

A072154


Coordination sequence for the planar net 4.6.12.


23



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

M. E. Davis, Ordered porous materials for emerging applications, Nature, 417 (Jun 20 2002), 813821 (gives structure).
Branko Gruenbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227247.


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]
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)]


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)/(1xx^5+x^6).  N. J. A. Sloane, Dec 20 2015
Conjecture: For n >= 7, a(n) = a(n1) + a(n5)  a(n6). Conjecture: a(5k) = 12k (k>0), a(5k+m) = 12k+2m+1 (k >= 0, 1 <= m < 5).  N. J. A. Sloane, Dec 20 2015
I now have a proof for all these empirical formulas and the conjectures.  N. J. A. Sloane, Dec 28 2015


CROSSREFS

Cf. A072149A072153.
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).
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



