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A250120 Coordination sequence for planar net 3.3.3.3.6 (also called the fsz net).

%I A250120 #161 Jun 02 2025 16:49:53
%S A250120 1,5,9,15,19,24,29,33,39,43,48,53,57,63,67,72,77,81,87,91,96,101,105,
%T A250120 111,115,120,125,129,135,139,144,149,153,159,163,168,173,177,183,187,
%U A250120 192,197,201,207,211,216,221,225,231,235
%N A250120 Coordination sequence for planar net 3.3.3.3.6 (also called the fsz net).
%C A250120 There are eleven uniform (or Archimedean) tilings (or planar nets), with vertex symbols 3^6, 3^4.6, 3^3.4^2, 3^2.4.3.4, 4^4, 3.4.6.4, 3.6.3.6, 6^3, 3.12^2, 4.6.12, and 4.8^2. Grünbaum and Shephard (1987) is the best reference.
%C A250120 a(n) is the number of vertices at graph distance n from any fixed vertex.
%C A250120 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 A250120 Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Fig. 2.1.5, p. 63.
%D A250120 Marjorie Senechal, Quasicrystals and geometry, Cambridge University Press, Cambridge, 1995, Fig. 1.10, Section 1.3, pp. 13-16.
%H A250120 Stefano Spezia, <a href="/A250120/b250120.txt">Table of n, a(n) for n = 0..10000</a> (first 512 terms from Maurizio Paolini)
%H A250120 Darrah Chavey, <a href="/A250120/a250120_2.png">Illustration of a(0)-a(12)</a>
%H A250120 Jean-Guillaume Eon, <a href="https://doi.org/10.3390/sym10020035">Symmetry and Topology: The 11 Uninodal Planar Nets Revisited</a>, Symmetry, 10 (2018), 13 pages, doi:10.3390/sym10020035. See Section 9.
%H A250120 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a> [updated version May 09 2020]
%H A250120 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">on arXiv</a>, arXiv:1803.08530 [math.CO], 2018-2019.
%H A250120 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 A250120 Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>
%H A250120 Bradley Klee, <a href="/A250120/a250120_1.png">Illustration of a(0)-a(7).</a>
%H A250120 Bradley Klee, <a href="/A250120/a250120_1.nb">Mathematica notebook for A250120</a>
%H A250120 Maurizio Paolini, <a href="/A250120/a250120.txt">C program for A250120</a>
%H A250120 Reticular Chemistry Structure Resource, <a href="http://rcsr.net/layers/fsz">fsz</a>
%H A250120 N. J. A. Sloane, <a href="/A250120/a250120.png">Initial hand-drawn illustration of a(0)-a(5)</a>
%H A250120 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 A250120 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 1</a>, <a href="https://vimeo.com/314790822">Part 2</a>, <a href="https://oeis.org/A320487/a320487.pdf">Slides.</a> (Mentions this sequence)
%H A250120 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).
%F A250120 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 A250120 This would imply that the sequence satisfies the recurrence:
%F A250120 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 A250120 (from Darrah Chavey)
%F A250120 and has generating function
%F A250120 (x^2+x+1)*(x^4+3*x^3+3*x+1)/((x^4+x^3+x^2+x+1)*(x-1)^2)
%F A250120 (from _N. J. A. Sloane_).
%F A250120 All the above conjectures are true - for proof see link to my article with Chaim Goodman-Strauss. - _N. J. A. Sloane_, Jan 14 2018; link added Mar 26 2018
%F A250120 a(n) ~ 24*n/5. - _Stefano Spezia_, May 08 2022
%F A250120 For n>0, a(n) = 2*(12*n + sqrt(1+2/sqrt(5))*sin(4*Pi*n/5) - sqrt(1-2/sqrt(5))*sin(2*Pi*n/5))/5. - _Natalia L. Skirrow_, Apr 13 2025
%t A250120 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 A250120 (C) /* 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 A250120 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 A250120 For partial sums of the present sequence, see A250121.
%K A250120 nonn,nice,easy
%O A250120 0,2
%A A250120 _N. J. A. Sloane_, Nov 23 2014
%E A250120 a(6)-a(10) from _Bradley Klee_, Nov 23 2014
%E A250120 a(11)-a(49) from Maurizio Paolini, Nov 23 2014