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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 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

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

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

Sean A. Irvine, Java implementation with explicit counting

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

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)

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 * A204206 A080751 A025218

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

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Last modified July 16 22:24 EDT 2018. Contains 312693 sequences. (Running on oeis4.)