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A301293 Expansion of (x^2+x+1)^2 / ((x^2+1)*(x-1)^2).

%I A301293 #37 Aug 30 2023 22:22:48
%S A301293 1,4,9,14,18,22,27,32,36,40,45,50,54,58,63,68,72,76,81,86,90,94,99,
%T A301293 104,108,112,117,122,126,130,135,140,144,148,153,158,162,166,171,176,
%U A301293 180,184,189,194,198,202,207,212,216,220,225,230,234,238,243,248,252
%N A301293 Expansion of (x^2+x+1)^2 / ((x^2+1)*(x-1)^2).
%C A301293 Appears to be coordination sequence for node of type 4^4 in "krm" 2-D tiling (or net).
%C A301293 Also appears to be coordination sequence for tetravalent node in "krk" 2-D tiling (or net).
%C A301293 Linear recurrence and g.f. confirmed by Shutov/Maleev link. - _Ray Chandler_, Aug 30 2023
%D A301293 Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, row 3, first tiling; also p. 66, row 3, first tiling.
%H A301293 Colin Barker, <a href="/A301293/b301293.txt">Table of n, a(n) for n = 0..1000</a>
%H A301293 Brian Galebach, <a href="http://probabilitysports.com/tilings.html">Collection of n-Uniform Tilings</a>. See Numbers 3 and 8 from the list of 20 2-uniform tilings.
%H A301293 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H A301293 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 A301293 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krm">The krm tiling (or net)</a>
%H A301293 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krk">The krk tiling (or net)</a>
%H A301293 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 A301293 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).
%F A301293 For explicit formula for a(n) see Maple code.
%F A301293 a(n) = 9*n/2 + (1 - (-1)^n)*i^(n*(n + 1))/4 for n>0, a(0)=1 and i=sqrt(-1). Therefore, for even n>0 a(n) = 9*n/2, otherwise a(n) = 9*n/2 - (-1)^((n-1)/2)/2. - _Bruno Berselli_, Mar 23 2018
%F A301293 a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4. - _Colin Barker_, Mar 23 2018
%p A301293 f:=proc(n) if n=0 then 1
%p A301293 elif (n mod 2) = 0 then 9*n/2
%p A301293 elif (n mod 4) = 1 then 18*(n-1)/4+4
%p A301293 else 18*(n-3)/4+14; fi; end;
%p A301293 s1:=[seq(f(n),n=0..60)];
%t A301293 Join[{1}, LinearRecurrence[{2, -2, 2, -1}, {4, 9, 14, 18}, 60]] (* _Jean-François Alcover_, Jan 08 2019 *)
%o A301293 (PARI) Vec((x^2+x+1)^2 / ((x^2+1)*(x-1)^2) + O(x^60)) \\ _Colin Barker_, Mar 23 2018
%Y A301293 Cf. A301291.
%Y A301293 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 A301293 nonn,easy
%O A301293 0,2
%A A301293 _N. J. A. Sloane_, Mar 23 2018