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%I #36 Sep 10 2016 20:04:23
%S 1,9,59,169,339,641,1075,1617,2381,3355,4533,5939,7645,9651,11933,
%T 14581,17631,21053,24871,29109,33863,39061,44775,51023,57817,65247,
%U 73193,81847,91113,101063,111691,123081,135155,148081,161763,176249,191611,207777,224861,242899,261837,281627,302653,324555,347405,371389,396495
%N Number of points of norm <= n in the bi-truncated cubic honeycomb (3-dimensional lattice, with truncated-octahedral cells).
%C 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).
%C 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.
%C 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).
%H Yuriy Sibirmovsky, <a href="/A276450/b276450.txt">Table of n, a(n) for n = 0..100</a>
%H N. G. Khlebtsov, <a href="http://dx.doi.org/10.1016/j.jqsrt.2012.12.027">T-matrix method in plasmonics: An overview</a>, J. Quantitative Spectroscopy & Radiative Transfer 123 (2013) 184-217.
%H Yuriy Sibirmovsky, <a href="/A276450/a276450.jpg">Coordinate axes and the cell arrangement</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bitruncated_cubic_honeycomb">Bitruncated cubic honeycomb</a>.
%e The origin has norm 0, so a(0)=1. Each cell has eight closest neighbors, touching along hexagonal faces. So a(1)=9.
%t rm=20;
%t CanonForm[A_]:=A-Min[A[[1]],A[[2]],A[[3]],A[[4]]]{1,1,1,1};
%t 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]]);
%t Do[S=0;
%t Do[A={j,k,l,m};
%t 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}];
%t Print[r," ",S],{r,0,rm}]
%Y Cf. A000605 (cubic lattice).
%K nonn
%O 0,2
%A _Yuriy Sibirmovsky_, Sep 02 2016
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