%I #73 Jan 05 2023 18:29:34

%S 1,3,4,6,8,12,14,15,18,21,22,24,28,30,30,33,38,39,38,42,48,48,46,51,

%T 58,57,54,60,68,66,62,69,78,75,70,78,88,84,78,87,98,93,86,96,108,102,

%U 94,105,118,111,102,114,128,120,110,123,138,129

%N Coordination sequence for planar net 3.12.12.

%C Also, growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^6 = 1 >. See Magma program in A298805. - _N. J. A. Sloane_, Feb 06 2018

%H Maurizio Paolini, <a href="/A250122/b250122.txt">Table of n, a(n) for n = 0..1021</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="/A250122/a250122.gif">Illustration of 3.12.12 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. See section 10 The 3.12^2 tiling.

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

%H Maurizio Paolini, <a href="/A250122/a250122.txt">C program for A250122</a>

%H Reticular Chemistry Structure Resource, <a href="http://rcsr.net/layers/hca">hca</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_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-3,4,-3,2,-1).

%F From _Joseph Myers_, Nov 28 2014: (Start)

%F Empirically,

%F a(4n) = 10n - 2 except for a(0) = 1

%F a(4n+1) = 9n + 3

%F a(4n+2) = 8n + 6 except for a(2) = 4

%F a(4n+3) = 9n + 6. (End)

%F If these are correct, the sequence has g.f.

%F -(-1 - x - x^2 - 3*x^3 + x^4 - 5*x^5 + 3*x^6 - 4*x^7 + 2*x^8)/((x - 1)^2*(x^2 + 1)^2). - _N. J. A. Sloane_, Nov 28 2014

%F All the above conjectures are true. - _N. J. A. Sloane_, Dec 31 2015

%F E.g.f.: (9*x*cosh(x) - 4*(2*cos(x) + x^2 - 3) + 9*x*sinh(x) - (x - 3)*sin(x))/4. - _Stefano Spezia_, Jan 05 2023

%t Join[{1, 3, 4}, LinearRecurrence[{2, -3, 4, -3, 2, -1}, {6, 8, 12, 14, 15, 18}, 100]] (* _Jean-François Alcover_, Aug 05 2018 *)

%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. A298805.

%K nonn,easy

%O 0,2

%A _Darrah Chavey_, Nov 23 2014

%E a(8) onwards from Maurizio Paolini and _Joseph Myers_ (independently), Nov 28 2014