0,3

a(n) = 1^2 + 2^2 + ... + (n-1)^2 + n^2 + (n-1)^2 + ... + 2^2 + 1^2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 28 2001

Series reversion of g.f. A(x) is Sum_{n>0} -A066357(n)(-x)^n.

Also as a(n)=(1/6)*(4*n^3+2*n), n>0: structured tetragonal diamond numbers (vertex structure 5) (Cf. A000447 - structured diamonds); and structured trigonal anti-prism numbers (vertex structure 5) (Cf. A100185 - structured anti-prisms). Cf. A100145 for more on structured polyhedral numbers. - James A. Record (james.record(AT)gmail.com), Nov. 7, 2004.

Schlaefli symbol for this polyhedron: {3,4}

If X is an n-set and Y and Z disjoint 2-subsets of X then a(n-4) is equal to the number of 5-subests of X intersecting both Y and Z. - Milan R. Janjic (agnus(AT)blic.net), Aug 26 2007

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.

H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.

T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (5).

B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

T. D. Noe, Table of n, a(n) for n=0..1000

Milan Janjic, Two Enumerative Functions

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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

X. Acloque, Polynexus Numbers and other mathematical wonders.

Hyun Kwang Kim, On Regular Polytope Numbers

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Partial sums of centered square numbers A001844. - Paul Barry (pbarry(AT)wit.ie), Jun 26 2003

G.f.: x(1+x)^2/(1-x)^4. a(n)=-a(-n)=(2n^3+n)/3.

a(n)=((n^5-(n-1)^5)-((n-1)^5-(n-2)^5))/30 - Xavier Acloque Oct 17 2003

a(n) is the sum of the products pq, where p and q are both positive and odd, and p+q=2n, e.g. a(4) = 7*1 + 5*3 + 3*5 + 1*7 = 44 - Jon Perry (perry(AT)globalnet.co.uk), May 17 2005

a(n) = 4C(n,3) + 2C(n,2) + n^2 - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Jul 06 2006

a(n) = C(n+2,3) + 2 C(n+1,3) + C(n,3)

al:=proc(s, n) binomial(n+s-1, s); end; be:=proc(d, n) local r; add( (-1)^r*binomial(d-1, r)*2^(d-1-r)*al(d-r, n), r=0..d-1); end; [seq(be(3, n), n=0..100)];

A005900:=(z+1)**2/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]

with (combinat):seq(fibonacci(4, 2*n)/12, n=0..40); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 21 2008

(PARI) a(n)=(2*n^3+n)/3

Sums of 2 consecutive terms give A001845. Cf. A001844.

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Cf. A022521.

Sequence in context: A096957 A035495 A061293 this_sequence A138357 A005712 A070893

Adjacent sequences: A005897 A005898 A005899 this_sequence A005901 A005902 A005903

nonn,easy

njas

More terms from Larry Reeves (larryr(AT)acm.org), May 03 2001