A250120 Coordination sequence for planar net 3.3.3.3.6 (also called the fsz net).
1, 5, 9, 15, 19, 24, 29, 33, 39, 43, 48, 53, 57, 63, 67, 72, 77, 81, 87, 91, 96, 101, 105, 111, 115, 120, 125, 129, 135, 139, 144, 149, 153, 159, 163, 168, 173, 177, 183, 187, 192, 197, 201, 207, 211, 216, 221, 225, 231, 235
Offset: 0
References
- Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Fig. 2.1.5, p. 63.
- Marjorie Senechal, Quasicrystals and geometry, Cambridge University Press, Cambridge, 1995, Fig. 1.10, Section 1.3, pp. 13-16.
Links
- Stefano Spezia, Table of n, a(n) for n = 0..10000 (first 512 terms from Maurizio Paolini)
- Darrah Chavey, Illustration of a(0)-a(12)
- Jean-Guillaume Eon, Symmetry and Topology: The 11 Uninodal Planar Nets Revisited, Symmetry, 10 (2018), 13 pages, doi:10.3390/sym10020035. See Section 9.
- Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers [updated version May 09 2020]
- 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 on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.
- Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
- Tom Karzes, Tiling Coordination Sequences
- Bradley Klee, Illustration of a(0)-a(7).
- Bradley Klee, Mathematica notebook for A250120
- Maurizio Paolini, C program for A250120
- Reticular Chemistry Structure Resource, fsz
- N. J. A. Sloane, Initial hand-drawn illustration of a(0)-a(5)
- N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Grünbaum and Shephard (1977)]
- N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part 1, Part 2, Slides. (Mentions this sequence)
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).
Crossrefs
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).
For partial sums of the present sequence, see A250121.
Programs
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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. */
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Mathematica
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 *)
Formula
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.
This would imply that the sequence satisfies the recurrence:
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 }
(from Darrah Chavey)
and has generating function
(x^2+x+1)*(x^4+3*x^3+3*x+1)/((x^4+x^3+x^2+x+1)*(x-1)^2)
(from N. J. A. Sloane).
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
a(n) ~ 24*n/5. - Stefano Spezia, May 08 2022
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
Extensions
a(6)-a(10) from Bradley Klee, Nov 23 2014
a(11)-a(49) from Maurizio Paolini, Nov 23 2014
Comments