

A296368


Coordination sequence for the Cairo or dual3.3.4.3.4 tiling with respect to a trivalent point.


25



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
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OFFSET

0,2


COMMENTS

There are two types of point in this tiling. This is the coordination sequence with respect to a point of degree 3.
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 CGSNJAS link for the proof.


REFERENCES

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Fig. 9.1.3, drawing P_524, 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
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. 632640. See Fig. 1 (left).
Chaim GoodmanStrauss and N. J. A. Sloane, A portion of the Cairo tiling
Chaim GoodmanStrauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121134, also on NJAS's home page. Also arXiv:1803.08530.
Tom Karzes, Tiling Coordination Sequences
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 GrünbaumShephard 1987 with Anumbers 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)


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 CGSNJAS 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 CGSNJAS link for a proof.  N. J. A. Sloane, Dec 30 2017)
G.f.: (x^6x^52*x^44*x^2x1)/((x^2+1)*(x1)^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(n1)  2*a(n2) + 2*a(n3)  a(n4) for n>6.
(End)


MATHEMATICA

Join[{1, 3, 8}, LinearRecurrence[{2, 2, 2, 1}, {12, 15, 20, 25}, 100]] (* JeanFrançois Alcover, Aug 05 2018 *)


PROG

(PARI) See Links section.


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.
Sequence in context: A280239 A344345 A310284 * A183991 A022806 A084162
Adjacent sequences: A296365 A296366 A296367 * A296369 A296370 A296371


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Dec 21 2017


EXTENSIONS

Terms a(8)a(20) and RCSR link from Davide M. Proserpio, Dec 22 2017
More terms from Rémy Sigrist, Dec 23 2017


STATUS

approved



