|
%I #61 Aug 31 2023 10:33:43
%S 1,4,5,6,12,14,15,18,21,26,28,26,31,38,37,38,44,46,47,50,53,58,60,58,
%T 63,70,69,70,76,78,79,82,85,90,92,90,95,102,101,102,108,110,111,114,
%U 117,122,124,122,127,134,133,134,140,142,143,146,149,154,156,154
%N Coordination sequence for a tetravalent node of type 3.4.3.12 in "cph" 2-D tiling (or net).
%C Linear recurrence and g.f. confirmed by Shutov/Maleev link. - _Ray Chandler_, Aug 31 2023
%D 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 Rémy Sigrist, <a href="/A301289/b301289.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 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 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H 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 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/cph">The cph tiling (or net)</a>
%H 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 Rémy Sigrist, <a href="/A301289/a301289.png">Illustration of first terms</a>
%H Rémy Sigrist, <a href="/A301289/a301289.gp.txt">PARI program for A301289</a>
%H N. J. A. Sloane, <a href="/A301289/a301289.pdf">Trunks and branches for determining coordination sequence, central view. Blue = trunks, red = branches, green = twigs.</a>
%H N. J. A. Sloane, <a href="/A301289/a301289_1.pdf">Trunks and branches, a different scan, truncated on right but otherwise shows quadrants I and IV in detail</a>
%H N. J. A. Sloane, <a href="/A301289/a301289_1.png">Page 1 of the proof of the theorem: counts for various classes of nodes</a> [Page 1 will be added as soon as I can find a wide-bed scanner]
%H N. J. A. Sloane, <a href="/A301289/a301289_2.png">Page 2 of the proof of the theorem: the corresponding generating functions</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2, -3, 4, -4, 4, -3, 2, -1).
%F Theorem: G.f. = (1+2*x+4*x^3+3*x^4+6*x^6-4*x^7+6*x^8-2*x^9) / ((1-x)^2*(1+x^2)*(1+x^2+x^4)).
%F The proof uses the coloring book method described in the Goodman-Strauss & Sloane article. The trunks and branches structure is shown in the first scan. (Not yet added.) The trunks are blue, the branches are red, and the twigs are green. There is mirror symmetry about the Y-axis, and quadrants I and II are essentially identical, as are quadrants III and IV. The counts of the various classes of nodes are given in the second scan, and the corresponding generating functions are in the third scan. Adding up the different terms gives the g.f. stated above. - _N. J. A. Sloane_, Apr 07 2018
%t Join[{1, 4}, LinearRecurrence[{2, -3, 4, -4, 4, -3, 2, -1}, {5, 6, 12, 14, 15, 18, 21, 26}, 100]] (* _Jean-François Alcover_, Aug 05 2018 *)
%o (PARI) See Links section.
%Y Cf. A301287.
%Y 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 nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Mar 23 2018
%E More terms from _Rémy Sigrist_, Mar 27 2018
|
