A296368 Coordination sequence for the Cairo or dual-3.3.4.3.4 tiling with respect to a trivalent point.
1, 3, 8, 12, 15, 20, 25, 28, 31, 36, 41, 44, 47, 52, 57, 60, 63, 68, 73, 76, 79, 84, 89, 92, 95, 100, 105, 108, 111, 116, 121, 124, 127, 132, 137, 140, 143, 148, 153, 156, 159, 164, 169, 172, 175, 180, 185, 188, 191, 196, 201, 204, 207, 212, 217, 220, 223, 228
Offset: 0
References
- 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.
- Herbert C. Moore, U.S. Patents 928,320 and 928,321, Patented July 20 1909. [Shows Cairo tiling.]
Links
- Rémy Sigrist, Table of n, a(n) for n = 0..1000
- Chaim Goodman-Strauss and N. J. A. Sloane, A portion of the Cairo tiling
- Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
- Chung, Ping Ngai, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, and Elena Wikner. Isoperimetric Pentagonal Tilings, Notices of the AMS 59, no. 5 (2012), pp. 632-640. See Fig. 1 (left).
- Tom Karzes, Tiling Coordination Sequences
- Frank Morgan, Optimal Pentagonal Tilings, Video, May 2021 [Has much to say about the Cairo tiling]
- Reticular Chemistry Structure Resource (RCSR), The mcm tiling (or net)
- Rémy Sigrist, PARI program for A296368
- N. J. A. Sloane, Illustration of initial terms (for a trivalent point)
- N. J. A. Sloane, Illustration of initial terms of coordination sequence 1,4,8,12,... for a tetravalent point
- N. J. A. Sloane, A tiling by rectangles which has the same graph and coordination sequences as the Cairo tiling (Seen on the streets of Piscataway, New Jersey, USA)
- N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
- N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)
- Index entries for linear recurrences with constant coefficients, signature (2, -2, 2, -1).
Crossrefs
For partial sums see A296909.
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).
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.
Programs
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Mathematica
Join[{1, 3, 8}, LinearRecurrence[{2, -2, 2, -1}, {12, 15, 20, 25}, 100]] (* Jean-François Alcover, Aug 05 2018 *) -
PARI
\\ See Links section.
Formula
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)
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)
G.f.: -(x^6-x^5-2*x^4-4*x^2-x-1)/((x^2+1)*(x-1)^2).
From Colin Barker, Dec 23 2017: (Start)
a(n) = (8*n - (-i)^n - i^n) / 2 for n>2, where i=sqrt(-1).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>6.
(End)
Extensions
Terms a(8)-a(20) and RCSR link from Davide M. Proserpio, Dec 22 2017
More terms from Rémy Sigrist, Dec 23 2017
Comments