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A276450 Number of points of norm <= n in the bi-truncated cubic honeycomb (3-dimensional lattice, with truncated-octahedral cells). 2
1, 9, 59, 169, 339, 641, 1075, 1617, 2381, 3355, 4533, 5939, 7645, 9651, 11933, 14581, 17631, 21053, 24871, 29109, 33863, 39061, 44775, 51023, 57817, 65247, 73193, 81847, 91113, 101063, 111691, 123081, 135155, 148081, 161763, 176249, 191611, 207777, 224861, 242899, 261837, 281627, 302653, 324555, 347405, 371389, 396495 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The lattice points coincide with the centers of the cells. Start from the origin. Draw four lines through the centers of the eight hexagonal faces of the cell and choose directions so the endpoints are vertices of a tetrahedron. The length of a unit vector is equal to the distance between the centers of the closest cells. Then every lattice point will have integer coordinates in this coordinate system. Denoting the coordinates by (a,b,c,d) we have (a,a,a,a)=(0,0,0,0), meaning the coordinates are not unique. To give unique coordinates to every point, at least one of a,b,c,d should be 0 and the others nonnegative. The squared Euclidean norm of a vector is a^2+b^2+c^2+d^2-(2/3)(ab+ac+ad+bc+bd+cd).

a(n) is the number of distinct points (a,b,c,d) where at least one of a,b,c,d is 0, the others are nonnegative integers, and a^2+b^2+c^2+d^2-2/3 (ab+ac+ad+bc+bd+cd) <= n^2.

Experimentally observed dense bcc clusters of gold contain 1, 9, 59, 169, 339, 701 and 1243 nanoparticles (N.G. Khlebtsov, Fig. 32 and text on p. 208). This exactly describes the number of points of norm <=n, but for the body-centered cubic lattice with the parameter equal to 2/sqrt(3).

LINKS

Yuriy Sibirmovsky, Table of n, a(n) for n = 0..100

N. G. Khlebtsov, T-matrix method in plasmonics: An overview, J. Quantitative Spectroscopy & Radiative Transfer 123 (2013) 184-217.

Yuriy Sibirmovsky, Coordinate axes and the cell arrangement.

Wikipedia, Bitruncated cubic honeycomb.

EXAMPLE

The origin has norm 0, so a(0)=1. Each cell has eight closest neighbors, touching along hexagonal faces. So a(1)=9.

MATHEMATICA

rm=20;

CanonForm[A_]:=A-Min[A[[1]], A[[2]], A[[3]], A[[4]]]{1, 1, 1, 1};

NormSq[A_]:=A[[1]]^2 + A[[2]]^2 + A[[3]]^2 + A[[4]]^2 - 2/3(A[[1]]A[[2]] + A[[2]]A[[3]] + A[[3]]A[[4]] + A[[4]]A[[1]] + A[[1]]A[[3]] + A[[2]]A[[4]]);

Do[S=0;

Do[A={j, k, l, m};

If[A-CanonForm[A]=={0, 0, 0, 0}&&NormSq[A]<=r^2, S+=1], {j, 0, r}, {k, 0, r}, {l, 0, r}, {m, 0, r}];

Print[r, " ", S], {r, 0, rm}]

CROSSREFS

Cf. A000605 (cubic lattice).

Sequence in context: A196293 A196211 A196679 * A276648 A308353 A280103

Adjacent sequences:  A276447 A276448 A276449 * A276451 A276452 A276453

KEYWORD

nonn

AUTHOR

Yuriy Sibirmovsky, Sep 02 2016

STATUS

approved

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Last modified September 19 23:31 EDT 2019. Contains 327207 sequences. (Running on oeis4.)