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A250120 Coordination sequence for planar net (also called the fsz net). 6132


%S 1,5,9,15,19,24,29,33,39,43,48,53,57,63,67,72,77,81,87,91,96,101,105,

%T 111,115,120,125,129,135,139,144,149,153,159,163,168,173,177,183,187,

%U 192,197,201,207,211,216,221,225,231,235

%N Coordination sequence for planar net (also called the fsz net).

%C There are eleven uniform (or Archimedean) tilings (or planar nets), with vertex symbols 3^6, 3^4.6, 3^3.4^2, 3^, 4^4,,, 6^3, 3.12^2, 4.6.12, and 4.8^2. Grünbaum and Shephard (1987) is the best reference.

%C a(n) is the number of vertices at graph distance n from any fixed vertex.

%C The Mathematica notebook can compute 30 or 40 iterations, and colors them with period 5. You could also change out images if you want to. These graphs are better for analyzing 5-iteration chunks of the pattern. You can see that under iteration all fragments of the circumferences are preserved in shape and translated outwards a distance approximately sqrt(21) (relative to small triangle edge), the length of a long diagonal of larger rhombus unit cell. The conjectured recurrence should follow from an analysis of how new pieces occur in between the translated pieces. - _Bradley Klee_, Nov 26 2014

%D Jean-Guillaume Eon, Symmetry and Topology: The 11 Uninodal Planar Nets Revisited, Symmetry, 10 (2018), 13 pages, doi:10.3390/sym10020035. See Section 9.

%D Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Fig. 2.1.5, p. 63.

%D Marjorie Senechal, Quasicrystals and geometry, Cambridge University Press, Cambridge, 1995, Fig. 1.10, Section 1.3, pp. 13-16.

%H Maurizio Paolini, <a href="/A250120/b250120.txt">Table of n, a(n) for n = 0..511</a>

%H Darrah Chavey, <a href="/A250120/a250120_2.png">Illustration of a(0)-a(12)</a>

%H Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>

%H C. Goodman-Strauss and N. J. A. Sloane, <a href="http://arxiv.org/abs/1803.08530">A Coloring Book Approach to Finding Coordination Sequences</a>, arXiv:1803.08530, March 2018.

%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 Bradley Klee, <a href="/A250120/a250120_1.png">Illustration of a(0)-a(7).</a>

%H Bradley Klee, <a href="/A250120/a250120_1.nb">Mathematica notebook for A250120</a>

%H Maurizio Paolini, <a href="/A250120/a250120.txt">C program for A250120</a>

%H Reticular Chemistry Structure Resource, <a href="http://rcsr.net/layers/fsz">fsz</a>

%H N. J. A. Sloane, <a href="/A250120/a250120.png">Initial hand-drawn illustration of a(0)-a(5)</a>

%H N. J. A. Sloane, <a href="/A008576/a008576.png">The uniform planar nets and their A-numbers</a> [Annotated scanned figure from Grünbaum and Shephard (1977)]

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 1, -1).

%F Based on the computations of Darrah Chavey, Bradley Klee, and Maurizio Paolini, there is a strong conjecture that the first differences of this sequence are 4, 4, 6, 4, 5, 5, 4, 6, 4, 5, 5, 4, 6, 4, 5, 5, ..., that is, 4 followed by (4,6,4,5,5) repeated.

%F This would imply that the sequence satisfies the recurrence:

%F for n > 2, a(n) = a(n-1) + { n == 0,3 (mod 5), 4; n == 4 (mod 5), 6; n == 1,2 (mod 5), 5 }

%F (from Darrah Chavey)

%F and has generating function

%F (x^2+x+1)*(x^4+3*x^3+3*x+1)/((x^4+x^3+x^2+x+1)*(x-1)^2)

%F (from _N. J. A. Sloane_).

%F All the above conjectures are true - for proof see "Coloring Book" link. - _N. J. A. Sloane_, Jan 14 2018; link added Mar 26 2018

%t CoefficientList[Series[(x^2+x+1)(x^4+3x^3+3x+1)/((x^4+x^3+x^2+x+1)(x-1)^2),{x,0,80}],x] (* or *) LinearRecurrence[{1,0,0,0,1,-1},{1,5,9,15,19,24,29},60] (* _Harvey P. Dale_, May 05 2018 *)

%o Comments on the C program (see link) from Maurizio Paolini, Nov 23 2014: Basically what I do is deform the net onto the integral lattice, connect nodes aligned either horizontally, vertically or diagonally from northeast to southwest, marking as UNREACHABLE the nodes with coordinates (i, j) satisfying i + 2*j = 0 mod 7. Then the code computes the distance from each node to the central node of the grid.

%Y List of coordination sequences for uniform planar nets: A008458 (the planar net, A008486 (6^3), A008574 ( and, A008576 (4.8.8), A008579 (, A008706 (, A072154 (4.6.12), A219529 (, A250120 (, A250122 (3.12.12).

%Y For partial sums of the present sequence, see A250121.

%K nonn,nice

%O 0,2

%A _N. J. A. Sloane_, Nov 23 2014

%E a(6)-a(10) from _Bradley Klee_, Nov 23 2014

%E a(11)-a(49) from Maurizio Paolini, Nov 23 2014

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Last modified August 21 18:08 EDT 2018. Contains 313955 sequences. (Running on oeis4.)