

A250122


Coordination sequence for planar net 3.12.12.


34



1, 3, 4, 6, 8, 12, 14, 15, 18, 21, 22, 24, 28, 30, 30, 33, 38, 39, 38, 42, 48, 48, 46, 51, 58, 57, 54, 60, 68, 66, 62, 69, 78, 75, 70, 78, 88, 84, 78, 87, 98, 93, 86, 96, 108, 102, 94, 105, 118, 111, 102, 114, 128, 120, 110, 123, 138, 129
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OFFSET

0,2


COMMENTS

Also, growth series for group with presentation < S, T : S^2 = T^3 = (S*T)^6 = 1 >. See Magma program in A298805.  N. J. A. Sloane, Feb 06 2018


LINKS

Maurizio Paolini, Table of n, a(n) for n = 0..1021
Agnes Azzolino, Regular and SemiRegular Tessellation Paper, 2011
Agnes Azzolino, Illustration of 3.12.12 tiling [From previous link]
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 arXiv:1803.08530.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227247.
Tom Karzes, Tiling Coordination Sequences
Maurizio Paolini, C program for A250122
Reticular Chemistry Structure Resource, hca
N. J. A. Sloane, The uniform planar nets and their Anumbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]


FORMULA

From Joseph Myers, Nov 28 2014: (Start)
Empirically,
a(4n) = 10n  2 except for a(0) = 1
a(4n+1) = 9n + 3
a(4n+2) = 8n + 6 except for a(2) = 4
a(4n+3) = 9n + 6. (End)
If these are correct, the sequence has g.f.
(1  x  x^2  3*x^3 + x^4  5*x^5 + 3*x^6  4*x^7 + 2*x^8)/((x  1)^2*(x^2 + 1)^2).  N. J. A. Sloane, Nov 28 2014
All the above conjectures are true. Details will be added soon.  N. J. A. Sloane, Dec 31 2015


MATHEMATICA

Join[{1, 3, 4}, LinearRecurrence[{2, 3, 4, 3, 2, 1}, {6, 8, 12, 14, 15, 18}, 100]] (* JeanFrançois Alcover, Aug 05 2018 *)


CROSSREFS

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: A050115 A167711 A037346 * A243653 A203444 A008864
Adjacent sequences: A250119 A250120 A250121 * A250123 A250124 A250125


KEYWORD

nonn


AUTHOR

Darrah Chavey, Nov 23 2014


EXTENSIONS

a(8) onwards from Maurizio Paolini and Joseph Myers (independently), Nov 28 2014


STATUS

approved



