

A008576


Coordination sequence for planar net 4.8.8.


43



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
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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, 188197, 203; translation in J. Math. Sci. (N.Y.) 129 (2005), no. 3, 39223926 [MR2023041]. See Table 1.


LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000
Agnes Azzolino, Regular and SemiRegular Tessellation Paper, 2011
Agnes Azzolino, Illustration of 4.8.8 tiling [From previous link]
Brian Galebach, kuniform tilings (k <= 6) and their Anumbers
Chaim GoodmanStrauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121134, also on NJAS's home page. Also arXiv:1803.08530.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227247.
Tom Karzes, Tiling Coordination Sequences
W. M. Meier and H. J. Moeck, Topology of 3D 4connected nets ..., J. Solid State Chem 27 1979 349355, esp. p. 351.
Reticular Chemistry Structure Resource, fes
N. J. A. Sloane, The uniform planar nets and their Anumbers [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))/((1x)^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(n1) + a(n3)  a(n4).  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 bfile.  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(n1).  Gary W. Adamson, Sep 12 2007


MAPLE

if n mod 3 = 0 then 8*n/3 elif n mod 3 = 1 then 8*(n1)/3+3 else 8*(n2)/3+5 fi;


MATHEMATICA

cspn[n_]:=Module[{c=Mod[n, 3]}, Which[c==0, (8n)/3, c==1, (8(n1))/3+3, True, (8(n2))/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

Cf. A042965.
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 A161696A161699, 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, A267167A267175.
Sequence in context: A026274 A137910 A022850 * A047622 A240603 A079392
Adjacent sequences: A008573 A008574 A008575 * A008577 A008578 A008579


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



