Search: id:A007290
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%I A007290 M1831
%S A007290 0,0,0,2,8,20,40,70,112,168,240,330,440,572,728,910,1120,1360,1632,
%T A007290 1938,2280,2660,3080,3542,4048,4600,5200,5850,6552,7308,8120,8990,9920,
%U A007290 10912,11968,13090,14280,15540,16872,18278,19760,21320,22960
%N A007290 2*C(n,3).
%C A007290 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
%C A007290 a(n+2)=(-1)*coefficient of X in Zagier's polynomial (n,n-1) - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Oct 12 2002
%C A007290 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
%C A007290 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
%C A007290 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]
%D A007290 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A007290 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.
%D A007290 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259.
%D A007290 O. Haxel et al., On the "Magic Numbers" in Nuclear Structure, Phys. Rev., 
               75 (1949), 1766.
%D A007290 V. B. Priezzhev, Series expansion for rectilinear polymers on the square 
               lattice, J. Phys. A 12 (1979), 2131-2139.
%H A007290 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A007290 V. Ladma, <a href="http://www.sweb.cz/vladimir_ladma/english/notes/texts/
               magicn.htm">Magic Numbers</a>
%F A007290 G.f.: 2*x^3*(1-x)^(-4).
%F A007290 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.
%F A007290 a(n) = sum of first n oblong numbers (cf. A002378). - Edward Weed (eweed(AT)gdrs.com), 
               Oct 22 2003
%F A007290 a(n)=numbperm(n,3)/3, n>=0 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 26 2007
%F A007290 a(n) = A000217(n) + A000330(n-2) for n > 2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Mar 20 2008
%p A007290 seq(numbperm (n,3)/3, n=0..42); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 26 2007
%p A007290 a:=n->sum(i^2-i, i=0..n):seq(a(n), n=-1..41); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 25 2008
%p A007290 a:=n->sum(j^2-j, j=0..n): seq(a(n), n=-1..41); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 08 2008
%p A007290 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]
%t A007290 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)
%t A007290 ...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]
%t A007290 Table[Sum[i^2 + i, {i, 0, n - 1}], {n, -1, 40}] [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jul 10 2009]
%Y A007290 A diagonal of A059419. Partial sums of A002378.
%Y A007290 A diagonal of A008291. Row 3 of A074650.
%Y A007290 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]
%Y A007290 Sequence in context: A025219 A032767 A032633 this_sequence A049031 A058037 
               A048096
%Y A007290 Adjacent sequences: A007287 A007288 A007289 this_sequence A007291 A007292 
               A007293
%K A007290 nonn,easy
%O A007290 0,4
%A A007290 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)

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