OFFSET
0,2
COMMENTS
"5-neighbor" means that each hexagon has a point in common with exactly five other hexagons.
This packing is actually the thinnest 5-neighbor packing in the plane using any centrally symmetric congruent polygons.
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.)
REFERENCES
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.
LINKS
Rémy Sigrist, Table of n, a(n) for n = 0..1000
Rémy Sigrist, Illustration of first terms
Rémy Sigrist, PARI program for A320498
N. J. A. Sloane, The packing and its graph. (The hexagons are shaded, the base point is marked D, and the green dots indicate the centers of large empty hexagrams.)
FORMULA
Conjectures from Colin Barker, Oct 25 2018: (Start)
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).
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.
(End)
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 22 2018
EXTENSIONS
Data corrected and extended by Rémy Sigrist, Oct 24 2018
STATUS
approved