A320498 - OEIS

%I #15 Oct 25 2018 19:47:35

%S 1,4,8,12,13,17,23,28,32,35,37,42,48,52,53,57,62,69,72,74,75,81,87,93,

%T 95,95,99,106,112,117,116,117,124,133,136,139,138,141,149,157,159,160,

%U 162,166,174,181,180,182,187,193,198,203,202,206,212,217,221,224

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

%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 D. (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="/A320498/b320498.txt">Table of n, a(n) for n = 0..1000</a>

%H Rémy Sigrist, <a href="/A320498/a320498.png">Illustration of first terms</a>

%H Rémy Sigrist, <a href="/A320498/a320498.gp.txt">PARI program for A320498</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 D, and the green dots indicate the centers of large empty hexagrams.)

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

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

%F a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) + a(n-16) - a(n-17) + a(n-18) - a(n-19) + a(n-20) - a(n-21) for n>24.

%F (End)

%o (PARI) See Links section.

%Y Cf. A320492, A320493, A320494, A320495, A320496, A320497.

%K nonn

%O 0,2

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

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