.12} with respect to a point of type 3.4.6.4.

Original entry on oeis.org

1, 4, 6, 7, 10, 14, 20, 24, 24, 23, 26, 34, 42, 44, 40, 37, 42, 54, 64, 64, 56, 51, 58, 74, 86, 84, 72, 65, 74, 94, 108, 104, 88, 79, 90, 114, 130, 124, 104, 93, 106, 134, 152, 144, 120, 107, 122, 154, 174, 164, 136, 121, 138, 174, 196, 184, 152, 135, 154, 194, 218

Offset: 0

N. J. A. Sloane, Dec 12 2015

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)
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
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)
Index entries for linear recurrences with constant coefficients, signature (4,-8,10,-8,4,-1).

See A265035 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[{4,-8,10,-8,4,-1},{1,4,6,7,10,14,20,24,24,23},100] (* Paolo Xausa, Nov 15 2023 *)

Based on the b-file, the g.f. appears to be (-2*x^9+6*x^8-8*x^7+7*x^6-2*x^5-2*x^4+5*x^3-2*x^2+1) / (x^6-4*x^5+8*x^4-10*x^3+8*x^2-4*x+1). - N. J. A. Sloane, Dec 14 2015

Extended by Joseph Myers, Dec 13 2015

b-file extended by Joseph Myers, Dec 18 2015

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

cp's OEIS Frontend

A299258 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.6.12 2D tiling (cf. A072154).

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A265078 Partial sums of A072154.

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A250120 Coordination sequence for planar net 3.3.3.3.6 (also called the fsz net).

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A008574 a(0) = 1, thereafter a(n) = 4n.

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A008486 Expansion of (1 + x + x^2)/(1 - x)^2.

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A008458 Coordination sequence for hexagonal lattice.

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