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%I #88 Aug 07 2022 21:29:20
%S 1,4,8,14,18,22,28,30,38,38,48,46,58,54,68,62,78,70,88,78,98,86,108,
%T 94,118,102,128,110,138,118,148,126,158,134,168,142,178,150,188,158,
%U 198,166,208,174,218,182,228,190,238,198,248,206,258,214,268,222,278
%N Coordination sequence for planar net 3.6.3.6. Spherical growth function for a certain reflection group in plane.
%C Interesting because coefficients never become monotonic.
%C Also the coordination sequence for a planar net made of densely packed circles. - _Yuriy Sibirmovsky_, Sep 11 2016
%C Described by J.-G. Eon (2014) as the coordination sequence of the Kagome net. - _N. J. A. Sloane_, Jan 03 2018
%D P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 161 (but beware errors).
%H T. D. Noe, <a href="/A008579/b008579.txt">Table of n, a(n) for n = 0..1000</a>
%H Pierre de La Harpe, and P. I. Grigorchuk, <a href="https://doi.org/10.1007/BF02671688">Local convexity of the growth function of finitely generated groups and question 5.2 in the Kourovka Notebook</a>, Algebra and Logic 37.6 (1998): 353-356.
%H Jean-Guillaume Eon, <a href="http://dx.doi.org/10.1107/S0108767301016609">Algebraic determination of generating functions for coordination sequences in crystal structures</a>, Acta Cryst. A58 (2002), 47-53. See p. 51.
%H Jean-Guillaume Eon, <a href="https://doi.org/10.1107/S0108767303022037">Topological density of nets: a direct calculation</a>, Acta Crystallographica Section A (Foundations of Crystallography), A60 (2014), 7-18; DOI: 10.1107/S0108767303022037. See Section 5.
%H Jean-Guillaume Eon, <a href="https://doi.org/10.3390/sym10020035">Symmetry and Topology: The 11 Uninodal Planar Nets Revisited</a>, Symmetry, 10 (2018), 13 pages, doi:10.3390/sym10020035. See Section 4.
%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">arXiv:1803.08530</a>.
%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 Reticular Chemistry Structure Resource, <a href="http://rcsr.net/layers/kgm">kgm</a>
%H Yuriy Sibirmovsky, <a href="/A008579/a008579_1.png">Illustration of initial terms with densely packed circles</a>.
%H N. J. A. Sloane, <a href="/A008579/a008579.png">Illustration of initial terms</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 (0,2,0,-1).
%F G.f.: (1 + 2*x)*(1 + 2*x + 2*x^2 + 2*x^3 - x^4)/(1 - x^2)^2.
%F From _R. J. Mathar_, Nov 26 2014: (Start)
%F a(2n) = A017365(n), n > 0.
%F a(2n+1) = A017137(n), n > 0. (End)
%F From _Stefano Spezia_, Aug 07 2022: (Start)
%F a(n) = (9 + (-1)^n)*n/2 - 2*(-1)^n for n > 1.
%F E.g.f.: 3 - 2*x + (4*x - 2)*cosh(x) + (5*x + 2)*sinh(x). (End)
%p f := n->if n mod 2 = 0 then 10*(n/2)-2 else 8*(n-1)/2+6 fi;
%t a[n_?EvenQ] := 10*n/2-2; a[n_?OddQ] := 8*(n-1)/2+6; a[0] = 1; a[1] = 4; Table[a[n], {n, 0, 45}] (* _Jean-François Alcover_, Nov 18 2011, after Maple *)
%t CoefficientList[Series[(1+2x)(1+2x+2x^2+2x^3-x^4)/(1-x^2)^2,{x,0,50}],x] (* or *) LinearRecurrence[{0,2,0,-1},{1,4,8,14,18,22},50] (* _Harvey P. Dale_, Sep 05 2018 *)
%o (Haskell)
%o a008579 0 = 1
%o a008579 1 = 4
%o a008579 n = (10 - 2*m) * n' + 8*m - 2 where (n',m) = divMod n 2
%o a008579_list = 1 : 4 : concatMap (\x -> map (* 2) [5*x-1,4*x+3]) [1..]
%o -- _Reinhard Zumkeller_, Nov 12 2012
%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 Cf. A017137, A017365.
%K nonn,nice,easy
%O 0,2
%A _N. J. A. Sloane_
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