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%I #14 Oct 23 2018 21:00:32
%S 1,4,11,15,22,28,32,39,45,48,56,62,65,73,78,82,90,95,99,106,112,116,
%T 123,129,132,140,146,149,157,162,166,174,179,183,190,196,200,207,213,
%U 216,224,230,233,241,246,250,258,263,267,274,280,284,291,297,300,308
%N Coordination sequence of thinnest 5-neighbor packing of the plane with congruent triangles with respect to a tetravalent point.
%C "5-neighbor" means that each triangle has a point in common with exactly five other triangles.
%C This packing is actually the thinnest 5-neighbor packing in the plane using any congruent convex polygons.
%C More formally, this sequence is the coordination sequence of the vertex-edge graph of the packing with respect to a tetavalent vertex. The base vertex is marked "B" in the figure (it is the midpoint of an edge of the large empty triangle).
%D William Moser and Janos Pach, Research Problems in Discrete Geometry: Packing and Covering, DIMACS Technical Report 93-32, May 1993. See Fig. 19.1a, page 32. There is an error in the figure: the triangle at the right of the bottom row should not be shaded. The figure shown here is correct.
%H Rémy Sigrist, <a href="/A320494/b320494.txt">Table of n, a(n) for n = 0..1000</a>
%H Rémy Sigrist, <a href="/A320494/a320494.png">Illustration of first terms</a>
%H Rémy Sigrist, <a href="/A320494/a320494.gp.txt">PARI program for A320494</a>
%H N. J. A. Sloane, <a href="/A320492/a320492.png">The packing and its graph.</a> (The triangles are shaded, the base point is marked B, and the green dots indicate the centers of large empty triangles.)
%F Conjectures from _Colin Barker_, Oct 23 2018: (Start)
%F G.f.: (1 + 4*x + 11*x^2 + 14*x^3 + 18*x^4 + 16*x^5 + 13*x^6 + 6*x^7 + 3*x^8 - 2*x^9) / ((1 - x)^2*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)).
%F a(n) = a(n-3) + a(n-5) - a(n-8) for n>9.
%F (End)
%o (PARI) See Links section.
%Y Cf. A320492, A320493, A320495, A320496, A320497, A320498.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Oct 21 2018
%E More terms from _Rémy Sigrist_, Oct 22 2018
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