A119869 Sizes of successive clusters in f.c.c. lattice centered at a lattice point.
1, 13, 19, 43, 55, 79, 87, 135, 141, 177, 201, 225, 249, 321, 321, 369, 381, 429, 459, 531, 555, 603, 627, 675, 683, 767, 791, 887, 935, 959, 959, 1055, 1061, 1157, 1205, 1253, 1289, 1409, 1433, 1481, 1505, 1553, 1601, 1721, 1745, 1865, 1865, 1961, 1985, 2093, 2123
Offset: 0
References
- 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.
Links
- N. J. A. Sloane, Table of n, a(n) for n = 0..9999
- Paul Bourke, Waterman Polyhedra.
- Paul Bourke, On-line generator
- Martin Kraus, Live Graphics3d
- Mirek Majewski, Dedicated commands in MuPAD
- Wouter Meeussen, ConvexHull3D package & demo-file.
- Mark Newbold, Waterman Polyhedra. CCPOLY Java Applet.
- Steve Waterman, Waterman Polyhedron.
- Steve Waterman, Missing numbers formula
Crossrefs
Programs
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Maple
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): 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
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Mathematica
a[n_] := Sum[SquaresR[3, 2k], {k, 0, n}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2012, after formula *) Accumulate[SquaresR[3,2*Range[0,70]]] (* Harvey P. Dale, Jun 01 2015 *)
Formula
Partial sums of A004015, which has an explicit generating function.
Extensions
Edited by N. J. A. Sloane, Aug 09 2006
Additional links from Steve Waterman, Nov 26 2006