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%I #21 Oct 25 2018 20:53:32
%S 1,4,6,12,14,19,22,28,32,34,39,44,46,52,54,58,62,69,69,75,77,82,87,93,
%T 92,98,100,107,111,117,114,122,123,132,134,140,137,146,148,156,157,
%U 163,160,171,172,180,179,187,183,196,195,203,202,211,208,220,218,226
%N Coordination sequence of thinnest 5-neighbor packing of the plane with congruent hexagons with respect to a point of type A.
%C "5-neighbor" means that each hexagon has a point in common with exactly five other hexagons.
%C This packing is actually the thinnest 5-neighbor packing in the plane using any centrally symmetric congruent polygons.
%C More formally, this sequence is the coordination sequence of the vertex-edge graph of the packing with respect to a vertex of type A. (The automorphism group of the tiling has four orbits on vertices, indicated by the letters A, B, C, D in the figure.)
%D William Moser and Janos Pach, Research Problems in Discrete Geometry: Packing and Covering, DIMACS Technical Report 93-32, May 1993. See Fig. 19.1b, page 32. There is an error in the figure: the hexagon at the right of the bottom row should not be shaded. The figure shown here is correct.
%H Rémy Sigrist, <a href="/A320495/b320495.txt">Table of n, a(n) for n = 0..1000</a>
%H Rémy Sigrist, <a href="/A320495/a320495_1.png">Illustration of first terms</a>
%H Rémy Sigrist, <a href="/A320495/a320495.gp.txt">PARI program for A320495</a>
%H N. J. A. Sloane, <a href="/A320495/a320495.png">The packing and its graph.</a> (The hexagons are shaded, the base point is marked A, and the green dots indicate the centers of large empty hexagrams.)
%F Based on the b-file, this appears to have g.f. = f/g, where
%F f = -2*x^24+x^23-x^22-x^21+5*x^20+6*x^19+4*x^18+5*x^17+
%F 13*x^16+16*x^15+9*x^14+8*x^13+13*x^12+16*x^11+11*x^10
%F +9*x^9+14*x^8+13*x^7+12*x^6+11*x^5+9*x^4+8*x^3+5*x^2+4*x+1
%F and
%F g = (1-x^2)*(1-x^6)*(1+x^4)*(1+x^8). - _N. J. A. Sloane_, Oct 25 2018
%o (PARI) See Links section.
%Y Cf. A320492, A320493, A320494, A320496, A320497, A320498.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Oct 22 2018
%E More terms from _Rémy Sigrist_, Oct 24 2018
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