A301674 - OEIS

%I #30 Feb 22 2025 22:14:15

%S 1,4,8,14,16,26,22,34,36,38,44,54,46,62,64,62,72,82,70,90,92,86,100,

%T 110,94,118,120,110,128,138,118,146,148,134,156,166,142,174,176,158,

%U 184,194,166,202,204,182,212,222,190,230,232,206,240,250,214,258,260

%N Coordination sequence for node of type V1 in "krs" 2-D tiling (or net).

%C Linear recurrence and g.f. confirmed by Shutov/Maleev link. - _Ray Chandler_, Aug 31 2023

%D Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, bottom row, 2nd tiling.

%H Rémy Sigrist, <a href="/A301674/b301674.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 6 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/krs">The krs 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="/A301674/a301674.png">Illustration of first terms</a>

%H Rémy Sigrist, <a href="/A301674/a301674.gp.txt">PARI program for A301674</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (-1,0,2,2,0,-1,-1).

%F (a) G.f. = -(2*x^8-x^7-5*x^6-18*x^5-20*x^4-20*x^3-12*x^2-5*x-1)/((x+1)*(x-1)^2*(x^2+x+1)^2). (b) Satisfies the recurrence {( - 2*n^5 - 13*n^4 - 22*n^3 + 7*n^2 + 30*n)*a(n) + ( - 2*n^5 - 13*n^4 - 25*n^3 + n^2 + 39*n)*a(n + 1) + ( - 6*n^2 + 6*n)*a(n + 2) + (2*n^5 + 7*n^4 + 7*n^3 - 7*n^2 - 9*n)*a(n + 3) + (2*n^5 + 7*n^4 + 4*n^3 - 7*n^2 - 6*n)*a(n + 4) = 0, a(0) = 1, a(1) = 4, a(2) = 8, a(3) = 14, a(4) = 16, a(5) = 26}. - _N. J. A. Sloane_, Mar 28 2018

%F Equivalent conjecture: 9*a(n) = 40*n -18*(-1)^n -6*(-1)^n*A076118(n+1) +6*A049347(n) -4*A049347(n-1). - _R. J. Mathar_, Apr 01 2018

%t LinearRecurrence[{-1,0,2,2,0,-1,-1},{1,4,8,14,16,26,22,34,36},100] (* _Paolo Xausa_, Nov 15 2023 *)

%o (PARI) See Links section.

%Y Cf. A301676.

%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,easy

%O 0,2

%A _N. J. A. Sloane_, Mar 25 2018

%E More terms from _Rémy Sigrist_, Mar 28 2018