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A301298 Expansion of (1 + 4*x + 4*x^2 + 4*x^3 + x^4)/((1 - x)*(1 - x^3)).

%I A301298 #39 Dec 30 2024 14:38:41
%S A301298 1,5,9,14,19,23,28,33,37,42,47,51,56,61,65,70,75,79,84,89,93,98,103,
%T A301298 107,112,117,121,126,131,135,140,145,149,154,159,163,168,173,177,182,
%U A301298 187,191,196,201,205,210,215,219,224,229,233,238,243,247,252,257,261
%N A301298 Expansion of (1 + 4*x + 4*x^2 + 4*x^3 + x^4)/((1 - x)*(1 - x^3)).
%C A301298 Coordination sequence for pentavalent node in the "krl" 2-D tiling (or net). (This is easily established using the "coloring book" method - see the Goodman-Strauss & Sloane link.)
%C A301298 Linear recurrence and g.f. confirmed by Shutov/Maleev link. - _Ray Chandler_, Aug 31 2023
%D A301298 Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 66, 3rd row, second tiling.
%H A301298 Vincenzo Librandi, <a href="/A301298/b301298.txt">Table of n, a(n) for n = 0..5000</a>
%H A301298 Brian Galebach, <a href="http://probabilitysports.com/tilings.html">Collection of n-Uniform Tilings</a>. See Number 4 from the list of 20 2-uniform tilings.
%H A301298 Brian Galebach, <a href="/A250120/a250120.html">k-uniform tilings (k <= 6) and their A-numbers</a>
%H A301298 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 A301298 Reticular Chemistry Structure Resource (RCSR), <a href="http://rcsr.net/layers/krl">The krl tiling (or net)</a>
%H A301298 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 A301298 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F A301298 G.f.: (1 + 4*x + 4*x^2 + 4*x^3 + x^4)/((1 - x)*(1 - x^3)).
%F A301298 a(n) = 5*n - floor((n + 1)/3) for n>0, a(0)=1. - _Bruno Berselli_, Mar 26 2018
%t A301298 CoefficientList[Series[(x^4 + 4 x^3 + 4 x^2 + 4 x + 1) / ((1 -x) (1 - x^3)), {x, 0, 60}], x] (* _Vincenzo Librandi_, Mar 26 2018 *)
%t A301298 LinearRecurrence[{1,0,1,-1},{1,5,9,14,19},60] (* _Harvey P. Dale_, Dec 30 2024 *)
%o A301298 (Magma) I:=[1,5,9,14,19]; [n le 5 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..80]]; // _Vincenzo Librandi_, Mar 26 2018
%o A301298 (Magma) [n eq 0 select 1 else 5*n-Floor((n+1)/3): n in [0..60]]; // _Bruno Berselli_, Mar 26 2018
%o A301298 (PARI) lista(nn) = {x='x+O('x^nn); Vec((x^4+4*x^3+4*x^2+4*x+1)/((1-x)*(1-x^3)))} \\ _Altug Alkan_, Mar 26 2018
%Y A301298 Cf. A298024.
%Y A301298 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 A301298 nonn,easy
%O A301298 0,2
%A A301298 _N. J. A. Sloane_, Mar 24 2018