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

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.

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

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)

A diagonal of A059419. Partial sums of A002378.

A diagonal of A008291. Row 3 of A074650.

Sequence in context: A025219 A032767 A032633 this_sequence A049031 A058037 A048096

Adjacent sequences: A007287 A007288 A007289 this_sequence A007291 A007292 A007293

nonn,easy

njas, Simon Plouffe (plouffe(AT)math.uqam.ca)