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A250120 Coordination sequence for planar net 3.3.3.3.6 (also called the fsz net). 6132
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

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

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

REFERENCES

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

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

Maurizio Paolini, Table of n, a(n) for n = 0..511

Darrah Chavey, Illustration of a(0)-a(12)

Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers

C. Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, arXiv:1803.08530, March 2018.

Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.

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)]

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 1, -1).

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 "Coloring Book" link. - N. J. A. Sloane, Jan 14 2018; link added Mar 26 2018

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 *)

PROG

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.

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.

Sequence in context: A315023 A315024 A315025 * A315026 A315027 A315028

Adjacent sequences:  A250117 A250118 A250119 * A250121 A250122 A250123

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Nov 23 2014

EXTENSIONS

a(6)-a(10) from Bradley Klee, Nov 23 2014

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

STATUS

approved

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Last modified August 15 15:14 EDT 2018. Contains 313778 sequences. (Running on oeis4.)