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A296368 Coordination sequence for the Cairo or dual-3.3.4.3.4 tiling with respect to a trivalent point.

%I A296368 #96 Feb 22 2025 22:26:08
%S A296368 1,3,8,12,15,20,25,28,31,36,41,44,47,52,57,60,63,68,73,76,79,84,89,92,
%T A296368 95,100,105,108,111,116,121,124,127,132,137,140,143,148,153,156,159,
%U A296368 164,169,172,175,180,185,188,191,196,201,204,207,212,217,220,223,228
%N A296368 Coordination sequence for the Cairo or dual-3.3.4.3.4 tiling with respect to a trivalent point.
%C A296368 There are two types of point in this tiling. This is the coordination sequence with respect to a point of degree 3.
%C A296368 The coordination sequence with respect to a point of degree 4 (see second illustration) is simply 1, 4, 8, 12, 16, 20, ..., the same as the coordination sequence for the 4.4.4.4 square grid (A008574). See the CGS-NJAS link for the proof.
%D A296368 Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Fig. 9.1.3, drawing P_5-24, page 480.
%D A296368 Herbert C. Moore, U.S. Patents 928,320 and 928,321, Patented July 20 1909. [Shows Cairo tiling.]
%H A296368 Rémy Sigrist, <a href="/A296368/b296368.txt">Table of n, a(n) for n = 0..1000</a>
%H A296368 Chaim Goodman-Strauss and N. J. A. Sloane, <a href="/A296368/a296368_3.png">A portion of the Cairo tiling</a>
%H A296368 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 A296368 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 (left).
%H A296368 Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>
%H A296368 Frank Morgan, <a href="https://www.youtube.com/watch?v=PpUx0nnWfKQ">Optimal Pentagonal Tilings</a>, Video, May 2021 [Has much to say about the Cairo tiling]
%H A296368 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/mcm">The mcm tiling (or net)</a>
%H A296368 Rémy Sigrist, <a href="/A296368/a296368.gp.txt">PARI program for A296368</a>
%H A296368 N. J. A. Sloane, <a href="/A296368/a296368.png">Illustration of initial terms (for a trivalent point)</a>
%H A296368 N. J. A. Sloane, <a href="/A296368/a296368_1.png">Illustration of initial terms of coordination sequence 1,4,8,12,... for a tetravalent point</a>
%H A296368 N. J. A. Sloane, <a href="/A296368/a296368_4.png">A tiling by rectangles which has the same graph and coordination sequences as the Cairo tiling</a> (Seen on the streets of Piscataway, New Jersey, USA)
%H A296368 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 A296368 N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, <a href="https://vimeo.com/314786942">Part I</a>, <a href="https://vimeo.com/314790822">Part 2</a>, <a href="https://oeis.org/A320487/a320487.pdf">Slides.</a> (Mentions this sequence)
%H A296368 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2, -2, 2, -1).
%F A296368 The simplest formula is: a(0)=1, a(1)=2, a(2)=8, and thereafter a(n) = 4n if n is odd, 4n - 1 if n == 0 (mod 4), and 4n+1 if n == 2 (mod 4). (See the CGS-NJAS link for proof. - _N. J. A. Sloane_, May 10 2018)
%F A296368 a(n + 4) = a(n) + 16 for any n >= 3. - _Rémy Sigrist_, Dec 23 2017 (See the CGS-NJAS link for a proof. - _N. J. A. Sloane_, Dec 30 2017)
%F A296368 G.f.: -(x^6-x^5-2*x^4-4*x^2-x-1)/((x^2+1)*(x-1)^2).
%F A296368 From _Colin Barker_, Dec 23 2017: (Start)
%F A296368 a(n) = (8*n - (-i)^n - i^n) / 2 for n>2, where i=sqrt(-1).
%F A296368 a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>6.
%F A296368 (End)
%t A296368 Join[{1, 3, 8}, LinearRecurrence[{2, -2, 2, -1}, {12, 15, 20, 25}, 100]] (* _Jean-François Alcover_, Aug 05 2018 *)
%o A296368 (PARI) \\ See Links section.
%Y A296368 For partial sums see A296909.
%Y A296368 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 A296368 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.
%K A296368 nonn,easy
%O A296368 0,2
%A A296368 _N. J. A. Sloane_, Dec 21 2017
%E A296368 Terms a(8)-a(20) and RCSR link from _Davide M. Proserpio_, Dec 22 2017
%E A296368 More terms from _Rémy Sigrist_, Dec 23 2017