%I #65 Mar 07 2024 10:58:36
%S 1,4,10,14,18,24,28,32,38,42,46,52,56,60,66,70,74,80,84,88,94,98,102,
%T 108,112,116,122,126,130,136,140,144,150,154,158,164,168,172,178,182,
%U 186,192,196,200,206,210,214,220,224,228,234,238,242,248,252,256,262
%N G.f.: (x^4+3*x^3+6*x^2+3*x+1)/((1-x)*(1-x^3)).
%C Coordination sequence for Dual(3^3.4^2) tiling with respect to a tetravalent node. This tiling is also called the prismatic pentagonal tiling, or the cem-d net. It is one of the 11 Laves tilings. (The identification of this coordination sequence with the g.f. in the definition was first conjectured by _Colin Barker_, Jan 22 2018.)
%C Also, coordination sequence for a tetravalent node in the "krl" 2-D tiling (or net).
%C Both of these identifications are easily established using the "coloring book" method - see the Goodman-Strauss & Sloane link.
%C For n>0, this is twice A047386 (numbers congruent to 0 or +-2 mod 7).
%C Linear recurrence and g.f. confirmed by Shutov/Maleev link. - _Ray Chandler_, Aug 31 2023
%D Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 66, 3rd row, second tiling. (For the krl tiling.)
%D B. Gruenbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987. See p. 96. (For the Dual(3^3.4^2) tiling.)
%H Rémy Sigrist, <a href="/A298024/b298024.txt">Table of n, a(n) for n = 0..1000</a>
%H Chung, Ping Ngai, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, and Elena Wikner. <a href="http://dx.doi.org/10.1090/noti838">Isoperimetric Pentagonal Tilings</a>, Notices of the AMS 59, no. 5 (2012), pp. 632-640. See Fig. 1 (right).
%H Brian Galebach, <a href="http://probabilitysports.com/tilings.html">Collection of n-Uniform Tilings</a>. See Number 4 from the list of 20 2-uniform tilings.
%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 Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>
%H Frank Morgan, <a href="https://www.youtube.com/watch?v=PpUx0nnWfKQ">Optimal Pentagonal Tilings</a>, Video, May 2021 [Mentions this tiling
%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/cem-d">The cem-d tiling (or net)</a>
%H Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krl">The krl tiling (or net)</a>
%H Anton Shutov and Andrey Maleev, <a href="https://doi.org/10.1515/zkri-2020-0002">Coordination sequences of 2-uniform graphs</a>, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
%H Rémy Sigrist, <a href="/A298024/a298024.png">Illustration of initial terms</a>
%H Rémy Sigrist, <a href="/A298024/a298024.gp.txt">PARI program for A298024</a>
%H N. J. A. Sloane, <a href="/A298024/a298024.pdf">Illustration of initial terms</a> [1 (black), 4 (black), 10 (black), 14 (red)]
%H N. J. A. Sloane, <a href="/A296368/a296368_2.png">Overview of coordination sequences of Laves tilings</a> [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F a(n) = a(n-1) + a(n-3) - a(n-4) for n>4. (Conjectured, correctly, by _Colin Barker_, Jan 22 2018.)
%t CoefficientList[Series[(x^4+3x^3+6x^2+3x+1)/((1-x)(1-x^3)),{x,0,60}],x] (* or *) LinearRecurrence[{1,0,1,-1},{1,4,10,14,18},80] (* _Harvey P. Dale_, Oct 03 2018 *)
%o (PARI) See Links section.
%Y Cf. A301298.
%Y See A298025 for partial sums, A298022 for a trivalent node.
%Y See also A047486.
%Y List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.
%Y Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jan 21 2018
%E More terms from _Rémy Sigrist_, Jan 21 2018
%E Entry revised by _N. J. A. Sloane_, Mar 25 2018