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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 (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

M. E. Davis, Ordered porous materials for emerging applications, Nature, 417 (Jun 20 2002), 813-821 (gives structure).

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

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]

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)]

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

Conjecture: For n >= 7, a(n) = a(n-1) + a(n-5) - a(n-6). 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. A072149-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).

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 May 29 14:18 EDT 2017. Contains 287247 sequences.