A316319 - OEIS

%I #31 Mar 11 2020 12:51:45

%S 1,3,7,14,25,38,51,63,75,87,99,111,123,135,147,159,171,183,195,207,

%T 219,231,243,255,267,279,291,303,315,327,339,351,363,375,387,399,411,

%U 423,435,447,459,471,483,495,507,519,531,543,555,567,579,591,603,615,627,639

%N Coordination sequence for a trivalent node in a chamfered version of the 3^6 triangular tiling of the plane.

%C 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 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 Colin Barker, <a href="/A316319/b316319.txt">Table of n, a(n) for n = 0..1000</a>

%H Rémy Sigrist, <a href="/A316319/a316319_1.png">Illustration of initial terms</a>

%H 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F a(n) = 12*n-21 = A017557(n-2) for n > 5.

%F From _Colin Barker_, Mar 11 2020: (Start)

%F G.f.: (1 + x + x^2)*(1 + x^2 + 2*x^3 + x^4 - x^5) / (1 - x)^2.

%F a(n) = 2*a(n-1) - a(n-2) for n>7.

%F (End)

%o (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 See A316320 for hexavalent node.

%Y See A250120 for links to thousands of other coordination sequences.

%Y Cf. A017557.

%K nonn,easy

%O 0,2

%A _Rémy Sigrist_ and _N. J. A. Sloane_, Jul 01 2018

%E Terms a(16) and beyond from _Andrey Zabolotskiy_, Sep 30 2019