%I #36 Nov 14 2023 16:36:21

%S 1,5,11,17,22,27,33,39,44,49,55,61,66,71,77,83,88,93,99,105,110,115,

%T 121,127,132,137,143,149,154,159,165,171,176,181,187,193,198,203,209,

%U 215,220,225,231,237,242,247,253,259,264,269,275,281,286,291,297,303

%N Coordination sequence for node of type V2 in "krc" 2-D tiling (or net).

%C Linear recurrence and g.f. confirmed by Shutov/Maleev link. - _Ray Chandler_, Aug 30 2023

%D 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 Rémy Sigrist, <a href="/A301710/b301710.txt">Table of n, a(n) for n = 0..1000</a> (first 100 terms from Davide M. Proserpio)

%H 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 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/krc">The krc 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="/A301710/a301710.png">Illustration of first terms</a>.

%H Rémy Sigrist, <a href="/A301710/a301710.gp.txt">PARI program for A301710</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2, -2, 2, -1).

%F G.f.: (x^4+3*x^3+3*x^2+3*x+1)/((x^2+1)*(x-1)^2); for n>0, a(2*t)=11*t, a(4*t+1)=22*t+5, a(4*t+3)=22*t+17. These should be easy to prove by the coloring book method (see link).

%F a(n) = ((-i)^(1+n) + i^(1+n) + 22*n) / 4 for n>0, where i=sqrt(-1) (conjectured). - _Colin Barker_, Apr 07 2018

%t LinearRecurrence[{2,-2,2,-1},{1,5,11,17,22},100] (* _Paolo Xausa_, Nov 14 2023 *)

%o (PARI) See Links section.

%Y Cf. A301708.

%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

%O 0,2

%A _N. J. A. Sloane_, Mar 26 2018

%E More terms from _Davide M. Proserpio_, Mar 28 2018