%I A001845 M4384 N1844
%S A001845 1,7,25,63,129,231,377,575,833,1159,1561,2047,2625,3303,4089,4991,6017,
%T A001845 7175,8473,9919,11521,13287,15225,17343,19649,22151,24857,27775,30913,
%U A001845 34279,37881,41727,45825,50183,54809,59711,64897,70375,76153,82239
%N A001845 Centered octahedral numbers (crystal ball sequence for cubic lattice).
%C A001845 Number of points in simple cubic lattice at n steps from origin.
%C A001845 If X is an n-set and Y_i (i=1,2,3) mutually disjoint 2-subsets of X then 
               a(n-6) is equal to the number of 6-subests of X intersecting each 
               Y_i (i=1,2,3). - Milan R. Janjic (agnus(AT)blic.net), Aug 26 2007
%C A001845 Equals binomial transform of [1, 6, 12, 8, 0, 0, 0,...] where (1, 6, 
               12, 8) = row 3 of the Chebyshev triangle A013609. - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Jul 19 2008
%D A001845 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001845 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001845 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D A001845 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. 
               (10).
%D A001845 R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, 
               SIAM Rev., 12 (1970), 277-279.
%H A001845 T. D. Noe, <a href="b001845.txt">Table of n, a(n) for n=0..1000</a>
%H A001845 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001845 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination 
               Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http:/
               /www.research.att.com/~njas/doc/ldl7.txt">Abstract</a>, <a href="http:/
               /www.research.att.com/~njas/doc/ldl7.pdf">pdf</a>, <a href="http:/
               /www.research.att.com/~njas/doc/ldl7.ps">ps</a>).
%H A001845 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A001845 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001845 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001845 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               OctahedralNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A001845 <a href="Sindx_Cor.html#crystal_ball">Index entries for crystal ball 
               sequences</a>
%F A001845 G.f.: (1+x)^3 /(1-x)^4. a(n) = (2*n+1)*(2*n^2+2*n+3)/3.
%F A001845 First differences of A014820(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               May 23 2006
%p A001845 (1/3)*(2*n+1)*(2*n^2+2*n+3);
%p A001845 A001845:=(z+1)**3/(z-1)**4; [Conjectured (correctly) by S. Plouffe in 
               his 1992 dissertation.]
%Y A001845 Sums of 2 consecutive terms give A008412.
%Y A001845 (1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, 
               A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, 
               A063492, A005917, A063493, A063494, A063495, A063496.
%Y A001845 Partial sums of A005899.
%Y A001845 Cf. A001846, A001847, A001848, etc., A014820, A013609.
%Y A001845 Sequence in context: A033814 A118395 A118396 this_sequence A127765 A155305 
               A155290
%Y A001845 Adjacent sequences: A001842 A001843 A001844 this_sequence A001846 A001847 
               A001848
%K A001845 nonn,easy,nice
%O A001845 0,2
%A A001845 N. J. A. Sloane (njas(AT)research.att.com).
%E A001845 More terms from Larry Reeves (larryr(AT)acm.org), Jul 17 2000