|
| |
|
|
A005897
|
|
a(n) = 6*n^2+2 for n>0, a(0)=1.
(Formerly M4497)
|
|
5
|
|
|
|
1, 8, 26, 56, 98, 152, 218, 296, 386, 488, 602, 728, 866, 1016, 1178, 1352, 1538, 1736, 1946, 2168, 2402, 2648, 2906, 3176, 3458, 3752, 4058, 4376, 4706, 5048, 5402, 5768, 6146, 6536, 6938, 7352, 7778, 8216, 8666, 9128, 9602, 10088, 10586
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Number of points on surface of 3-dimensional cube in which each face has a square grid of dots drawn on it (with n+1 points along each edge, including the corners).
Coordination sequence for b.c.c. lattice.
Binomial transform of [1, 7, 11, 1, -1, 1, -1, 1,...]. - Gary W. Adamson, Oct 22 2007
First differences of A005898 centered cube numbers: n^3 + (n+1)^3. -Jonathan Vos Post, Feb 06 2011
Apart from the first term, numbers of the form (r^2+2*s^2)*n^2+2 = (r*n)^2+(s*n-1)^2+(s*n+1)^2: in this case is r=2, s=1. After 8, all terms are in A000408. - Bruno Berselli, Feb 07 2012
|
|
|
REFERENCES
|
H. S. M. Coxeter, ``Polyhedral numbers,'' in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
Gmelin Handbook of Inorg. and Organomet. Chem., 8th Ed., 1994, TYPIX search code (194) hP4
R. W. Marks and R. B. Fuller, The Dymaxion World of Buckminster Fuller. Anchor, NY, 1973, p. 46.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
|
|
|
LINKS
|
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
R. W. Grosse-Kunstleve, Coordination Sequences and Encyclopedia of Integer Sequences
R. W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, Algebraic Description of Coordination Sequences and Exact Topological Densities for Zeolites, Acta Cryst., A52 (1996), pp. 879-889.
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences related to b.c.c. lattice
Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).
|
|
|
FORMULA
|
G.f.: (1+x)*(1+4*x+x^2)/(1-x)^3. - Simon Plouffe (see MAPLE line)
a(0) = 1, a(n) = (n+1)^3 - (n-1)^3. - Ilya Nikulshin (ilyanik(AT)gmail.com), Aug 11 2009
a(0)=1, a(1)=8, a(2)=26, a(3)=56, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Oct 25 2011
|
|
|
EXAMPLE
|
For n = 1 we get the 8 corners of the cube; for n = 2 each face has 9 points, for a total of 8 + 12 + 6 = 26.
|
|
|
MAPLE
|
A005897:=-(z+1)*(z**2+4*z+1)/(z-1)**3; [Conjectured (correctly) by Simon Plouffe in his 1992 dissertation.]
|
|
|
MATHEMATICA
|
Join[{1}, 6Range[50]^2+2] (* or *) Join[{1}, LinearRecurrence[{3, -3, 1}, {8, 26, 56}, 50]] (* From Harvey P. Dale, Oct 25 2011 *)
|
|
|
PROG
|
(MAGMA) [1], [6*n^2 + 2: n in [1..50]]; // Vincenzo Librandi, Oct 26 2011
|
|
|
CROSSREFS
|
Cf. A206399.
Sequence in context: A126264 A225274 A085690 * A215097 A111694 A129111
Adjacent sequences: A005894 A005895 A005896 * A005898 A005899 A005900
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
N. J. A. Sloane, rwgk(AT)cci.lbl.gov (R.W. Grosse-Kunstleve)
|
|
|
STATUS
|
approved
|
| |
|
|
