OFFSET
0,2
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).
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
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 *)
PROG
(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
CROSSREFS
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.
KEYWORD
nonn,easy
AUTHOR
STATUS
approved