

A072154


Coordination sequence for the planar net 4.6.12.


30



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

There is only one type of node in this structure: each node meets a square, a hexagon and a 12gon.
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_45 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, 188197, 203; translation in J. Math. Sci. (N.Y.) 129 (2005), no. 3, 39223926 [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 SemiRegular 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), 813821 (gives structure).
Brian Galebach, kuniform tilings (k <= 6) and their Anumbers
Chaim GoodmanStrauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121134, also on NJAS's home page. Also on arXiv, arXiv:1803.08530 [math.CO], 20182019.
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), 227247.
Sean A. Irvine, Java implementation with explicit counting
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource, fxt
N. J. A. Sloane, AlPO_45 structure, after Davis
N. J. A. Sloane, The uniform planar nets and their Anumbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
N. J. A. Sloane, The subgraph H used in the proof of the formulas


FORMULA

Empirical g.f.: (x+1)^2*(x^2x+1)*(x^2+x+1)/((x1)^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(n1) + a(n5)  a(n6), 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 GoodmanStrauss & 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) 12gons. All the 12gons (both red and blue) have a unique closest point to P.
If the closest point in a 12gon is at distance d from P, then the contributions of the 12 points of the 12gon 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 12gons (along the trunks) are especially easy to count, because there is a unique blue 12gon at shortest distance d from P for d = 1,2,3,4,...
(End)


MATHEMATICA

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


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.
Sequence in context: A139130 A219087 A186705 * A309269 A204206 A080751
Adjacent sequences: A072151 A072152 A072153 * A072155 A072156 A072157


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Jun 28 2002


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


STATUS

approved



