A008576 Coordination sequence for planar net 4.8.8.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

References

• 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.

Links

• 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).

Crossrefs

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.

Programs

Maple

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;

Mathematica

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 *)

PARI

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

Formula

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