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%I A301708 #45 Jun 12 2024 09:40:24
%S A301708 1,6,11,16,22,28,33,38,44,50,55,60,66,72,77,82,88,94,99,104,110,116,
%T A301708 121,126,132,138,143,148,154,160,165,170,176,182,187,192,198,204,209,
%U A301708 214,220,226,231,236,242,248,253,258,264,270,275,280,286,292,297,302
%N A301708 Coordination sequence for node of type V1 in "krc" 2-D tiling (or net).
%C A301708 Linear recurrence and g.f. confirmed by Shutov/Maleev link. - _Ray Chandler_, Aug 30 2023
%D A301708 Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, 1st row, 1st tiling.
%H A301708 Rémy Sigrist, <a href="/A301708/b301708.txt">Table of n, a(n) for n = 0..1000</a> (first 100 terms from Davide M. Proserpio)
%H A301708 Brian Galebach, <a href="http://probabilitysports.com/tilings.html">Collection of n-Uniform Tilings</a>. See Number 15 from the list of 20 2-uniform tilings.
%H A301708 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>.
%H A301708 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 A301708 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krc">The krc tiling (or net)</a>.
%H A301708 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 A301708 Rémy Sigrist, <a href="/A301708/a301708.png">Illustration of first terms</a>.
%H A301708 Rémy Sigrist, <a href="/A301708/a301708.gp.txt">PARI program for A301708</a>.
%H A301708 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).
%F A301708 G.f. = (x^4+4*x^3+x^2+4*x+1)/((x^2+1)*(x-1)^2); for n>0, a(2*t)=11*t, a(4*t+1)=22*t+6, a(4*t+3)=22*t+16. These should be easy to prove by the coloring book method (see link).
%F A301708 Conjecture: a(n) = (i*((-i)^n - i^n) + 22*n) / 4 where i=sqrt(-1). - _Colin Barker_, Apr 07 2018
%F A301708 E.g.f.: (2 + 11*exp(x)*x + sin(x))/2. - _Stefano Spezia_, Jun 08 2024
%t A301708 LinearRecurrence[{2,-2,2,-1},{1,6,11,16,22},100] (* _Paolo Xausa_, Nov 14 2023 *)
%o A301708 (PARI) \\ See Links section.
%Y A301708 Cf. A301710.
%Y A301708 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 A301708 nonn,easy
%O A301708 0,2
%A A301708 _N. J. A. Sloane_, Mar 26 2018
%E A301708 More terms from _Davide M. Proserpio_, Mar 28 2018

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