0,4

Number of acute triangles made from the vertices of a regular n-polygon when n is even (cf. A000330). - Sen-Peng You (giawgwan(AT)single.url.com.tw), Apr 05 2001

a(n+2)=(-1)*coefficient of X in Zagier's polynomial (n,n-1) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 12 2002

Definite integrals of certain products of 2 derivatives of (orthogonal) Chebyshev polynomials of the 2nd kind are pi-multiple of this sequence. For even (p+q): Integrate[ D[ChebyshevU[p, x], x] D[ChebyshevU[q, x], x] (1 - x^2)^(1/2), {x,-1,1}] / Pi = a(n), where n=Min[p,q]. Example: a(3)=20 because Integrate[ D[ChebyshevU[3, x], x] D[ChebyshevU[5, x], x] (1 - x^2)^(1/2), {x,-1,1}]/Pi = 20 since 3=Min[3,5] and 3+5 is even. - Christoph Pacher (Christoph.Pacher(AT)arcs.ac.at), Dec 16 2004

If Y is a 2-subset of an n-set X then, for n>=3, a(n-1) is the number of 3-subsets and 4-subsets of X having exactly one element in common with Y. - Milan R. Janjic (agnus(AT)blic.net), Dec 28 2007

a(n) is also the number of proper colorings of the cycle graph Csub3(also the complete graph Ksub3) when n colors are available. [From Gary E. Stevens (StevensG(AT)Hartwick.edu), Dec 28 2008]

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

L. Berzolari, Allgemeine Theorie der Ho"heren Ebenen Algebraischen Kurven, Encyclopa"die der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Band III_2. Heft 3, Leipzig: B.G. Teubner, 1906. p. 352.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259.

O. Haxel et al., On the "Magic Numbers" in Nuclear Structure, Phys. Rev., 75 (1949), 1766.

V. B. Priezzhev, Series expansion for rectilinear polymers on the square lattice, J. Phys. A 12 (1979), 2131-2139.

Index entries for sequences related to linear recurrences with constant coefficients

V. Ladma, Magic Numbers

G.f.: 2*x^3*(1-x)^(-4).

a(n) = a(n-1)*n/(n-3) = a(n-1)+A002378(n-2) = 2*A000292(n-3) = sum_{i = 0 to n} (i*(i+1)) = n(n+1)(n+2)/3. - Henry Bottomley (se16(AT)btinternet.com), Jun 02 2000 Formula corrected by Iain Paterson (paterson(AT)ihs.ac.at), Apr 19 2006.

a(n) = sum of first n oblong numbers (cf. A002378). - Edward Weed (eweed(AT)gdrs.com), Oct 22 2003

a(n)=numbperm(n,3)/3, n>=0 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007

a(n) = A000217(n) + A000330(n-2) for n > 2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 20 2008

seq(numbperm (n, 3)/3, n=0..42); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007

a:=n->sum(i^2-i, i=0..n):seq(a(n), n=-1..41); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2008

a:=n->sum(j^2-j, j=0..n): seq(a(n), n=-1..41); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 08 2008

with(finance):seq(add(cashflows([k^2, k, 0], 0 ), k=1..n), n=-2..45); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 22 2008]

Table[Integrate[ D[ChebyshevU[n, x], x] D[ChebyshevU[n, x], x] (1 - x^2)^(1/2), {x, -1, 1}]/Pi, {n, 1, 20}] (Pacher)

...and/or...lst={0}; s=0; Do[s+=n^2-n; AppendTo[lst, s], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 30 2008]

Table[Sum[i^2 + i, {i, 0, n - 1}], {n, -1, 40}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 10 2009]

A diagonal of A059419. Partial sums of A002378.

A diagonal of A008291. Row 3 of A074650.

A145066 "s+=n^2+1", A051925 "s+=n^2-1", A145067 "s+=n^2-2", A145068 "s+=n^2+2" [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 30 2008]

Adjacent sequences: A007287 A007288 A007289 this_sequence A007291 A007292 A007293

Sequence in context: A025219 A032767 A032633 this_sequence A049031 A058037 A048096

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)