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A301686 Coordination sequence for node of type V1 in "krh" 2-D tiling (or net).

%I A301686 #36 Jun 13 2024 16:52:28
%S A301686 1,5,7,14,17,19,28,29,31,42,41,43,56,53,55,70,65,67,84,77,79,98,89,91,
%T A301686 112,101,103,126,113,115,140,125,127,154,137,139,168,149,151,182,161,
%U A301686 163,196,173,175,210,185,187,224,197,199,238,209,211,252,221,223
%N A301686 Coordination sequence for node of type V1 in "krh" 2-D tiling (or net).
%C A301686 Linear recurrence and g.f. confirmed by Shutov/Maleev link. - _Ray Chandler_, Aug 30 2023
%D A301686 Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, 2nd row, 1st tiling.
%H A301686 Rémy Sigrist, <a href="/A301686/b301686.txt">Table of n, a(n) for n = 0..1000</a>
%H A301686 Brian Galebach, <a href="http://probabilitysports.com/tilings.html">Collection of n-Uniform Tilings</a>. See Number 12 from the list of 20 2-uniform tilings.
%H A301686 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H A301686 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 A301686 Rémy Sigrist, <a href="/A301686/a301686.png">Illustration of first terms</a>
%H A301686 Rémy Sigrist, <a href="/A301686/a301686.gp.txt">PARI program for A301686</a>
%H A301686 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krh">The krh tiling (or net)</a>
%H A301686 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).
%F A301686 G.f.: -(-x^6-5*x^5-7*x^4-12*x^3-7*x^2-5*x-1)/(x^6-2*x^3+1). - _N. J. A. Sloane_, Mar 28 2018
%F A301686 a(n) = 2*(19*n + n*A099837(n+3) + 9*A049347(n+2)/2)/9 for n > 0. - _Stefano Spezia_, Jun 08 2024
%t A301686 LinearRecurrence[{0,0,2,0,0,-1},{1,5,7,14,17,19,28},100] (* _Paolo Xausa_, Nov 16 2023 *)
%o A301686 (PARI) \\ See Links section.
%Y A301686 Cf. A301688.
%Y A301686 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.
%Y A301686 Cf. A049347, A099837.
%K A301686 nonn,easy
%O A301686 0,2
%A A301686 _N. J. A. Sloane_, Mar 25 2018
%E A301686 More terms from _Rémy Sigrist_, Mar 26 2018