## 32 Citations

### Proper n-Cell Polycubes in n - 3 Dimensions

- Mathematics, Computer ScienceCOCOON
- 2011

A formula is proved for the number of polycubes of size n that are proper in (n - 3) dimensions, which is said to be proper in d dimensions if the convex hull of the centers of its cubes is d-dimensional.

### Polycubes with Small Perimeter Defect

- MathematicsAnnals of Combinatorics
- 2022

A few formulae enumerating polycubes with a fixed deviation from the maximum possible perimeter are presented.

### Formulae and Growth Rates of Animals on Cubical and Triangular Lattices

- Mathematics
- 2017

A polyomino of size n consists of n squares joined along their edges. A popular example is the computer game Tetris, which features polyominoes of size 4. A d-dimensional polycube of size n is a…

### The Perimeter of Proper Polycubes

- Mathematics, Computer ScienceJ. Integer Seq.
- 2017

Formulas are derived for the number of polycubes of size $n$ and perimeter $t$ that are proper in $n-1 and$n-2$ dimensions that complement computer based enumerations of perimeter polynomials in percolation problems.

### Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

- Mathematics, Computer ScienceAlgorithmica
- 2022

The method for the best known upper bound on A2(n) is revisited and extended and the number of d-dimensional polycubes with n cubes is defined.

### C O ] 1 0 M ay 2 01 7 The Perimeter of Proper Polycubes

- Mathematics, Computer Science
- 2018

We derive formulas for the number of polycubes of size n and perimeter t that are proper in n − 1 and n − 2 dimensions. These formulas complement computer based enumerations of perimeter polynomials…

### Automatic Proofs for Formulae Enumerating Proper Polycubes

- MathematicsElectron. Notes Discret. Math.
- 2015

## References

SHOWING 1-10 OF 32 REFERENCES

### Counting d-Dimensional Polycubes and nonrectangular Planar polyominoes

- MathematicsInt. J. Comput. Geom. Appl.
- 2006

This paper describes a generalization of Redelmeier’s algorithm for counting two-dimensional rectangular polyominoes and computed the number of distinct 3-D polycubes of size 18, which is the first tabulation of this value.

### Counting Polyominoes on Twisted Cylinders

- Mathematics
- 2004

We improve the lower bounds on Klarner's constant, which describes the exponential growth rate of the number of polyominoes (connected subsets of grid squares) with a given number of squares. We…

### Percolation processes in d-dimensions

- Physics
- 1976

Series data for the mean cluster size for site mixtures on a d-dimensional simple hypercubical lattice are presented. Numerical evidence for the existence of a critical dimension for the cluster…

### Counting Polyominoes: A Parallel Implementation for Cluster Computing

- Computer Science, MathematicsInternational Conference on Computational Science
- 2003

This work has developed a parallel algorithm for the enumeration of polyominoes, which are connected sets of lattice cells joined at an edge, and implements the finite-lattice method and associated transfer-matrix calculations in a very efficient parallel setup.

### Cell Growth Problems

- MathematicsCanadian Journal of Mathematics
- 1967

The square lattice is the set of all points of the plane whose Cartesian coordinates are integers. A cell of the square lattice is a point-set consisting of the boundary and interior points of a unit…

### The Self-Avoiding-Walk and Percolation Critical Points in High Dimensions

- MathematicsCombinatorics, Probability and Computing
- 1995

It is proved existence of an asymptotic expansion in the inverse dimension, to all orders, for the connective constant for self-avoiding walks on Z d, and the method uses the lace expansion.

### A pattern theorem for lattice clusters

- Mathematics
- 1999

We consider general classes of lattice clusters, including various kinds of animals and trees on different lattices. We prove that if a given local configuration (“pattern”) of sites and bonds can…