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A301287 Coordination sequence for node of type 3.12.12 in "cph" 2-D tiling (or net).

%I A301287 #71 Aug 31 2023 10:30:19
%S A301287 1,3,6,7,8,15,18,17,20,25,28,29,30,35,40,39,40,47,50,49,52,57,60,61,
%T A301287 62,67,72,71,72,79,82,81,84,89,92,93,94,99,104,103,104,111,114,113,
%U A301287 116,121,124,125,126,131,136,135,136,143,146,145,148,153,156,157,158
%N A301287 Coordination sequence for node of type 3.12.12 in "cph" 2-D tiling (or net).
%C A301287 Linear recurrence and g.f. confirmed by Shutov/Maleev link. - _Ray Chandler_, Aug 31 2023
%D A301287 Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 66, bottom row, first tiling.
%H A301287 Rémy Sigrist, <a href="/A301287/b301287.txt">Table of n, a(n) for n = 0..1000</a>
%H A301287 Brian Galebach, <a href="http://probabilitysports.com/tilings.html">Collection of n-Uniform Tilings</a>. See Number 2 from the list of  20 2-uniform tilings.
%H A301287 Brian Galebach, <a href="/A301287/a301287.pdf">Enlarged illustration of tiling, suitable for coloring</a> (taken from the web site in the previous link)
%H A301287 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H A301287 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 A301287 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/cph">The cph tiling (or net)</a>
%H A301287 Anton Shutov and Andrey Maleev, <a href="https://doi.org/10.1515/zkri-2020-0002">Coordination sequences of 2-uniform graphs</a>, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
%H A301287 Rémy Sigrist, <a href="/A301287/a301287.png">Illustration of initial terms</a>
%H A301287 Rémy Sigrist, <a href="/A301287/a301287.gp.txt">PARI program for A301287</a>
%H A301287 N. J. A. Sloane, <a href="/A301287/a301287_1.pdf">Trunks and branches for determining coordination sequence, central view. Blue = trunks, red = branches, green = twigs.</a>
%H A301287 N. J. A. Sloane, <a href="/A301287/a301287_2.pdf">Trunks and branches, a different scan, truncated on right but otherwise shows quadrants I and IV in detail</a>
%H A301287 N. J. A. Sloane, <a href="/A301287/a301287_2.png">Trunks and branches in first quadrant, in full.</a>
%H A301287 N. J. A. Sloane, <a href="/A301287/a301287_3.png">Trunks and branches in fourth quadrant, in full.</a>
%H A301287 N. J. A. Sloane, <a href="/A301287/a301287_4.png">Details of proof (page 1)</a>
%H A301287 N. J. A. Sloane, <a href="/A301287/a301287_5.png">Details of proof (page 2)</a>
%H A301287 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,2,-1,1,-1).
%F A301287 G.f. = -(2*x^8-2*x^7-x^6-4*x^5-2*x^4-2*x^3-4*x^2-2*x-1) / ((x^2+1)*(x^2+x+1)*(x-1)^2). _N. J. A. Sloane_, Mar 28 2018 (This is now a theorem. - _N. J. A. Sloane_, Apr 05 2018)
%F A301287 Equivalent conjecture: 3*a(n) = 8*n+2*A057078(n+1)+3*A228826(n+2). - _R. J. Mathar_, Mar 31 2018 (This is now a theorem. - _N. J. A. Sloane_, Apr 05 2018)
%F A301287 Theorem: G.f. = (1+2*x+4*x^2+2*x^3+2*x^4+4*x^5+1*x^6+2*x^7-2*x^8) / ((1-x)*(1+x^2)*(1-x^3)).
%F A301287 Proof. This follows by applying the coloring book method described in the Goodman-Strauss & Sloane article. The trunks and branches structure is shown in the links, and the details of the proof (by calculating the generating function) are on the next two scanned pages. - _N. J. A. Sloane_, Apr 05 2018
%t A301287 Join[{1, 3, 6}, LinearRecurrence[{1, -1, 2, -1, 1, -1}, {7, 8, 15, 18, 17, 20}, 100]] (* _Jean-François Alcover_, Aug 05 2018 *)
%o A301287 (PARI) See Links section.
%Y A301287 Cf. A301289.
%Y A301287 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 A301287 nonn,easy
%O A301287 0,2
%A A301287 _N. J. A. Sloane_, Mar 23 2018
%E A301287 More terms from _Rémy Sigrist_, Mar 27 2018