A320496 - OEIS

%I #15 Oct 25 2018 20:51:20

%S 1,2,6,9,16,17,22,25,31,34,37,43,47,49,56,56,60,65,72,74,79,79,83,90,

%T 97,97,102,101,108,114,122,119,125,123,133,139,145,142,148,148,158,

%U 162,168,164,173,172,183,184,191,186,198,197,206,207,214,211,223,220

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

%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 B. (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="/A320496/b320496.txt">Table of n, a(n) for n = 0..1000</a>

%H Rémy Sigrist, <a href="/A320496/a320496.png">Illustration of first terms</a>

%H Rémy Sigrist, <a href="/A320496/a320496.gp.txt">PARI program for A320496</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 B, and the green dots indicate the centers of large empty hexagrams.)

%F The b-file suggests that this sequence has g.f. = f/g, where

%F f = -4*x^27+2*x^26-2*x^25+2*x^23+2*x^22+6*x^21+7*x^20+16*x^19+19*x^18+20*x^17+

%F  22*x^16+22*x^15+25*x^14+24*x^13+25*x^12+26*x^11+21*x^10+25*x^9+25*x^8+23*x^7+

%F  21*x^6+17*x^5+16*x^4+9*x^3+6*x^2+2*x+1

%F and

%F g = x^22-x^16-x^6+1. - _N. J. A. Sloane_, Oct 25 2018

%o (PARI) See Links section.

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