A038590 Sizes of clusters in hexagonal lattice A_2 centered at lattice point.

1, 7, 13, 19, 31, 37, 43, 55, 61, 73, 85, 91, 97, 109, 121, 127, 139, 151, 163, 169, 187, 199, 211, 223, 235, 241, 253, 265, 271, 283, 295, 301, 313, 337, 349, 361, 367, 379, 385, 397, 409, 421, 433, 439, 451, 463, 475, 499, 511, 517, 535, 547

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

The sequence can be approximated by a linear function with a correction term. See link for the function and representation of the deviation. The structures in the difference function can also be set to music after scaling. Some MIDI examples are linked. - Hugo Pfoertner, Mar 16 2024

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
• B. K. Teo and N. J. A. Sloane, Atomic Arrangements and Electronic Requirements for Close-Packed Circular and Spherical Clusters, Inorganic Chemistry, 25 (1986), pp. 2315-2322. See Table IV.

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..10000
• G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
• Hugo Pfoertner, A038590(n) - n*(30.1066 - 26.4407/n^0.0792978), Plot of difference to fit.
• Hugo Pfoertner, Audio conversion of difference to fitted function, MIDI example 1.
• Hugo Pfoertner, Audio conversion of difference to fitted function, MIDI example 2, terms starting at n=6000.
• Hugo Pfoertner, Audio conversion of difference to fitted function, MIDI example 3, shifted scaling.

Crossrefs

Cf. A004016, A035019, A038161, A038589.

Formula

Unique(A038589). Or, partial sums of A035019.