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Coordination sequence for the planar net 4.6.12.
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%I #101 Jan 05 2023 16:12:38

%S 1,3,5,7,9,12,15,17,19,21,24,27,29,31,33,36,39,41,43,45,48,51,53,55,

%T 57,60,63,65,67,69,72,75,77,79,81,84,87,89,91,93,96,99,101,103,105,

%U 108,111,113,115,117,120,123,125,127,129,132,135,137

%N Coordination sequence for the planar net 4.6.12.

%C There is only one type of node in this structure: each node meets a square, a hexagon and a 12-gon.

%C The coordination sequence with respect to a particular node gives the number of nodes that can be reached from that node in n steps along edges.

%C Also, coordination sequence for the aluminophosphate AlPO_4-5 structure.

%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, line "p6m" (but beware typos).

%H Sean A. Irvine, <a href="/A072154/b072154.txt">Table of n, a(n) for n = 0..999</a>

%H Joerg Arndt, <a href="/A072154/a072154.pdf">The 4.6.12 planar net</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="/A072154/a072154_1.gif">Larger illustration of 4.6.12 planar net</a> [From previous link]

%H M. E. Davis, <a href="https://www.nature.com/articles/nature00785">Ordered porous materials for emerging applications</a>, Nature, 417 (Jun 20 2002), 813-821 (gives structure).

%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 4.7 p. 23.

%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 Sean A. Irvine, <a href="/A072154/a072154.java.txt">Java implementation with explicit counting</a>

%H Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>

%H Reticular Chemistry Structure Resource, <a href="http://rcsr.net/layers/fxt">fxt</a>

%H N. J. A. Sloane, <a href="/A072154/a072154.gif">AlPO_4-5 structure, after Davis</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 N. J. A. Sloane, <a href="/A072154/a072154.png">The subgraph H used in the proof of the formulas</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).

%F Empirical g.f.: (x+1)^2*(x^2-x+1)*(x^2+x+1)/((x-1)^2*(x^4+x^3+x^2+x+1)). - _Colin Barker_, Nov 18 2012

%F This empirical g.f. can also be written as (1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + x^6)/(1 - x - x^5 + x^6). - _N. J. A. Sloane_, Dec 20 2015

%F Theorem: For n >= 7, a(n) = a(n-1) + a(n-5) - a(n-6), and a(5k) = 12k (k > 0), a(5k+m) = 12k + 2m + 1 (k >= 0, 1 <= m < 5). This also implies the conjectured g.f.'s. - _N. J. A. Sloane_, conjectured Dec 20 2015, proved Jan 20 2018.

%F Notes on the proof, from _N. J. A. Sloane_, Jan 20 2018 (Start)

%F The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.

%F The figure is divided into 6 sectors by the blue trunks. In the interior of each sector, working outwards from the base point P at the origin, there are successively 1,2,3,4,... (red) 12-gons. All the 12-gons (both red and blue) have a unique closest point to P.

%F If the closest point in a 12-gon is at distance d from P, then the contributions of the 12 points of the 12-gon to a(d), a(d+1), ..., a(d+6) are 1,2,2,2,2,2,1, respectively.

%F The rest of the proof is now a matter of simple counting.

%F The blue 12-gons (along the trunks) are especially easy to count, because there is a unique blue 12-gon at shortest distance d from P for d = 1,2,3,4,...

%F (End)

%F a(n) = 2*(6*n + sqrt(1 + 2/sqrt(5))*sin(2*n*Pi/5) + sqrt(1 - 2/sqrt(5))*sin(4*n*Pi/5))/5 for n > 0. - _Stefano Spezia_, Jan 05 2023

%t Join[{1}, LinearRecurrence[{1, 0, 0, 0, 1, -1}, {3, 5, 7, 9, 12, 15}, 100]] (* _Jean-François Alcover_, Dec 13 2018 *)

%Y Cf. A072149, A072150, A072151, A072152, A072153.

%Y For partial sums see A265078.

%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 See also A301730.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jun 28 2002

%E More terms from _Sean A. Irvine_, Sep 29 2011

%E Thanks to Darrah Chavey for pointing out that this is the planar net 4.6.12. - _N. J. A. Sloane_, Nov 24 2014