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

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