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A301291 Expansion of (x^4+3*x^3+x^2+3*x+1)/((x^2+1)*(x-1)^2).

%I A301291 #36 Aug 30 2023 22:24:47
%S A301291 1,5,9,13,18,23,27,31,36,41,45,49,54,59,63,67,72,77,81,85,90,95,99,
%T A301291 103,108,113,117,121,126,131,135,139,144,149,153,157,162,167,171,175,
%U A301291 180,185,189,193,198,203,207,211,216,221,225,229,234,239,243,247,252
%N A301291 Expansion of (x^4+3*x^3+x^2+3*x+1)/((x^2+1)*(x-1)^2).
%C A301291 Appears to be coordination sequence for node of type 3^3.4^2 in "krm" 2-D tiling (or net).
%C A301291 Also appears to be coordination sequence for pentavalent node in "krk" 2-D tiling (or net).
%C A301291 Linear recurrence and g.f. confirmed by Shutov/Maleev link. - _Ray Chandler_, Aug 30 2023
%D A301291 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 A301291 Colin Barker, <a href="/A301291/b301291.txt">Table of n, a(n) for n = 0..1000</a>
%H A301291 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 A301291 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H A301291 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 A301291 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krm">The krm tiling (or net)</a>
%H A301291 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krk">The krk tiling (or net)</a>
%H A301291 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 A301291 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).
%F A301291 For explicit formula for a(n) see Maple code.
%F A301291 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
%F A301291 E.g.f.: (2 + 9*x*exp(x) + sin(x))/2. - _Stefano Spezia_, Jan 31 2023
%p A301291 f:=proc(n) if n=0 then 1
%p A301291 elif (n mod 2) = 0 then 9*n/2
%p A301291 elif (n mod 4) = 1 then 18*(n-1)/4+5
%p A301291 else 18*(n-3)/4+13; fi; end;
%p A301291 s1:=[seq(f(n),n=0..60)];
%t A301291 Join[{1}, LinearRecurrence[{2, -2, 2, -1}, {5, 9, 13, 18}, 60]] (* _Jean-François Alcover_, Jan 08 2019 *)
%o A301291 (PARI) Vec((x^4+3*x^3+x^2+3*x+1)/((x^2+1)*(x-1)^2) + O(x^60)) \\ _Colin Barker_, Mar 23 2018
%Y A301291 Cf. A301293.
%Y A301291 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 A301291 nonn,easy
%O A301291 0,2
%A A301291 _N. J. A. Sloane_, Mar 23 2018