A072154 Coordination sequence for the planar net 4.6.12.

1, 3, 5, 7, 9, 12, 15, 17, 19, 21, 24, 27, 29, 31, 33, 36, 39, 41, 43, 45, 48, 51, 53, 55, 57, 60, 63, 65, 67, 69, 72, 75, 77, 79, 81, 84, 87, 89, 91, 93, 96, 99, 101, 103, 105, 108, 111, 113, 115, 117, 120, 123, 125, 127, 129, 132, 135, 137

Offset: 0

Author

N. J. A. Sloane, Jun 28 2002

Keywords

• nonn
• easy

Comments

There is only one type of node in this structure: each node meets a square, a hexagon and a 12-gon.

The coordination sequence with respect to a particular node gives the number of nodes that can be reached from that node in n steps along edges.

Also, coordination sequence for the aluminophosphate AlPO_4-5 structure.

References

• A. V. Shutov, On the number of words of a given length in plane crystallographic groups (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 302 (2003), Anal. Teor. Chisel i Teor. Funkts. 19, 188--197, 203; translation in J. Math. Sci. (N.Y.) 129 (2005), no. 3, 3922-3926 [MR2023041]. See Table 1, line "p6m" (but beware typos).

Links

• Sean A. Irvine, Table of n, a(n) for n = 0..999
• Joerg Arndt, The 4.6.12 planar net
• Agnes Azzolino, Regular and Semi-Regular Tessellation Paper, 2011.
• Agnes Azzolino, Larger illustration of 4.6.12 planar net [From previous link]
• M. E. Davis, Ordered porous materials for emerging applications, Nature, 417 (Jun 20 2002), 813-821 (gives structure).
• Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
• 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.
• Rostislav Grigorchuk and Cosmas Kravaris, On the growth of the wallpaper groups, arXiv:2012.13661 [math.GR], 2020. See section 4.7 p. 23.
• Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
• Sean A. Irvine, Java implementation with explicit counting
• Tom Karzes, Tiling Coordination Sequences
• Reticular Chemistry Structure Resource, fxt
• N. J. A. Sloane, AlPO_4-5 structure, after Davis
• N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
• N. J. A. Sloane, The subgraph H used in the proof of the formulas
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Crossrefs

Cf. A072149, A072150, A072151, A072152, A072153.

For partial sums see A265078.

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).

See also A301730.

Programs

Mathematica

Join[{1}, LinearRecurrence[{1, 0, 0, 0, 1, -1}, {3, 5, 7, 9, 12, 15}, 100]] (* Jean-François Alcover, Dec 13 2018 *)

Formula

Empirical g.f.: (x+1)^2*(x^2-x+1)*(x^2+x+1)/((x-1)^2*(x^4+x^3+x^2+x+1)). - Colin Barker, Nov 18 2012

This empirical g.f. can also be written as (1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + x^6)/(1 - x - x^5 + x^6). - N. J. A. Sloane, Dec 20 2015

Theorem: For n >= 7, a(n) = a(n-1) + a(n-5) - a(n-6), and a(5k) = 12k (k > 0), a(5k+m) = 12k + 2m + 1 (k >= 0, 1 <= m < 5). This also implies the conjectured g.f.'s. - N. J. A. Sloane, conjectured Dec 20 2015, proved Jan 20 2018.

Notes on the proof, from N. J. A. Sloane, Jan 20 2018 (Start)

The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.

The figure is divided into 6 sectors by the blue trunks. In the interior of each sector, working outwards from the base point P at the origin, there are successively 1,2,3,4,... (red) 12-gons. All the 12-gons (both red and blue) have a unique closest point to P.

If the closest point in a 12-gon is at distance d from P, then the contributions of the 12 points of the 12-gon to a(d), a(d+1), ..., a(d+6) are 1,2,2,2,2,2,1, respectively.

The rest of the proof is now a matter of simple counting.

The blue 12-gons (along the trunks) are especially easy to count, because there is a unique blue 12-gon at shortest distance d from P for d = 1,2,3,4,...

(End)

a(n) = 2*(6*n + sqrt(1 + 2/sqrt(5))*sin(2*n*Pi/5) + sqrt(1 - 2/sqrt(5))*sin(4*n*Pi/5))/5 for n > 0. - Stefano Spezia, Jan 05 2023

Extensions

More terms from Sean A. Irvine, Sep 29 2011

Thanks to Darrah Chavey for pointing out that this is the planar net 4.6.12. - N. J. A. Sloane, Nov 24 2014