%I A005894 M3850
%S A005894 1,5,15,35,69,121,195,295,425,589,791,1035,1325,1665,2059,2511,
%T A005894 3025,3605,4255,4979,5781,6665,7635,8695,9849,11101,12455,13915,
%U A005894 15485,17169,18971,20895,22945,25125,27439,29891,32485,35225,38115
%N A005894 Centered tetrahedral numbers.
%C A005894 Binomial transform of (1,4,6,4,0,0,0,.......) - Paul Barry (pbarry(AT)wit.ie), 
               Jul 01 2003
%C A005894 If X is an n-set and Y a fixed 4-subset of X then a(n-4) is equal to 
               the number of 4-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), 
               Jul 30 2007
%D A005894 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005894 S. 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.
%D A005894 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. 
               (10).
%D A005894 B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral 
               clusters, Inorgan. Chem. 24 (1985), 4545-4558.
%H A005894 T. D. Noe, <a href="b005894.txt">Table of n, a(n) for n=0..1000</a>
%H A005894 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A005894 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 A005894 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.
%F A005894 a(n)=(1/3)*(2*n+1)*(n^2+n+3). G.f.: (1-x^4)/(1-x)^5.
%F A005894 a(n)=C(n, 0)+4C(n, 1)+6C(n, 2)+4C(n, 3) - Paul Barry (pbarry(AT)wit.ie), 
               Jul 01 2003
%F A005894 a(n) is the sum of 4 consecutive tetrahedral (or pyramidal) numbers: 
               C(n+3,3) = (n+1)(n+2)(n+3)/6 = A000292(n). a(n) = A000292(n-3) + 
               A000292(n-2) + A000292(n-1) + A000292(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               May 20 2006
%F A005894 binomial(n+6,n+3)+binomial(n+5,n+2)+binomial(n+4,n+1)+binomial(n+3,n).
%p A005894 A005894:=(z+1)*(1+z**2)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 
               dissertation.]
%Y A005894 (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 A005894 Cf. A000292.
%Y A005894 Sequence in context: A061829 A063382 A069983 this_sequence A015622 A000750 
               A008487
%Y A005894 Adjacent sequences: A005891 A005892 A005893 this_sequence A005895 A005896 
               A005897
%K A005894 nonn,easy,nice
%O A005894 0,2
%A A005894 N. J. A. Sloane (njas(AT)research.att.com).