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Coordination sequence for node of type V2 in "krj" 2-D tiling (or net).
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%I #27 Nov 16 2023 08:44:33

%S 1,5,10,16,22,27,32,37,42,48,54,59,64,69,74,80,86,91,96,101,106,112,

%T 118,123,128,133,138,144,150,155,160,165,170,176,182,187,192,197,202,

%U 208,214,219,224,229,234,240,246,251,256,261,266,272,278,283,288,293,298,304,310,315,320,325,330,336

%N Coordination sequence for node of type V2 in "krj" 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, 2nd row, 3rd tiling.

%H Ray Chandler, <a href="/A301697/b301697.txt">Table of n, a(n) for n = 0..1000</a>

%H Brian Galebach, <a href="http://probabilitysports.com/tilings.html">Collection of n-Uniform Tilings</a>. See Number 17 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 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krj">The krj 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 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-2,2,-1).

%F G.f.: -(-x^6 -3*x^5 -2*x^4 -4*x^3 -2*x^2 -3*x -1) /(x^6 -2*x^5 +2*x^4 -2*x^3 +2*x^2 -2*x +1) = 1 + x*(5+6*x^2+5*x^4) / ( (x^2-x+1) *(1+x+x^2) *(x-1)^2 ). - _N. J. A. Sloane_, Mar 29 2018

%F Equivalent conjecture: 6*a(n) = 32*n -3*A010892(n-1) + A049347(n-1) for n>0. - _R. J. Mathar_, Nov 26 2018

%t LinearRecurrence[{2,-2,2,-2,2,-1},{1,5,10,16,22,27,32},100] (* _Paolo Xausa_, Nov 16 2023 *)

%Y Cf. A219529.

%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 a(11)-a(100) from _Davide M. Proserpio_, Mar 28 2018