|
%I #21 Dec 27 2018 08:26:22
%S 1,13,19,43,55,79,87,135,141,177,201,225,249,321,321,369,381,429,459,
%T 531,555,603,627,675,683,767,791,887,935,959,959,1055,1061,1157,1205,
%U 1253,1289,1409,1433,1481,1505,1553,1601,1721,1745,1865,1865,1961,1985,2093,2123
%N Sizes of successive clusters in f.c.c. lattice centered at a lattice point.
%D N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
%H N. J. A. Sloane, <a href="/A119869/b119869.txt">Table of n, a(n) for n = 0..9999</a>
%H Paul Bourke, <a href="http://paulbourke.net/geometry/waterman/">Waterman Polyhedra.</a>
%H Paul Bourke, <a href="http://paulbourke.net/geometry/waterman/gen/">On-line generator</a>
%H Martin Kraus, <a href="http://www.martin-kraus.org/LiveGraphics3D">Live Graphics3d</a>
%H Mirek Majewski, <a href="http://www.mupad.com/mathpad/2006/majewski/">Dedicated commands in MuPAD</a>
%H Wouter Meeussen, <a href="http://users.pandora.be/Wouter.Meeussen/ConvexHull3D.m">ConvexHull3D package & demo-file.</a>
%H Mark Newbold, <a href="http://dogfeathers.com/java/ccppoly.html">Waterman Polyhedra.</a> CCPOLY Java Applet.
%H Steve Waterman, <a href="http://watermanpolyhedron.com/watermanpolyhedra1.html">Waterman Polyhedron.</a>
%H Steve Waterman, <a href="http://watermanpolyhedron.com/MISSING.html">Missing numbers formula</a>
%F Partial sums of A004015, which has an explicit generating function.
%p maxd:=20001: read format: temp0:=trunc(evalf(sqrt(maxd)))+2: a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od: th2:=series(a,q,maxd): a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od: th3:=series(a,q,maxd): th4:=series(subs(q=-q,th3),q,maxd):
%p t1:=series((th3^3+th4^3)/2,q,maxd): t1:=series(subs(q=sqrt(q),t1),q,floor(maxd/2)): t2:=seriestolist(t1): t4:=0; for n from 1 to nops(t2) do t4:=t4+t2[n]; lprint(n-1, t4); od: # _N. J. A. Sloane_, Aug 09 2006
%t a[n_] := Sum[SquaresR[3, 2k], {k, 0, n}]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jul 12 2012, after formula *)
%t Accumulate[SquaresR[3,2*Range[0,70]]] (* _Harvey P. Dale_, Jun 01 2015 *)
%Y Cf. A004015, A004215, A055039.
%Y Cf. A055039 [missing polyhedra]. Properties of Waterman polyhedra: A119870 [vertices], A119871 [faces], A119872 [edges], A119873 [volume]. Waterman polyhedra with different centers: A119874, A119875, A119876, A119877, A119878.
%K nonn,nice
%O 0,2
%A _Hugo Pfoertner_, May 26 2006
%E Edited by _N. J. A. Sloane_, Aug 09 2006
%E Additional links from Steve Waterman, Nov 26 2006
|
