A320497 - OEIS

%I #15 Oct 25 2018 17:25:24

%S 1,4,8,10,14,16,22,28,32,36,36,42,48,50,54,58,60,68,74,72,76,80,86,94,

%T 96,94,98,106,112,116,118,116,122,132,138,138,140,138,148,158,160,160,

%U 162,164,174,180,182,182,186,190,200,202,204,204,212,216,222,224

%N Coordination sequence of thinnest 5-neighbor packing of the plane with congruent hexagons with respect to a point of type C.

%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 C. (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="/A320497/b320497.txt">Table of n, a(n) for n = 0..1000</a>

%H Rémy Sigrist, <a href="/A320497/a320497.png">Illustration of first terms</a>

%H Rémy Sigrist, <a href="/A320497/a320497.gp.txt">PARI program for A320497</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 C, and the green dots indicate the centers of large empty hexagrams.)

%F Conjectures from _Colin Barker_, Oct 25 2018: (Start)

%F G.f.: (1 + 4*x + 8*x^2 + 10*x^3 + 14*x^4 + 16*x^5 + 21*x^6 + 24*x^7 + 24*x^8 + 26*x^9 + 22*x^10 + 26*x^11 + 26*x^12 + 22*x^13 + 22*x^14 + 22*x^15 + 23*x^16 + 22*x^17 + 18*x^18 + 12*x^19 + 8*x^20 + 6*x^21 + 5*x^22 + 2*x^23 - 2*x^24 - 4*x^25) / (x^22 - x^16 - x^6 + 1).

%F a(n) = a(n-6) + a(n-16) - a(n-22) for n>25.

%F (End)

%o (PARI) See Links section.

%Y Cf. A320492, A320493, A320494, A320495, A320496, A320498.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Oct 22 2018

%E Data corrected and extended by _Rémy Sigrist_, Oct 24 2018