A005897 (Points on surface of cube)

[N. J. A. Sloane]

The entry is , in full,

%I A005897 M4497
%S A005897 1,8,26,56,98,152,218,296,386,488,602,728,866,1016,1178,1352,1538,
%T A005897 1736,1946,2168,2402,2648,2906,3176,3458,3752,4058,4376,4706,5048,
%U A005897 5402,5768,6146,6536,6938,7352,7778,8216,8666,9128,9602,10088,10586
%N A005897 Points on surface of cube: 6n^2 + 2 (coordination sequence for b.c.c. lattice).
%D A005897 H. S. M. Coxeter, ``Polyhedral numbers,'' in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
%D A005897 Gmelin Handbook of Inorg. and Organomet. Chem., 8th Ed., 1994, TYPIX search code (194) hP4
%D A005897 R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.
%D A005897 B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
%H A005897 R. W. Grosse-Kunstleve, <a href="http://cci.lbl.gov/~rwgk/EIS/CS.html">Coordination Sequences and Encyclopedia of Integer Sequences</a>
%H A005897 R. W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/ac96cs/">Algebraic Description of Coordination Sequences and Exact Topological Densities for Zeolites</a>, Acta Cryst., A52 (1996), pp. 87
%H A005897 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ba.html#bcc">Index entries for sequences related to b.c.c. lattice</a>
%K A005897 nonn,easy,nice
%O A005897 0,2
%A A005897 njas, rwgk(AT)cci.lbl.gov (R.W. Grosse-Kunstleve)


My paper with Teo discusses this and many similar questions.

To get 6n^2 + 2 (for n>0) take a cube, divide
each edge into n+1 equally spaced points
(so for n = 2 we get *----*----* )
and fill in each face with a square mesh through these
points. The total number of points is
8 + 12(n-1) + 6(n-1)^2 = 6n^2+2.

NJAS