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Coordination sequence for node of type V2 in "krn" 2-D tiling (or net).
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%I #27 Nov 15 2023 12:18:15

%S 1,4,9,15,20,26,32,36,40,46,52,56,62,68,72,76,82,88,92,98,104,108,112,

%T 118,124,128,134,140,144,148,154,160,164,170,176,180,184,190,196,200,

%U 206,212,216,220,226,232,236,242,248,252,256,262,268,272,278,284,288,292,298,304,308,314,320,324,328

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

%H Ray Chandler, <a href="/A301680/b301680.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 9 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/krn">The krn 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_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,1,-1).

%F G.f.: -(x^11-x^9-x^8-3*x^7-6*x^6-6*x^5-5*x^4-6*x^3-5*x^2-3*x-1)/(x^8-x^7-x+1). - _N. J. A. Sloane_, Mar 29 2018

%F Equivalent conjecture: 7*a(n) = 36*n-2*b(n) for n>3, where b(n>=1) = 4, 1, -2, 2, -1, -4, 0 (continued 7-periodic). - _R. J. Mathar_, Mar 30 2018

%t LinearRecurrence[{1,0,0,0,0,0,1,-1},{1,4,9,15,20,26,32,36,40,46,52,56},100] (* _Paolo Xausa_, Nov 15 2023 *)

%Y Cf. A301678.

%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 25 2018

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