This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A316319 #31 Mar 11 2020 12:51:45 %S A316319 1,3,7,14,25,38,51,63,75,87,99,111,123,135,147,159,171,183,195,207, %T A316319 219,231,243,255,267,279,291,303,315,327,339,351,363,375,387,399,411, %U A316319 423,435,447,459,471,483,495,507,519,531,543,555,567,579,591,603,615,627,639 %N A316319 Coordination sequence for a trivalent node in a chamfered version of the 3^6 triangular tiling of the plane. %C A316319 Let E denote the lattice of Eisenstein integers u + v*w in the plane, with each point joined to its six neighbors. Here u and v are ordinary integers and w = (-1+sqrt(-3))/2 is a complex cube root of unity. Let theta = w - w^2 = sqrt(-3). Then theta*E is a sublattice of E of index 3 (Conway-Sloane, Fig. 7.2). The tiling considered in this sequence is obtained by replacing each node in theta*E by a small hexagon. %D A316319 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd. ed., 1993. See Fig. 7.2, page 199. %H A316319 Colin Barker, <a href="/A316319/b316319.txt">Table of n, a(n) for n = 0..1000</a> %H A316319 Rémy Sigrist, <a href="/A316319/a316319_1.png">Illustration of initial terms</a> %H A316319 N. J. A. Sloane, <a href="/A316319/a316319.png">The graph of the tiling.</a> (The red dots indicate the nodes of the sublattice theta*E.) %H A316319 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1). %F A316319 a(n) = 12*n-21 = A017557(n-2) for n > 5. %F A316319 From _Colin Barker_, Mar 11 2020: (Start) %F A316319 G.f.: (1 + x + x^2)*(1 + x^2 + 2*x^3 + x^4 - x^5) / (1 - x)^2. %F A316319 a(n) = 2*a(n-1) - a(n-2) for n>7. %F A316319 (End) %o A316319 (PARI) Vec((1 + x + x^2)*(1 + x^2 + 2*x^3 + x^4 - x^5) / (1 - x)^2 + O(x^50)) \\ _Colin Barker_, Mar 11 2020 %Y A316319 See A316320 for hexavalent node. %Y A316319 See A250120 for links to thousands of other coordination sequences. %Y A316319 Cf. A017557. %K A316319 nonn,easy %O A316319 0,2 %A A316319 _Rémy Sigrist_ and _N. J. A. Sloane_, Jul 01 2018 %E A316319 Terms a(16) and beyond from _Andrey Zabolotskiy_, Sep 30 2019