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A301682 Coordination sequence for node of type V1 in "krg" 2-D tiling (or net).

%I A301682 #31 Jun 12 2024 17:18:43
%S A301682 1,6,6,18,18,18,36,30,30,54,42,42,72,54,54,90,66,66,108,78,78,126,90,
%T A301682 90,144,102,102,162,114,114,180,126,126,198,138,138,216,150,150,234,
%U A301682 162,162,252,174,174,270,186,186,288,198,198,306,210,210,324,222,222,342,234,234,360,246,246,378,258
%N A301682 Coordination sequence for node of type V1 in "krg" 2-D tiling (or net).
%C A301682 Linear recurrence and g.f. confirmed by Shutov/Maleev link. - _Ray Chandler_, Aug 30 2023
%D A301682 Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 66, 2nd row, 2nd tiling.
%H A301682 Ray Chandler, <a href="/A301682/b301682.txt">Table of n, a(n) for n = 0..1000</a>
%H A301682 Brian Galebach, <a href="http://probabilitysports.com/tilings.html">Collection of n-Uniform Tilings</a>. See Number 10 from the list of 20 2-uniform tilings.
%H A301682 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H A301682 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H A301682 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krg">The krg tiling (or net)</a>
%H A301682 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 A301682 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).
%F A301682 G.f.: -(-x^6-6*x^5-6*x^4-16*x^3-6*x^2-6*x-1)/(x^6-2*x^3+1). - _N. J. A. Sloane_, Mar 29 2018
%F A301682 a(n) = 2*(7*n + n*A099837(n+3) + 3*A049347(n+2))/3 for n > 0. - _Stefano Spezia_, Jun 08 2024
%t A301682 LinearRecurrence[{0,0,2,0,0,-1},{1,6,6,18,18,18,36},100] (* _Paolo Xausa_, Nov 15 2023 *)
%Y A301682 Cf. A301684.
%Y A301682 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 A301682 Cf. A049347, A099837.
%K A301682 nonn,easy
%O A301682 0,2
%A A301682 _N. J. A. Sloane_, Mar 25 2018
%E A301682 a(11)-a(100) from _Davide M. Proserpio_, Mar 28 2018