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A301672 Coordination sequence for node of type V2 in "krr" 2-D tiling (or net).

%I A301672 #33 Jun 13 2024 15:42:03
%S A301672 1,4,8,13,17,20,25,30,33,37,42,46,50,54,58,63,67,70,75,80,83,87,92,96,
%T A301672 100,104,108,113,117,120,125,130,133,137,142,146,150,154,158,163,167,
%U A301672 170,175,180,183,187,192,196,200,204,208,213,217,220,225,230,233
%N A301672 Coordination sequence for node of type V2 in "krr" 2-D tiling (or net).
%C A301672 Linear recurrence and g.f. confirmed by Shutov/Maleev link. - _Ray Chandler_, Aug 30 2023
%D A301672 Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, bottom row, 1st tiling.
%H A301672 Rémy Sigrist, <a href="/A301672/b301672.txt">Table of n, a(n) for n = 0..1000</a>
%H A301672 Brian Galebach, <a href="http://probabilitysports.com/tilings.html">Collection of n-Uniform Tilings</a>. See Number 7 from the list of 20 2-uniform tilings.
%H A301672 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H A301672 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krr">The krr tiling (or net)</a>
%H A301672 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 A301672 Rémy Sigrist, <a href="/A301672/a301672.png">Illustration of first terms</a>
%H A301672 Rémy Sigrist, <a href="/A301672/a301672.gp.txt">PARI program for A301672</a>
%H A301672 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,2,-1,1,-1).
%F A301672 Based on the b-file, the g.f. appears to be
%F A301672 (x^3+2*x^2+x+1)*(x^3+x^2+2*x+1) / ((1-x)*(1+x^2)*(1-x^3)). - _N. J. A. Sloane_, Mar 25 2018
%F A301672 a(n) = (75*n - 9*A163805(n+2) + 6*A049347(n+2))/18 for n > 0. - _Stefano Spezia_, Jun 08 2024
%t A301672 LinearRecurrence[{1,-1,2,-1,1,-1},{1,4,8,13,17,20,25},100] (* _Paolo Xausa_, Nov 15 2023 *)
%o A301672 (PARI) \\ See Links section.
%Y A301672 Cf. A301670.
%Y A301672 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 A301672 Cf. A049347, A163805.
%K A301672 nonn,easy
%O A301672 0,2
%A A301672 _N. J. A. Sloane_, Mar 25 2018
%E A301672 More terms from _Rémy Sigrist_, Mar 25 2018