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%I A265035 #73 Nov 15 2023 08:05:43
%S A265035 1,3,6,9,11,14,17,21,25,28,30,32,35,39,43,46,48,50,53,57,61,64,66,68,
%T A265035 71,75,79,82,84,86,89,93,97,100,102,104,107,111,115,118,120,122,125,
%U A265035 129,133,136,138,140,143,147,151,154,156,158,161,165,169,172,174,176
%N A265035 Coordination sequence of 2-uniform tiling {3.4.6.4, 4.6.12} with respect to a point of type 4.6.12.
%C A265035 _Joseph Myers_ (Dec 14 2015) reports that "My program for coordination sequences requires describing the tiling structure under translation, listing all edges in the form: (class1, 0, 0) has an edge to (class2, x, y). The present tiling has 18 orbits of vertices under translation and 30 orbits of edges under translation (each of which is described in both directions).  So in principle it could generate the other 19 2-uniform tilings, but without a cross check with hand-computed terms there's a risk of e.g. missing some edges, and a fair amount of work producing all the descriptions of translation classes of edges."
%C A265035 Linear recurrence and g.f. confirmed by Shutov/Maleev link. - _Ray Chandler_, Aug 31 2023
%D A265035 Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See page 67, 4th row, 3rd tiling.
%D A265035 Otto Krötenheerdt, Die homogenen Mosaike n-ter Ordnung in der euklidischen Ebene, I, II, III, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math-Natur. Reihe, 18 (1969), 273-290; 19 (1970), 19-38 and 97-122. [Includes classification of 2-uniform tilings]
%D A265035 Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166.
%H A265035 Joseph Myers, <a href="/A265035/b265035.txt">Table of n, a(n) for n = 0..20000</a>
%H A265035 Miguel Carlos Fernández-Cabo, <a href="http://dx.doi.org/10.12988/imf.2014.45103">Artisan Procedures to Generate Uniform Tilings</a>, International Mathematical Forum, Vol. 9, 2014, no. 23, 1109-1130. [Background information]
%H A265035 Brian Galebach, <a href="http://probabilitysports.com/tilings.html">Collection of n-Uniform Tilings</a>. See Number 1 from the list of 20 2-uniform tilings.
%H A265035 Brian Galebach, <a href="/A265035/a265035.png">The tiling {3.4.6.4, 4.6.12}</a>, Number 1 from list of 20 2-uniform tilings. (From the previous link)
%H A265035 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H A265035 Chaim Goodman-Strauss and N. J. A. Sloane, <a href="https://doi.org/10.1107/S2053273318014481">A Coloring Book Approach to Finding Coordination Sequences</a>, Acta Cryst. A75 (2019), 121-134, also <a href="http://NeilSloane.com/doc/Cairo_final.pdf">on NJAS's home page</a>. Also <a href="http://arxiv.org/abs/1803.08530">arXiv:1803.08530</a>.
%H A265035 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krt">The krt tiling (or net)</a>
%H A265035 N. J. A. Sloane, <a href="/A265035/a265035_1.png">Illustration of initial terms of A265035</a> (point of type 4.6.12)
%H A265035 N. J. A. Sloane, <a href="/A265035/a265035_2.png">Illustration of initial terms of A265036</a> (point of type 3.4.6.4)
%H A265035 N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, <a href="https://vimeo.com/314786942">Part I</a>, <a href="https://vimeo.com/314790822">Part 2</a>, <a href="https://oeis.org/A320487/a320487.pdf">Slides.</a> (Mentions this sequence)
%H A265035 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,3,-1).
%F A265035 Based on the b-file, the g.f. appears to be (1+x^2+2*x^5-2*x^6+2*x^7-x^8)/(1-3*x+4*x^2-3*x^3+x^4). This matches the first 1000 terms, so is probably correct. - _N. J. A. Sloane_, Dec 14 2015
%F A265035 Conjectured g.f. is equivalent to a(n) = 3*n - A010892(n+1) for n >= 5. - _R. J. Mathar_, Oct 09 2020
%t A265035 LinearRecurrence[{3,-4,3,-1},{1,3,6,9,11,14,17,21,25},100] (* _Paolo Xausa_, Nov 15 2023 *)
%Y A265035 See A265036 for the other type of point.
%Y A265035 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).
%Y A265035 Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
%K A265035 nonn
%O A265035 0,2
%A A265035 _N. J. A. Sloane_, Dec 12 2015
%E A265035 Extended by _Joseph Myers_, Dec 13 2015
%E A265035 b-file extended by _Joseph Myers_, Dec 18 2015

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