A008458 -id:A008458 - cp's OEIS frontend

A022144 Coordination sequence for root lattice B_2.

1, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360

Offset: 0

Michael Baake (mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de)

Equivalently, the coordination sequence for a point of degree 8 in the tiling of the Euclidean plane by right triangles (with angles Pi/2, Pi/4, Pi/4). These triangles are fundamental regions for the Coxeter group (2,4,4). In the notation of Conway et al. 2008 this is the tiling *442. The coordination sequence for a point of degree 4 is given by A234275. - N. J. A. Sloane, Dec 28 2015
Number of points of L_infinity norm n in the simple square lattice Z^2. - N. J. A. Sloane, Apr 15 2008
Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 24 ).
Number of 4 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (11;0) and (01;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 i1Sergey Kitaev, Nov 11 2004
These numbers correspond to the number of primes in the shells of a prime spiral. In a(2) there are 8 primes surrounding 2 in a prime spiral. - Enoch Haga, Apr 06 2000

			1 + 8*x + 16*x^2 + 24*x^3 + 32*x^4 + 40*x^5 + 48*x^6 + 56*x^7 + ...

J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 191.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
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.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
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 linear recurrences with constant coefficients, signature (2,-1).

Apart from initial term, the same as A008590.
Cf. A234275.
For partial sums see A016754.
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.

[0^n+8*n: n in [0..50] ]; // Vincenzo Librandi, Aug 21 2011

Join[{1}, LinearRecurrence[{2, -1}, {8, 16}, 50]] (* Jean-François Alcover, Jan 07 2019 *)

def A022144(n): return n<<3 if n else 1 # Chai Wah Wu, Mar 11 2025

a(n) = [x^(2*n)] ((1 + x)/(1 - x))^2.
G.f. for coordination sequence of B_n lattice: Sum_{i=0..n} binomial(2*n+1, 2*i)*z^i - 2*n*z*(1+z)^(n-1)/(1-z)^n. [Bacher et al.]
a(n) = (2*n+1)^2 - (2*n-1)^2. Binomial transform of [1, 7, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Dec 27 2007
a(n) = 0^n + 8*n. - Vincenzo Librandi, Aug 21 2011
G.f.: 1 + 8*x/(1-x)^2. - R. J. Mathar, Feb 16 2018
Sum_{i=0..n} a(i) = (2*n+1)^2 = A016754(n). - Chunqing Liu, Jan 12 2020
E.g.f.: 1 + 8*x*exp(x). - Stefano Spezia, Apr 05 2021

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

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

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

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

T. D. Noe, Table of n, a(n) for n = 0..1000
Pierre de La Harpe, and P. I. Grigorchuk, Local convexity of the growth function of finitely generated groups and question 5.2 in the Kourovka Notebook, Algebra and Logic 37.6 (1998): 353-356.
Jean-Guillaume Eon, Algebraic determination of generating functions for coordination sequences in crystal structures, Acta Cryst. A58 (2002), 47-53. See p. 51.
Jean-Guillaume Eon, Topological density of nets: a direct calculation, Acta Crystallographica Section A (Foundations of Crystallography), A60 (2014), 7-18; DOI: 10.1107/S0108767303022037. See Section 5.
Jean-Guillaume Eon, Symmetry and Topology: The 11 Uninodal Planar Nets Revisited, Symmetry, 10 (2018), 13 pages, doi:10.3390/sym10020035. See Section 4.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource, kgm
Yuriy Sibirmovsky, Illustration of initial terms with densely packed circles.
N. J. A. Sloane, Illustration of initial terms
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).

Cf. A017137, A017365.

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

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

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

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

A103881 Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 1, 12, 12, 2, 1, 20, 42, 18, 2, 1, 30, 110, 92, 24, 2, 1, 42, 240, 340, 162, 30, 2, 1, 56, 462, 1010, 780, 252, 36, 2, 1, 72, 812, 2562, 2970, 1500, 362, 42, 2, 1, 90, 1332, 5768, 9492, 7002, 2570, 492, 48, 2, 1, 110, 2070, 11832, 26474, 27174, 14240, 4060, 642, 54, 2, 1, 132, 3080, 22530, 66222, 91112, 65226, 26070, 6040, 812, 60, 2

Offset: 1

Ralf Stephan, Feb 20 2005

T(n,k) is the number of integer sequences of length n+1 with sum zero and sum of absolute values 2k. - R. H. Hardin, Feb 23 2009

			Array begins:
  1,   2,     2,      2,       2,        2,         2,          2, ... A040000;
  1,   6,    12,     18,      24,       30,        36,         42, ... A008458;
  1,  12,    42,     92,     162,      252,       362,        492, ... A005901;
  1,  20,   110,    340,     780,     1500,      2570,       4060, ... A008383;
  1,  30,   240,   1010,    2970,     7002,     14240,      26070, ... A008385;
  1,  42,   462,   2562,    9492,    27174,     65226,     137886, ... A008387;
  1,  56,   812,   5768,   26474,    91112,    256508,     623576, ... A008389;
  1,  72,  1332,  11832,   66222,   271224,    889716,    2476296, ... A008391;
  1,  90,  2070,  22530,  151560,   731502,   2777370,    8809110, ... A008393;
  1, 110,  3080,  40370,  322190,  1815506,   7925720,   28512110, ... A008395;
  1, 132,  4422,  68772,  643632,  4197468,  20934474,   85014204, ... A035837;
  1, 156,  6162, 112268, 1219374,  9129276,  51697802,  235895244, ... A035838;
  1, 182,  8372, 176722, 2206932, 18827718, 120353324,  614266354, ... A035839;
  1, 210, 11130, 269570, 3838590, 37060506, 265953170, 1511679210, ... A035840;
  ...
Antidiagonals:
  1;
  1,  2;
  1,  6,    2;
  1, 12,   12,    2;
  1, 20,   42,   18,    2;
  1, 30,  110,   92,   24,    2;
  1, 42,  240,  340,  162,   30,    2;
  1, 56,  462, 1010,  780,  252,   36,   2;
  1, 72,  812, 2562, 2970, 1500,  362,  42,  2;
  1, 90, 1332, 5768, 9492, 7002, 2570, 492, 48,  2;

Muniru A Asiru, Table of n, a(n) for n = 1..5050 (antidiagonals 1 to 100, flattened)
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Arun Padakandla, P. R. Kumar, and Wojciech Szpankowski, On the Discrete Geometry of Differential Privacy via Ehrhart Theory, November 2017.
Arun Padakandla, P. R. Kumar, and Wojciech Szpankowski, Preserving Privacy and Fidelity via Ehrhart Theory, July 2017.
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Rows include A040000, A008458, A005901, A008383, A008385, A008387, A008389, A008391, A008393, A008395, A035837, A035838, A035839, A035840, A035841 - A035876.
Columns include A002376, A001621.
Main diagonal is in A103882.
Cf. A047085, A103884, A103903, A103998, A143007.

T:=Flat(List([1..12],n->Concatenation([1],List([1..n-1],k->Sum([1..n],i->Binomial(n-k+1,i)*Binomial(k-1,i-1)*Binomial(n-i,k)))))); # Muniru A Asiru, Oct 14 2018

A103881:= func< n,k | k le 0 select 1 else (&+[Binomial(n-k+1, j)*Binomial(k-1, j-1)*Binomial(n-j, k): j in [1..n-k]]) >;
[A103881(n,k): k in [0..n-1], n in [1..15]]; // G. C. Greubel, Oct 16 2018; May 24 2023

T:=proc(n,k) option remember; local i;
if k=0 then 1 else
add( binomial(n+1,i)*binomial(k-1,i-1)*binomial(n-i+k,k),i=1..n); fi;
end:
g:=n->[seq(T(n-i,i),i=0..n-1)]:
for n from 1 to 14 do lprint(op(g(n))); od:

T[n_, k_]:= (n+1)*(n+k-1)!*HypergeometricPFQ[{1-k,1-n,-n}, {2,-n-k+1}, 1]/(k!*(n-1)!); T[, 0]=1; Flatten[Table[T[n-k, k], {n,12}, {k,0,n-1}]] (* _Jean-François Alcover, Dec 27 2012 *)

A103881(n,k) = if(k==0, 1, sum(j=1, n-k, binomial(n-k+1, j)*binomial(k-1, j-1)*binomial(n-j, k)));
for(n=1, 15, for(k=0, n-1, print1(A103881(n,k), ", "))) \\ G. C. Greubel, Oct 16 2018; May 24 2023

def A103881(n,k): return 1 if k==0 else (n-k+1)*binomial(n-1,k)*hypergeometric([k-n,1+k-n,1-k], [2,1-n], 1).simplify()
flatten([[A103881(n,k) for k in range(n)] for n in range(1,16)]) # G. C. Greubel, May 24 2023

T(n,k) = Sum_{i=1..n} C(n+1, i)*C(k-1, i-1)*C(n-i+k, k), T(n,0)=1.
G.f. of n-th row: (Sum_{i=0..n} C(n, i)^2*x^i)/(1-x)^n.
From G. C. Greubel, May 24 2023: (Start)
T(n, k) = Sum_{j=0..n} binomial(n,j)^2 * binomial(n+k-j-1, n-1) (array).
T(n, k) = (n+1)*binomial(n+k-1,k)*hypergeometric([-n,1-n,1-k], [2,1-n-k], 1), with T(n, k) = 1 (array).
t(n, k) = (n-k+1)*binomial(n-1,k)*hypergeometric([k-n,1+k-n,1-k], [2,1-n], 1), with t(n, 0) = 1 (antidiagonals).
Sum_{k=0..n-1} t(n, k) = A047085(n). (End)
From Peter Bala, Jul 09 2023: (Start)
T(n,k) = [x^k] Legendre_P(n, (1 + x)/(1 - x)).
(n+1)*T(n+1,k) = (n+1)*T(n+1,k-1) + (2*n+1)*(T(n,k) + T(n,k-1)) - n*(T(n-1,k) - T(n-1,k-1)). (End)

Corrected by N. J. A. Sloane, Dec 15 2012, at the suggestion of Manuel Blum

A250122 Coordination sequence for planar net 3.12.12.

Original entry on oeis.org

1, 3, 4, 6, 8, 12, 14, 15, 18, 21, 22, 24, 28, 30, 30, 33, 38, 39, 38, 42, 48, 48, 46, 51, 58, 57, 54, 60, 68, 66, 62, 69, 78, 75, 70, 78, 88, 84, 78, 87, 98, 93, 86, 96, 108, 102, 94, 105, 118, 111, 102, 114, 128, 120, 110, 123, 138, 129

Offset: 0

Darrah Chavey, Nov 23 2014

Also, growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^6 = 1 >. See Magma program in A298805. - N. J. A. Sloane, Feb 06 2018

Maurizio Paolini, Table of n, a(n) for n = 0..1021
Agnes Azzolino, Regular and Semi-Regular Tessellation Paper, 2011.
Agnes Azzolino, Illustration of 3.12.12 tiling [From previous link]
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also on arXiv, arXiv:1803.08530 [math.CO], 2018-2019. See section 10 The 3.12^2 tiling.
Rostislav Grigorchuk and Cosmas Kravaris, On the growth of the wallpaper groups, arXiv:2012.13661 [math.GR], 2020. See section 4.6 p. 23.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Tom Karzes, Tiling Coordination Sequences
Maurizio Paolini, C program for A250122
Reticular Chemistry Structure Resource, hca
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).

List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).

Cf. A298805.

Join[{1, 3, 4}, LinearRecurrence[{2, -3, 4, -3, 2, -1}, {6, 8, 12, 14, 15, 18}, 100]] (* Jean-François Alcover, Aug 05 2018 *)

From Joseph Myers, Nov 28 2014: (Start)
Empirically,
a(4n) = 10n - 2 except for a(0) = 1
a(4n+1) = 9n + 3
a(4n+2) = 8n + 6 except for a(2) = 4
a(4n+3) = 9n + 6. (End)
If these are correct, the sequence has g.f.
-(-1 - x - x^2 - 3*x^3 + x^4 - 5*x^5 + 3*x^6 - 4*x^7 + 2*x^8)/((x - 1)^2*(x^2 + 1)^2). - N. J. A. Sloane, Nov 28 2014
All the above conjectures are true. - N. J. A. Sloane, Dec 31 2015
E.g.f.: (9*x*cosh(x) - 4*(2*cos(x) + x^2 - 3) + 9*x*sinh(x) - (x - 3)*sin(x))/4. - Stefano Spezia, Jan 05 2023

a(8) onwards from Maurizio Paolini and Joseph Myers (independently), Nov 28 2014

A072154 Coordination sequence for the planar net 4.6.12.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 15, 17, 19, 21, 24, 27, 29, 31, 33, 36, 39, 41, 43, 45, 48, 51, 53, 55, 57, 60, 63, 65, 67, 69, 72, 75, 77, 79, 81, 84, 87, 89, 91, 93, 96, 99, 101, 103, 105, 108, 111, 113, 115, 117, 120, 123, 125, 127, 129, 132, 135, 137

Offset: 0

N. J. A. Sloane, Jun 28 2002

There is only one type of node in this structure: each node meets a square, a hexagon and a 12-gon.
The coordination sequence with respect to a particular node gives the number of nodes that can be reached from that node in n steps along edges.
Also, coordination sequence for the aluminophosphate AlPO_4-5 structure.

A. V. Shutov, On the number of words of a given length in plane crystallographic groups (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 302 (2003), Anal. Teor. Chisel i Teor. Funkts. 19, 188--197, 203; translation in J. Math. Sci. (N.Y.) 129 (2005), no. 3, 3922-3926 [MR2023041]. See Table 1, line "p6m" (but beware typos).

Sean A. Irvine, Table of n, a(n) for n = 0..999
Joerg Arndt, The 4.6.12 planar net
Agnes Azzolino, Regular and Semi-Regular Tessellation Paper, 2011.
Agnes Azzolino, Larger illustration of 4.6.12 planar net [From previous link]
M. E. Davis, Ordered porous materials for emerging applications, Nature, 417 (Jun 20 2002), 813-821 (gives structure).
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.
Rostislav Grigorchuk and Cosmas Kravaris, On the growth of the wallpaper groups, arXiv:2012.13661 [math.GR], 2020. See section 4.7 p. 23.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Sean A. Irvine, Java implementation with explicit counting
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource, fxt
N. J. A. Sloane, AlPO_4-5 structure, after Davis
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
N. J. A. Sloane, The subgraph H used in the proof of the formulas
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A072149, A072150, A072151, A072152, A072153.
For partial sums see A265078.
List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).
See also A301730.

Join[{1}, LinearRecurrence[{1, 0, 0, 0, 1, -1}, {3, 5, 7, 9, 12, 15}, 100]] (* Jean-François Alcover, Dec 13 2018 *)

Empirical g.f.: (x+1)^2*(x^2-x+1)*(x^2+x+1)/((x-1)^2*(x^4+x^3+x^2+x+1)). - Colin Barker, Nov 18 2012
This empirical g.f. can also be written as (1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + x^6)/(1 - x - x^5 + x^6). - N. J. A. Sloane, Dec 20 2015
Theorem: For n >= 7, a(n) = a(n-1) + a(n-5) - a(n-6), and a(5k) = 12k (k > 0), a(5k+m) = 12k + 2m + 1 (k >= 0, 1 <= m < 5). This also implies the conjectured g.f.'s. - N. J. A. Sloane, conjectured Dec 20 2015, proved Jan 20 2018.
Notes on the proof, from N. J. A. Sloane, Jan 20 2018 (Start)
The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.
The figure is divided into 6 sectors by the blue trunks. In the interior of each sector, working outwards from the base point P at the origin, there are successively 1,2,3,4,... (red) 12-gons. All the 12-gons (both red and blue) have a unique closest point to P.
If the closest point in a 12-gon is at distance d from P, then the contributions of the 12 points of the 12-gon to a(d), a(d+1), ..., a(d+6) are 1,2,2,2,2,2,1, respectively.
The rest of the proof is now a matter of simple counting.
The blue 12-gons (along the trunks) are especially easy to count, because there is a unique blue 12-gon at shortest distance d from P for d = 1,2,3,4,...
(End)
a(n) = 2*(6*n + sqrt(1 + 2/sqrt(5))*sin(2*n*Pi/5) + sqrt(1 - 2/sqrt(5))*sin(4*n*Pi/5))/5 for n > 0. - Stefano Spezia, Jan 05 2023

More terms from Sean A. Irvine, Sep 29 2011

Thanks to Darrah Chavey for pointing out that this is the planar net 4.6.12. - N. J. A. Sloane, Nov 24 2014

A008706 Coordination sequence for 3.3.3.4.4 planar net.

Original entry on oeis.org

1, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260, 265, 270, 275

Offset: 0

N. J. A. Sloane

Also the Engel expansion of exp^(1/5); cf. A006784 for the Engel expansion definition. - Benoit Cloitre, Mar 03 2002

			G.f. = 1 + 5*x + 10*x^2 + 15*x^3 + 20*x^4 + 25*x^5 + 30*x^6 + 35*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource, cem
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A006784, A048476 (binomial Transf.)
Essentially the same as A008587.
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).
First differences of A005891.

[0^n+5*n: n in [0..50] ]; // Vincenzo Librandi, Aug 21 2011

Join[{1}, LinearRecurrence[{2, -1}, {5, 10}, 100]] (* Jean-François Alcover, Dec 13 2018 *)

a(n)=0^n+5*n \\ Charles R Greathouse IV, Mar 19 2015

From Paul Barry, Jul 21 2003: (Start)
G.f.: (1 + 3*x + x^2)/(1 - x)^2.
a(n) = 0^n + 5n. (End)
G.f.: A(x) + 1, where A(x) is the g.f. of A008587. - Gennady Eremin, Feb 21 2021
E.g.f.: 1 + 5*x*exp(x). - Stefano Spezia, Jan 05 2023

A019557 Coordination sequence for G_2 lattice.

Original entry on oeis.org

1, 12, 30, 48, 66, 84, 102, 120, 138, 156, 174, 192, 210, 228, 246, 264, 282, 300, 318, 336, 354, 372, 390, 408, 426, 444, 462, 480, 498, 516, 534, 552, 570, 588, 606, 624, 642, 660, 678, 696, 714, 732, 750, 768, 786, 804, 822, 840, 858, 876, 894, 912, 930, 948, 966, 984, 1002, 1020, 1038, 1056

Offset: 0

Michael Baake (mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de)

Also, coordination sequence of Dual(3.12.12) tiling with respect to a 12-valent node. - N. J. A. Sloane, Jan 22 2018

For n > 1, also the number of minimum vertex colorings of the n-Andrásfai graph. - Eric W. Weisstein, Mar 03 2024

			From _Peter M. Chema_, Mar 20 2016: (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          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           12                30                 48
Compare to A003154, A045946, and A270700. (End)

Michael Baake and Uwe Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), pp. 253-256
Roland Bacher, Pierre de la Harpe, and Boris Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Séries 1) (1997), pp. 1137-1142.
Roland Bacher, Pierre de la Harpe, and Boris Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, Annales de l'institut Fourier, 49 no. 3 (1999), pp. 727-762.
Tom Karzes, Tiling Coordination Sequences.
N. J. A. Sloane, Illustration of layers 0,1,2 in the graph of the Dual(3.12.12) tiling. Centered at a 12-valent node. Note that some of the blue edges are not part of the underlying graph.
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]
Eric Weisstein's World of Mathematics, Andrásfai Graph.
Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000290, A003154, A008706, A016789, A045946, A270700.
For partial sums see A082040.
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.

CoefficientList[Series[(1 + 10 x + 7 x^2)/(1 - x)^2, {x, 0, 59}], x] (* Michael De Vlieger, Mar 21 2016 *)

my(x='x+O('x^100)); Vec((1+10*x+7*x^2)/(1-x)^2) \\ Altug Alkan, Mar 20 2016

a(n) = 18*n - 6, n >= 1.
G.f.: (1 + 10*x + 7*x^2)/(1-x)^2.
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 6*exp(x)*(3*x - 1) + 7.
a(n) = 6*A016789(n-1) for n >= 1.
a(n) = 2*a(n-1) - a(n-2) for n >= 3. (End)

A296368 Coordination sequence for the Cairo or dual-3.3.4.3.4 tiling with respect to a trivalent point.

Original entry on oeis.org

1, 3, 8, 12, 15, 20, 25, 28, 31, 36, 41, 44, 47, 52, 57, 60, 63, 68, 73, 76, 79, 84, 89, 92, 95, 100, 105, 108, 111, 116, 121, 124, 127, 132, 137, 140, 143, 148, 153, 156, 159, 164, 169, 172, 175, 180, 185, 188, 191, 196, 201, 204, 207, 212, 217, 220, 223, 228

Offset: 0

N. J. A. Sloane, Dec 21 2017

There are two types of point in this tiling. This is the coordination sequence with respect to a point of degree 3.

The coordination sequence with respect to a point of degree 4 (see second illustration) is simply 1, 4, 8, 12, 16, 20, ..., the same as the coordination sequence for the 4.4.4.4 square grid (A008574). See the CGS-NJAS link for the proof.

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Fig. 9.1.3, drawing P_5-24, page 480.
Herbert C. Moore, U.S. Patents 928,320 and 928,321, Patented July 20 1909. [Shows Cairo tiling.]

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Chaim Goodman-Strauss and N. J. A. Sloane, A portion of the Cairo tiling
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.
Chung, Ping Ngai, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, and Elena Wikner. Isoperimetric Pentagonal Tilings, Notices of the AMS 59, no. 5 (2012), pp. 632-640. See Fig. 1 (left).
Tom Karzes, Tiling Coordination Sequences
Frank Morgan, Optimal Pentagonal Tilings, Video, May 2021 [Has much to say about the Cairo tiling]
Reticular Chemistry Structure Resource (RCSR), The mcm tiling (or net)
Rémy Sigrist, PARI program for A296368
N. J. A. Sloane, Illustration of initial terms (for a trivalent point)
N. J. A. Sloane, Illustration of initial terms of coordination sequence 1,4,8,12,... for a tetravalent point
N. J. A. Sloane, A tiling by rectangles which has the same graph and coordination sequences as the Cairo tiling (Seen on the streets of Piscataway, New Jersey, USA)
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)
Index entries for linear recurrences with constant coefficients, signature (2, -2, 2, -1).

For partial sums see A296909.
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.

Join[{1, 3, 8}, LinearRecurrence[{2, -2, 2, -1}, {12, 15, 20, 25}, 100]] (* Jean-François Alcover, Aug 05 2018 *)

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

The simplest formula is: a(0)=1, a(1)=2, a(2)=8, and thereafter a(n) = 4n if n is odd, 4n - 1 if n == 0 (mod 4), and 4n+1 if n == 2 (mod 4). (See the CGS-NJAS link for proof. - N. J. A. Sloane, May 10 2018)
a(n + 4) = a(n) + 16 for any n >= 3. - Rémy Sigrist, Dec 23 2017 (See the CGS-NJAS link for a proof. - N. J. A. Sloane, Dec 30 2017)
G.f.: -(x^6-x^5-2*x^4-4*x^2-x-1)/((x^2+1)*(x-1)^2).
From Colin Barker, Dec 23 2017: (Start)
a(n) = (8*n - (-i)^n - i^n) / 2 for n>2, where i=sqrt(-1).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>6.
(End)

Terms a(8)-a(20) and RCSR link from Davide M. Proserpio, Dec 22 2017

More terms from Rémy Sigrist, Dec 23 2017

cp's OEIS Frontend

A022144 Coordination sequence for root lattice B_2.

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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.

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

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

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A103881 Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n.

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A250122 Coordination sequence for planar net 3.12.12.

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