cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A119870 Number of vertices of the root-n Waterman polyhedron.

Original entry on oeis.org

12, 6, 24, 12, 24, 32, 48, 54, 36, 24, 48, 24, 72, 72, 48, 60, 48, 54, 72, 72, 72, 72, 48, 56, 132, 96, 120, 96, 72, 72, 96, 102, 96, 96, 120, 84, 120, 144, 96, 72, 120, 72, 168, 168, 120, 120, 144, 168, 108, 126, 168, 72, 144, 152, 144, 144, 192, 120, 144, 144
Offset: 1

Views

Author

Hugo Pfoertner, May 26 2006

Keywords

Comments

The root-n Waterman polyhedron is the convex hull of the intersection of a closed ball of radius sqrt(2*n) with the lattice of sphere-center points of a cubic close packing. [Probably the f.c.c. lattice is intended here. - N. J. A. Sloane, Aug 09 2006]
The basic sphere center series of Waterman polyhedra is obtained by choosing a sphere center as the center of the closed ball. Other choices are possible. An example is given in A119874 ... A119878. For n in A055039 no lattice points are hit; the corresponding polyhedra are the same as for n-1.

Crossrefs

Cf. A119870, A119875 [vertices of void-centered Waterman polyhedron].
Cf. A055039 [missing polyhedra]. Properties of Waterman polyhedra: A119870 [vertices], A119871 [faces], A119872 [edges], A119873 [volume]. Waterman polyhedra with different center: A119874, A119875, A119876, A119877, A119878.

A119869 Sizes of successive clusters in f.c.c. lattice centered at a lattice point.

Original entry on oeis.org

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

Views

Author

Hugo Pfoertner, May 26 2006

Keywords

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

Crossrefs

Cf. A004015, A004215, A055039.
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.

Programs

  • 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
  • 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

A119874 Sizes of successive clusters in f.c.c. lattice centered at an octahedral hole.

Original entry on oeis.org

6, 14, 38, 38, 68, 92, 116, 116, 164, 188, 236, 236, 266, 298, 370, 370, 418, 466, 490, 490, 586, 610, 682, 682, 736, 784, 856, 856, 904, 976, 1048, 1048, 1144, 1168, 1264, 1264, 1312, 1368, 1464, 1464, 1566, 1638, 1686, 1686, 1830, 1878, 1926, 1926, 1974
Offset: 0

Views

Author

Hugo Pfoertner, Jun 05 2006

Keywords

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

Crossrefs

Cf. A005887.
Cf. A119869, Properties of Waterman polyhedra of void center type: A119875 [vertices], A119876 [faces], A119877 [edges], A119878 [volume].

Programs

  • 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),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

Formula

Partial sums of A005887, which has an explicit generating function.

Extensions

Edited by N. J. A. Sloane, Aug 09 2006

A119872 Number of edges of the root-n Waterman polyhedron as defined in A119870.

Original entry on oeis.org

24, 12, 48, 24, 36, 72, 72, 120, 72, 36, 96, 48, 144, 144, 72, 96, 108, 96, 144, 120, 144, 120, 72, 120, 264, 156, 240, 168, 120, 120, 144, 192, 216, 192, 192, 216, 252, 264, 168, 120, 264, 144, 336, 336, 216, 216, 264, 288, 240, 300, 312, 144, 288, 408, 264, 264
Offset: 1

Views

Author

Hugo Pfoertner, May 26 2006

Keywords

Comments

For more information see A119870.

Crossrefs

Cf. A119869, A119877 [edges of void-centered Waterman polyhedron].
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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