A022144 -id:A022144 - cp's OEIS frontend

A298028 Coordination sequence of Dual(3.6.3.6) tiling with respect to a trivalent node.

1, 3, 12, 9, 24, 15, 36, 21, 48, 27, 60, 33, 72, 39, 84, 45, 96, 51, 108, 57, 120, 63, 132, 69, 144, 75, 156, 81, 168, 87, 180, 93, 192, 99, 204, 105, 216, 111, 228, 117, 240, 123, 252, 129, 264, 135, 276, 141, 288, 147, 300, 153, 312, 159, 324, 165, 336, 171, 348, 177, 360, 183, 372, 189, 384, 195

Offset: 0

N. J. A. Sloane, Jan 21 2018

Also known as the kgd net.

This is one of the Laves tilings.

Robert Israel, Table of n, a(n) for n = 0..10000
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource (RCSR), The kgd tiling (or net)
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]
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008579, A135556 (partial sums), A298026 (trivalent point).
If the initial 1 is changed to 0 we get A165988 (but we need both sequences, just as we have both A008574 and A008586).
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.

f3:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 6*n else 3*n; fi; end;
[seq(f3(n),n=0..80)];

Join[{1}, LinearRecurrence[{0, 2, 0, -1}, {3, 12, 9, 24}, 80]] (* Jean-François Alcover, Mar 23 2020 *)

a(0)=1; a(2*k) = 12*k, a(2*k+1) = 6*k+3.
G.f.: 1 + 3*x*(x^2+4*x+1)/(1-x^2)^2. - Robert Israel, Jan 21 2018
a(n) = 3*A022998(n), n>0. - R. J. Mathar, Jan 29 2018

A298029 Coordination sequence of Dual(3.4.6.4) tiling with respect to a trivalent node.

Original entry on oeis.org

1, 3, 6, 12, 18, 33, 39, 51, 57, 69, 75, 87, 93, 105, 111, 123, 129, 141, 147, 159, 165, 177, 183, 195, 201, 213, 219, 231, 237, 249, 255, 267, 273, 285, 291, 303, 309, 321, 327, 339, 345, 357, 363, 375, 381, 393, 399, 411, 417, 429, 435, 447, 453, 465, 471, 483, 489, 501, 507, 519, 525, 537, 543, 555

Offset: 0

N. J. A. Sloane, Jan 21 2018

Also known as the deltoidal trihexagonal tiling, or the mta net.
In the Ferreol link this is described as the dual to the Diana tiling. - N. J. A. Sloane, May 24 2020
This is one of the Laves tilings.

Colin Barker, Table of n, a(n) for n = 0..1000
Robert Ferreol, Pavage de diane et son dual
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.
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource (RCSR), The mta tiling (or net)
N. J. A. Sloane, The Dual(3.4.6.4) tiling
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]
Wikipedia, Laves tilings
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A007310, A008574, A298030 (partial sums), A298031 (for a tetravalent node), A298033 (hexavalent node), A306771.

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, 6, 12, 18}, LinearRecurrence[{1, 1, -1}, {33, 39, 51}, 60]] (* Jean-François Alcover, Jan 07 2019 *)
Join[{1,3,6,12,18},Table[If[EvenQ[n],9n-15,9n-12],{n,5,70}]] (* Harvey P. Dale, Aug 25 2019 *)

Vec((1 + 2*x + 2*x^2 + 4*x^3 + 3*x^4 + 9*x^5 - 3*x^7) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018

Theorem: For n >= 5, if n is even then a(n) = 9*n-15, otherwise a(n) = 9*n-12. The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. - N. J. A. Sloane, Jan 24 2018
G.f.: -(3*x^7 - 9*x^5 - 3*x^4 - 4*x^3 - 2*x^2 - 2*x - 1)/((1 - x)*(1 - x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>7. - Colin Barker, Jan 25 2018
a(n) = (3/2)*(6*n - (-1)^n - 9) for n>4. - Bruno Berselli, Jan 25 2018
a(n) = 3*A007310(n-1), n>4. - R. J. Mathar, Jan 29 2018

A298031 Coordination sequence of Dual(3.4.6.4) tiling with respect to a tetravalent node.

Original entry on oeis.org

1, 4, 10, 16, 30, 36, 48, 54, 66, 72, 84, 90, 102, 108, 120, 126, 138, 144, 156, 162, 174, 180, 192, 198, 210, 216, 228, 234, 246, 252, 264, 270, 282, 288, 300, 306, 318, 324, 336, 342, 354, 360, 372, 378, 390, 396, 408, 414, 426, 432, 444, 450, 462, 468, 480, 486, 498, 504, 516, 522, 534, 540

Offset: 0

N. J. A. Sloane, Jan 21 2018; extended with formula, Jan 24 2018

Also known as the mta net.
This is one of the Laves tilings.
In the Ferreol link this is described as the dual to the Diana tiling. - N. J. A. Sloane, May 24 2020

Colin Barker, Table of n, a(n) for n = 0..1000
Robert Ferreol, Pavage de diane et son dual
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.
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource (RCSR), The mta tiling (or net)
N. J. A. Sloane, The Dual(3.4.6.4) tiling
N. J. A. Sloane, The subgraph H shown in one quadrant of the graph of the tiling.
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]
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A008574, A298032 (partial sums), A298029 (for a trivalent node), A298033 (hexavalent node).

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.

f4:=proc(n) local L; L:=[1,4,10,16];
if n<4 then L[n+1] elif (n mod 2) = 0 then 9*n-6 else 9*n-9; fi;
end;
[seq(f4(n),n=0..80)];

Join[{1, 4, 10, 16}, LinearRecurrence[{1, 1, -1}, {30, 36, 48}, 62]] (* Jean-François Alcover, Apr 23 2018 *)

Vec((1 + 3*x + 5*x^2 + 3*x^3 + 8*x^4 - 2*x^6) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018

Theorem: For n >= 4, a(n) = 9*n-6 if n is even, otherwise a(n) = 9*n-9.
The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.
G.f.: -(2*x^6 - 8*x^4 - 3*x^3 - 5*x^2 - 3*x - 1) / ((1 - x)*(1 - x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>4. - Colin Barker, Jan 25 2018
a(n) = 6*A007494(n-1), n>3. - R. J. Mathar, Jan 29 2018

A298033 Coordination sequence of the Dual(3.4.6.4) tiling with respect to a hexavalent node.

Original entry on oeis.org

1, 6, 12, 24, 30, 42, 48, 60, 66, 78, 84, 96, 102, 114, 120, 132, 138, 150, 156, 168, 174, 186, 192, 204, 210, 222, 228, 240, 246, 258, 264, 276, 282, 294, 300, 312, 318, 330, 336, 348, 354, 366, 372, 384, 390, 402, 408, 420, 426, 438, 444, 456, 462, 474, 480, 492, 498, 510, 516, 528, 534, 546, 552

Offset: 0

N. J. A. Sloane, Jan 21 2018, corrected Jan 24 2018

Also known as the mta net.

This is one of the Laves tilings.

Colin Barker, Table of n, a(n) for n = 0..1000
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.
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource (RCSR), The mta tiling (or net)
N. J. A. Sloane, The Dual(3.4.6.4) tiling
N. J. A. Sloane, The subgraph H shown in one 60-degree sector of the graph of the tiling.
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]
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

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. A008574, A038764 (partial sums), A298029 (coordination sequence for a trivalent node), A298031 (coordination sequence for a tetravalent node).

f6:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 9*n-6 else 9*n-3; fi; end;
[seq(f6(n),n=0..80)];

Join[{1}, LinearRecurrence[{1, 1, -1}, {6, 12, 24}, 62]] (* Jean-François Alcover, Apr 23 2018 *)

Vec((1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018

apply( {A298033(n)=if(n,n*3\/2*6-6,1)}, [0..66]) \\ M. F. Hasler, Jan 11 2022

Theorem: For n>0, a(n) = 9*n-6 if n is even, a(n) = 9*n-3 if n is odd.
The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.
G.f.: (1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)*(1 - x^2)).
First differences are 1, 5, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, ...
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3. - Colin Barker, Jan 25 2018
a(n) = 6*floor((3n-1)/2) for n > 0. - M. F. Hasler, Jan 11 2022

A017569 a(n) = 12*n + 4.

Original entry on oeis.org

4, 16, 28, 40, 52, 64, 76, 88, 100, 112, 124, 136, 148, 160, 172, 184, 196, 208, 220, 232, 244, 256, 268, 280, 292, 304, 316, 328, 340, 352, 364, 376, 388, 400, 412, 424, 436, 448, 460, 472, 484, 496, 508, 520, 532, 544, 556, 568, 580, 592, 604, 616, 628

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(46).
Number of 6 X n 0-1 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 i1A008574; m=3: A016933; m=4: A022144; m=5: A017293. - Sergey Kitaev, Nov 13 2004
Except for 4, exponents e such that x^e - x^2 + 1 is reducible.
If Y and Z are 2-blocks of a (3n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 28 2007
Terms are perfect squares iff n is a generalized octagonal number (A001082), then n = k*(3*k-2) and a(n) = (2*(3*k-1))^2. - Bernard Schott, Feb 26 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Milan Janjic, Two Enumerative Functions.
Tanya Khovanova, Recursive Sequences.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), Article A21, 20pp.
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).

Cf. A008594, A017533, A017545, A089911.
Cf. A016777, A016933, A017293, A022144.
Cf. A001082.

a017569 = (+ 4) . (* 12)  -- Reinhard Zumkeller, Jul 05 2013

[12*n+4: n in [0..50]]; // Vincenzo Librandi, May 04 2011

12*Range[0,200]+4 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

A089911(a(n)) = 3. - Reinhard Zumkeller, Jul 05 2013
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/36 + log(2)/12. - Amiram Eldar, Dec 12 2021
From Stefano Spezia, Feb 25 2023: (Start)
O.g.f.: 4*(1 + 2*x)/(1 - x)^2.
E.g.f.: 4*exp(x)*(1 + 3*x). (End)
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 2*a(n-1) - a(n-2).
a(n) = 2*A016933(n) = 4*A016777(n) = A016777(4*n+1). (End)

A010006 Coordination sequence for C_3 lattice: a(n) = 16*n^2 + 2 (n>0), a(0)=1.

Original entry on oeis.org

1, 18, 66, 146, 258, 402, 578, 786, 1026, 1298, 1602, 1938, 2306, 2706, 3138, 3602, 4098, 4626, 5186, 5778, 6402, 7058, 7746, 8466, 9218, 10002, 10818, 11666, 12546, 13458, 14402, 15378, 16386, 17426, 18498, 19602, 20738, 21906, 23106, 24338, 25602, 26898

Offset: 0

N. J. A. Sloane, mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de (Michael Baake)

If Y_i (i=1,2,3) are 2-blocks of a (2n+1)-set X then a(n-1) is the number of 5-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Oct 28 2007

Also sequence found by reading the segment (1, 18) together with the line from 18, in the direction 18, 66, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

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, 1997; 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.
Milan Janjic, Two Enumerative Functions
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399. For the coordination sequences of other C_n lattices see A022144 (C_2), A010006 (C_3), A019560 - A019564 (C_4 through C_8), A035746 - A035787 (C_9 through C_50). Cf. A137513.

[1] cat [16*n^2+2: n in [1..50]]; // Vincenzo Librandi, Feb 20 2012

Join[{1},Table[16n^2+2,{n,50}]] (* Harvey P. Dale, Oct 15 2012 *)

A010006(n)=16*n^2+2-!n   \\ M. F. Hasler, Feb 14 2012

a(0)=1, a(n) = 16*n^2 + 2, n >= 1.
G.f.: (1+x)*(1+14*x+x^2)/(1-x)^3.
G.f. for coordination sequence of C_n lattice: (1/(1-z)^n)*Sum_{i=0..n} binomial(2*n, 2*i)*z^i.
E.g.f.: (x*(x+1)*16+2)*e^x - 1. - Gopinath A. R., Feb 14 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=18, a(2)=66, a(3)=146. - Harvey P. Dale, Oct 15 2012
G.f. for sequence with interpolated zeros: cosh(6*arctanh(x)) = (1/2)*( ((1 - x)/(1 + x))^3 + ((1 + x)/(1 - x))^3) = 1 + 18*x^2 + 66*x^4 + 146*x^6 + .... More generally, cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. Note that exp(t*arctanh(x)) is the e.g.f. for the Mittag_Leffler polynomials. See A137513. - Peter Bala, Apr 09 2017
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(2)/16*Pi*coth( Pi*sqrt(2)/4) = 1.095237238050... - R. J. Mathar, May 07 2024
a(n) = 2*A081585(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A069129(n)+A069129(n+1). - R. J. Mathar, May 07 2024

A103884 Square array A(n,k) read by antidiagonals: row n gives coordination sequence for lattice C_n.

Original entry on oeis.org

1, 1, 8, 1, 18, 16, 1, 32, 66, 24, 1, 50, 192, 146, 32, 1, 72, 450, 608, 258, 40, 1, 98, 912, 1970, 1408, 402, 48, 1, 128, 1666, 5336, 5890, 2720, 578, 56, 1, 162, 2816, 12642, 20256, 14002, 4672, 786, 64, 1, 200, 4482, 27008, 59906, 58728, 28610, 7392, 1026, 72

Offset: 2

Ralf Stephan, Feb 20 2005

			Array begins:
  1,   8,    16,     24,      32,       40,        48, ... A022144;
  1,  18,    66,    146,     258,      402,       578, ... A010006;
  1,  32,   192,    608,    1408,     2720,      4672, ... A019560;
  1,  50,   450,   1970,    5890,    14002,     28610, ... A019561;
  1,  72,   912,   5336,   20256,    58728,    142000, ... A019562;
  1,  98,  1666,  12642,   59906,   209762,    596610, ... A019563;
  1, 128,  2816,  27008,  157184,   658048,   2187520, ... A019564;
  1, 162,  4482,  53154,  374274,  1854882,   7159170, ... A035746;
  1, 200,  6800,  97880,  822560,  4780008,  21278640, ... A035747;
  1, 242,  9922, 170610, 1690370, 11414898,  58227906, ... A035748;
  1, 288, 14016, 284000, 3281280, 25534368, 148321344, ... A035749;
  ...
Antidiagonals, T(n, k), begins as:
  1;
  1,   8;
  1,  18,   16;
  1,  32,   66,   24;
  1,  50,  192,  146,   32;
  1,  72,  450,  608,  258,   40;
  1,  98,  912, 1970, 1408,  402,  48;
  1, 128, 1666, 5336, 5890, 2720, 578, 56;

G. C. Greubel, Antidiagonals n = 2..50, 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).
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Rows include A022144, A010006, A019560, A019561, A019562, A019563, A019564, A035746, A035747, A035748, A035749, A035750 - A035787.
Columns include A001105, A035598, A035600, A035602, A035604, A035606.
Main diagonal is in A103885.
Cf. A103881, A103993, A108998.

A103884:= func< n,k | k eq 0 select 1 else 2*(&+[2^j*Binomial(n-k,j+1)*Binomial(2*k-1,j) : j in [0..2*k-1]]) >;
[A103884(n,k): k in [0..n-2], n in [2..12]]; // G. C. Greubel, May 23 2023

nmin = 2; nmax = 11; t[n_, 0]= 1; t[n_, k_]:= 2n*Hypergeometric2F1[1-2k, 1-n, 2, 2]; tnk= Table[ t[n, k], {n, nmin, nmax}, {k, 0, nmax-nmin}]; Flatten[ Table[ tnk[[ n-k+1, k ]], {n, 1, nmax-nmin+1}, {k, 1, n} ] ] (* Jean-François Alcover, Jan 24 2012, after formula *)

def A103884(n,k): return 1 if k==0 else 2*sum(2^j*binomial(n-k,j+1)*binomial(2*k-1,j) for j in range(2*k))
flatten([[A103884(n,k) for k in range(n-1)] for n in range(2,13)]) # G. C. Greubel, May 23 2023

A(n,k) = Sum_{i=1..2*k} 2^i*C(n, i)*C(2*k-1, i-1), A(n,0) = 1 (array).
G.f. of n-th row: (Sum_{i=0..n} C(2*n, 2*i)*x^i)/(1-x)^n.
T(n, k) = A(n-k, k) (antidiagonals).
T(n, n-2) = A022144(n-2).
T(n, k) = 2*(n-k)*Hypergeometric2F1([1+k-n, 1-2*k], [2], 2), T(n, 0) = 1, for n >= 2, 0 <= k <= n-2. - G. C. Greubel, May 23 2023
From  Peter Bala, Jul 09 2023: (Start)
T(n,k) = [x^k] Chebyshev_T(n, (1 + x)/(1 - x)), where Chebyshev_T(n, x) denotes the n-th Chebyshev polynomial of the first kind.
T(n+1,k) = T(n+1,k-1)  + 2*T(n,k) + 2*T(n,k-1) + T(n-1,k) - T(n-1,k-1). (End)

Definition clarified by N. J. A. Sloane, May 25 2023

A019560 Coordination sequence for C_4 lattice.

Original entry on oeis.org

1, 32, 192, 608, 1408, 2720, 4672, 7392, 11008, 15648, 21440, 28512, 36992, 47008, 58688, 72160, 87552, 104992, 124608, 146528, 170880, 197792, 227392, 259808, 295168, 333600, 375232, 420192, 468608

Offset: 0

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

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
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.
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A103884 (row 4). For coordination sequences of other C_n lattices see A022144 (C_2), A010006 (C3), A019560 - A019564 (C_4 through C_8), A035746 - A035787 (C_9 through C_50).

Cf. A008412, A137513.

[1] cat [(32/3)*n*(1 + 2*n^2): n in [1..40]]; // Vincenzo Librandi, Apr 10 2017

Join[{1}, Table[(32/3) n (1 + 2 n^2), {n, 30}]] (* Vincenzo Librandi, Apr 10 2017 *)

a(n) = (32/3)*n*(1 + 2*n^2) for n>0.
G.f.: (1 + 28*x + 70*x^2 + 28*x^3 + x^4)/(1 - x)^4.
G.f. for sequence with interpolated zeros: cosh(8*arctanh(x)) = 1/2*(((1 + x)/(1 - x))^4 + ((1 - x)/(1 + x))^4) = 1 + 32*x^2 + 192*x^4 + 608*x^6 + .... Cf. A057813. - Peter Bala, Apr 09 2017
a(n) = A008412(2*n). - Seiichi Manyama, Jun 08 2018

A110907 Number of points in the standard root system version of the D_3 (or f.c.c.) lattice having L_infinity norm n.

Original entry on oeis.org

1, 12, 50, 108, 194, 300, 434, 588, 770, 972, 1202, 1452, 1730, 2028, 2354, 2700, 3074, 3468, 3890, 4332, 4802, 5292, 5810, 6348, 6914, 7500, 8114, 8748, 9410, 10092, 10802, 11532, 12290, 13068, 13874, 14700, 15554, 16428, 17330, 18252, 19202

Offset: 0

N. J. A. Sloane, Apr 15 2008

This lattice consists of all points (x,y,z) where x,y,z are integers with an even sum.
The L_infinity norm of a vector is the largest component in absolute value.
The sequence for the D_k lattice has the terms ((2*n+1)^k-(2*n-1)^k)/2, if k is even, and the terms ((2n+1)^k-(2*n-1)^k)/2+(-1)^n if k is odd (like here for k=3). The sequence for A_2 is A008458, for A_3 A010006, for A_4 the first differences of A083669. A_5 is 2+2*n^2*(25+44*n^2) if n>0, and 1 if n=0. - R. J. Mathar, Feb 09 2010

			a(0) = 1: 000
a(1) = 12: +-1 +-1 0, where the 0 can be in any of the three coordinates
a(2) = 50: +-2 0 0 (6), +-2 +-1 +-1 (24), +-2 +-2 0 (12), +-2 +-2 +-2 (8).

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, Chap. 4.

R. J. Mathar, Point counts of D_k and some A_k and E_k integer lattices inside hypercubes arXiv:1002.3844 [math.GT], 2010.
G. Nebe and N. J. A. Sloane, Home page for this lattice
Index entries for sequences related to f.c.c. lattice
Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 1).

Cf. A117216, A022144, A010014, A175112 (D_5), A175114 (D_6).

A110907 := proc(n) a :=0 ; for x from -n to n do for y from -n to n do for z from -n to n do if type(x+y+z,'even') then m := max( abs(x),abs(y),abs(z)) ; if m = n then a := a+1 ; end if; end if; end do ; end do ; end do ; a ; end proc: seq(A110907(n),n=0..40) ; # R. J. Mathar, Feb 03 2010

a[0] = 1; a[n_] := 1 + (-1)^n + 12*n^2;
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 16 2017, after R. J. Mathar *)

From R. J. Mathar, Feb 03 2010: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n>4.
a(n) = 1 + (-1)^n + 12*n^2, n>0.
G.f.: 1 - 2*x*(6 + 13*x + 4*x^2 + x^3)/((1+x)*(x-1)^3). (End)

I would like to get analogous sequences for A_2, A_4, A_5, ..., D_4 (see A117216), D_5, ..., E_6, E_7, E_8.
Extended by R. J. Mathar, Feb 03 2010
Removed the "conjectured" attribute from formulas - R. J. Mathar, Feb 27 2010

A317297 a(n) = (n - 1)(4n^2 - 8*n + 5).

Original entry on oeis.org

0, 5, 34, 111, 260, 505, 870, 1379, 2056, 2925, 4010, 5335, 6924, 8801, 10990, 13515, 16400, 19669, 23346, 27455, 32020, 37065, 42614, 48691, 55320, 62525, 70330, 78759, 87836, 97585, 108030, 119195, 131104, 143781, 157250, 171535, 186660, 202649, 219526, 237315, 256040, 275725, 296394, 318071

Offset: 1

Omar E. Pol, Sep 01 2018

Conjecture: For n > 1, a(n) is the maximum eigenvalue of a 2*(n-1) X 2*(n-1) square matrix M defined as M[i,j,n] = j + n*(i-1) if i is odd and M[i,j,n] = n*i - j + 1 if i is even (see A317614). - Stefano Spezia, Dec 27 2018

Connections can be made to A022144 and A010014. Namely, a formula for A022144 is (2*n+1)^2 - (2*n-1)^2. A formula for A010014 is (2*n+1)^3 - (2*n-1)^3. The general form can be represented by (2*n+1)^d - (2*n-1)^d, where d designates the number of dimensions. When d is 4, a(n) = ((2*(n-1)+1)^4 - (2*(n-1)-1)^4)/16, namely the general form shifted by 1 and divided by 16 is a(n). - Yigit Oktar, Aug 16 2024

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First bisection of A006003.
Nonzero terms give the row sums of A007607.
Conjecture: 0 together with a bisection of A246697.
Cf. A219086 (partial sums).
Cf. A008595, A033430, A135453, A317614.
Cf. A010014, A022144 (see comments)

Table[(n - 1) (4 n^2 - 8 n + 5), {n, 1, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 5, 34, 111}, 50] (* or *) CoefficientList[Series[x (5 + 14 x + 5 x^2)/(1 - x)^4, {x, 0, 50}], x] (* Stefano Spezia, Sep 01 2018 *)

PARI
```
a(n) = (n - 1)*(4*n^2 - 8*n + 5)
```

concat(0, Vec(x^2*(5 + 14*x + 5*x^2)/(1 - x)^4 + O(x^50))) \\ Colin Barker, Sep 01 2018

a(n) = 4*n^3 - 12*n^2 + 13*n - 5 = A033430(n) - A135453(n) + A008595(n) - 5.
G.f.: x^2*(5 + 14*x + 5*x^2)/(1 - x)^4. - Colin Barker, Sep 01 2018
a(n) = 4*a(n - 1) - 6*a(n - 2) + 4*a(n - 3) - a(n - 4) for n > 4. - Stefano Spezia, Sep 01 2018
E.g.f.: exp(x)*(5*x + 12*x^2 + 4*x^3). - Stefano Spezia, Jan 15 2019
a(n) = ((2*(n-1)+1)^4 - (2*(n-1)-1)^4)/16. - Yigit Oktar, Aug 16 2024

cp's OEIS Frontend

A298028 Coordination sequence of Dual(3.6.3.6) tiling with respect to a trivalent node.

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A298029 Coordination sequence of Dual(3.4.6.4) tiling with respect to a trivalent node.

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A298031 Coordination sequence of Dual(3.4.6.4) tiling with respect to a tetravalent node.

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A298033 Coordination sequence of the Dual(3.4.6.4) tiling with respect to a hexavalent node.

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A017569 a(n) = 12*n + 4.

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A010006 Coordination sequence for C_3 lattice: a(n) = 16*n^2 + 2 (n>0), a(0)=1.

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A103884 Square array A(n,k) read by antidiagonals: row n gives coordination sequence for lattice C_n.

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A317297 a(n) = (n - 1)(4n^2 - 8*n + 5).