

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
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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

Reticular Chemistry Structure Resource, fxt


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.
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)
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


MATHEMATICA

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


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS

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



