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%I #86 Aug 07 2022 22:11:55
%S 1,3,5,8,11,13,16,19,21,24,27,29,32,35,37,40,43,45,48,51,53,56,59,61,
%T 64,67,69,72,75,77,80,83,85,88,91,93,96,99,101,104,107,109,112,115,
%U 117,120,123,125,128,131,133
%N Coordination sequence for planar net 4.8.8.
%C Also, growth series for the affine Coxeter (or Weyl) groups B_2. - _N. J. A. Sloane_, Jan 11 2016
%D 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).
%D 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.
%H T. D. Noe, <a href="/A008576/b008576.txt">Table of n, a(n) for n = 0..1000</a>
%H Agnes Azzolino, <a href="http://www.mathnstuff.com/math/spoken/here/2class/150/display.htm">Regular and Semi-Regular Tessellation Paper</a>, 2011
%H Agnes Azzolino, <a href="/A008576/a008576.gif">Illustration of 4.8.8 tiling</a> [From previous link]
%H Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H Chaim Goodman-Strauss and N. J. A. Sloane, <a href="https://doi.org/10.1107/S2053273318014481">A Coloring Book Approach to Finding Coordination Sequences</a>, Acta Cryst. A75 (2019), 121-134, also <a href="http://NeilSloane.com/doc/Cairo_final.pdf">on NJAS's home page</a>. Also <a href="http://arxiv.org/abs/1803.08530">on arXiv</a>, arXiv:1803.08530 [math.CO], 2018-2019.
%H Rostislav Grigorchuk and Cosmas Kravaris, <a href="https://arxiv.org/abs/2012.13661">On the growth of the wallpaper groups</a>, arXiv:2012.13661 [math.GR], 2020. See section 4.5 p. 22.
%H Branko Grünbaum and Geoffrey C. Shephard, <a href="http://www.jstor.org/stable/2689529">Tilings by regular polygons</a>, Mathematics Magazine, 50 (1977), 227-247.
%H Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>
%H W. M. Meier and H. J. Moeck, <a href="http://dx.doi.org/10.1016/0022-4596(79)90177-4">Topology of 3-D 4-connected nets ...</a>, J. Solid State Chem 27 1979 349-355, esp. p. 351.
%H Reticular Chemistry Structure Resource, <a href="http://rcsr.net/layers/fes">fes</a>
%H N. J. A. Sloane, <a href="/A008576/a008576.png">The uniform planar nets and their A-numbers</a> [Annotated scanned figure from Gruenbaum and Shephard (1977)]
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F G.f.: ((1+x)^2*(1+x^2))/((1-x)^2*(1+x+x^2)). - _Ralf Stephan_, Apr 24 2004
%F 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
%F a(0)=1; thereafter a(3k)=8k, a(3k+1)=8k+3, a(3k+2)=8k+5. - _N. J. A. Sloane_, Dec 22 2015
%F 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
%F 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
%F a(n) = (2/9)*(12*n + (3/2)*A102283(n)) for n > 0. - _Stefano Spezia_, Aug 07 2022
%p 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;
%t 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 *)
%o (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
%Y Cf. A042965, A102283.
%Y 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).
%Y For partial sums see A008577.
%Y 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.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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