Cubes in {0,1,...,n}3

Abstract

The main aim of this paper is to describe a procedure for calculating the number of cubes that have coordinates in the set ¹0; 1; : : : ; nº. For this purpose we continue and, at the same time, revise some of the work begun in a sequence of papers about equilateral triangles and regular tetrahedra all having integer coordinates for their vertices. We adapt the code that was included in a paper by the first author and was used to calculate the number of regular tetrahedra with vertices in ¹0; 1; : : : ; nº 3 . The idea is based on the theoretical results obtained by the first author with A. Markov. We then extend the sequence A098928 in the Online Encyclopedia of Integer Sequences to the first one hundred terms.

Kepler's conjecture and phase transitions in the high-density hard-core model on $\mathbb{Z}^3$

“…In case when we use the above construction we specify the corresponding values q, h(q), the family of meshes τ (q) i,hk,j , 0 ≤ j ≤ r, and the type of sequence {j k } allowing us to obtain a PC for a given D 2 via (2. 16). An important class is formed by periodic layered PCs where the corresponding sequence {j k } = j 0 ...j p is obtained by repeating a finite string j 0 ...j p .…”

“…19), (2.20). The rhombic lattice α (8,16) i,0,0 lies in lQ i,0 . Further, consider the following 2 sites at distance 3/2 from the origin:…”

“…Cf. [16]. The symmetries of cubic ℓ-sub-lattices and hence FCC ℓ-sub-lattices are important for our considerations because if a sub-lattice generates an EPGM then every Z 3 -symmetric image of this sub-lattice also generates an EPGM.…”