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

Starting with "1" = binomial transform of [1, 5, 8, 4, 0, 0, 0,...] where (1, 5, 8, 4) = row 3 of the Chebyshev triangle A081277. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 19 2008

a(n) = largest coefficient of (1+...+x^(n-1)) ^ 4 [From Ron Hardin (rhhardin(AT)att.net), Jul 23 2009]

Convolution square root of (1 + 6x + 19x^3 + ...) = (1 + 3x + 5x^2 + 7x^3 + ...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 27 2009]

Starting with offset 1 = the triangular series convolved with [1, 3, 4, 4, 4,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 28 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

X. Acloque, Polynexus Numbers and other mathematical wonders.

Milan Janjic, Two Enumerative Functions

Hyun Kwang Kim, On Regular Polytope Numbers

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.

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)

G.f.: sage: taylor( mul( x*(x^2+2*x+1)/(x-1)^4 for i in xrange(1,2)),x,0,40)# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 03 2009]

sum(1/((2*n^3+n)*(1/3)),n=1..infinity)=3*gamma+3*Psi((I*(1/2))*sqrt(2))-(1/2)*(3*I)*Pi*coth((1/2)*Pi*sqrt(2))-(1/2)*(3*I)*sqrt(2)=1.27818515909094617954039094836757133842390... [From Stephen Crowley (crow(AT)crowlogic.net), Jul 14 2009]

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; [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.

Cf. A081277.

Sequence in context: A096957 A035495 A061293 this_sequence A138357 A005712 A070893

Adjacent sequences: A005897 A005898 A005899 this_sequence A005901 A005902 A005903

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

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