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A250122 Coordination sequence for planar net 3.12.12. 20
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 (list; graph; refs; listen; history; text; internal format)
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 Semi-Regular Tessellation Paper, 2011

Agnes Azzolino, Illustration of 3.12.12 tiling [From previous link]

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

Maurizio Paolini, C program for A250122

Reticular Chemistry Structure Resource, hca

N. J. A. Sloane, The uniform planar nets and their A-numbers [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]] (* Jean-François 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,changed

AUTHOR

Darrah Chavey, Nov 23 2014

EXTENSIONS

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

STATUS

approved

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Last modified August 18 00:28 EDT 2018. Contains 313817 sequences. (Running on oeis4.)