%I A005581 M1744
%S A005581 0,0,2,7,16,30,50,77,112,156,210,275,352,442,546,665,800,952,1122,1311,
%T A005581 1520,1750,2002,2277,2576,2900,3250,3627,4032,4466,4930,5425,5952,6512,
%U A005581 7106,7735,8400,9102,9842,10621,11440,12300,13202,14147,15136,16170
%N A005581 (n-1)*n*(n+4)/6.
%C A005581 A class of Boolean functions of n variables and rank 2.
%C A005581 Also, number of inscribable triangles within a (n+4)-gon sharing with 
               them its vertices but not its sides. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Nov 14 2003
%C A005581 a(n) = A111808(n,3) for n>2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 17 2005
%C A005581 G.f.: (x^2)*(2-x)/(1-x)^4.
%C A005581 If X is an n-set and Y a fixed 2-subset of X then a(n-2) is equal to 
               the number of (n-3)-subsets of X intersecting Y. - Milan R. Janjic 
               (agnus(AT)blic.net), Jul 30 2007
%C A005581 The sequence starting with offset 2 = binomial transform of [2, 5, 4, 
               1, 0, 0, 0,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 
               20 2009]
%D A005581 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 797.
%D A005581 V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post 
               classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 
               (translated in Discrete Mathematics and Applications, 9, (1999), 
               no. 6).
%D A005581 J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 
               177.
%D A005581 A. McLeod and W. O. J. Moser, Counting cyclic binary strings, Math. Mag., 
               80 (No. 1, 2007), 29-37.
%D A005581 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005581 A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with 
               Elementary Solutions. Vol. I. Combinatorial Analysis and Probability 
               Theory. New York: Dover Publications, Inc., 1987, p. 13, #51 (the 
               case k=3) (First published: San Francisco: Holden-Day, Inc., 1964)
%H A005581 T. D. Noe, <a href="b005581.txt">Table of n, a(n) for n=0..1000</a>
%H A005581 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A005581 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A005581 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005581 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005581 C. Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, 
               Explorations and Formulas of the Euler/Pascal Cube</a>.
%H A005581 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TrinomialCoefficient.html">Trinomial Coefficient</a>
%H A005581 <a href="Sindx_Bo.html#Boolean">Index entries for sequences related to 
               Boolean functions</a>
%F A005581 G.f.: (x^2)*(2-x)/(1-x)^4.
%F A005581 a(n)=binomial(n+2, n-1)+binomial(n+1, n-1).
%F A005581 Convolution of {1, 2, 3, ...} with {2, 3, 4, ...} - Jon Perry (perry(AT)globalnet.co.uk), 
               Jun 25 2003
%F A005581 a(n+2)=2*te(n)-te(n-1), e.g. a(5)=2*te(3)-te(2)=2*20-10=30, where te(n) 
               are the tetrahedral numbers A000292 - Jon Perry (perry(AT)globalnet.co.uk), 
               Jul 23 2003
%F A005581 C(3+n,3)-C(1+n,1) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 
               07 2006
%F A005581 a(n) is the coefficient of x^3 in the expansion of (1+x+x^2)^n. For example, 
               a(1)=0 since (1+x+x^2)^1=1+x+x^2. - Peter C. Heinig (algorithms(AT)gmx.de), 
               Apr 09 2007
%F A005581 E.g.f.: (x^2 +x^3/6)* exp(x). - MIchael Somos Apr 13 2007
%p A005581 A005581 := n->(n-1)*n*(n+4)/6;
%p A005581 a:=n->sum ((j+3)*j/2,j=0..n): seq(a(n),n=-1..44); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Dec 17 2006
%p A005581 seq((n+3)*binomial(n,3)/n, n=1..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 28 2007
%p A005581 A005581:=-(-2+z)/(z-1)**4; [S. Plouffe in his 1992 dissertation.]
%p A005581 seq(sum(binomial(n,m), m=1..3)+n^2,n=-1..44); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 19 2008
%t A005581 Table[(n - 1)*n*(n + 4)/6, {n, 0, 40}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 10 2006
%o A005581 (PARI) {a(n)= n* (n+4)* (n-1)/6} /* MIchael Somos Apr 13 2007 */
%Y A005581 Cf. A005582. a(n)= A027907(n, 3), n >= 0 (fourth column of trinomial 
               coefficients).
%Y A005581 Cf. A000292.
%Y A005581 A005586(n)= -a(-4-n).
%Y A005581 Sequence in context: A070169 A162420 A130883 this_sequence A064468 A074470 
               A023550
%Y A005581 Adjacent sequences: A005578 A005579 A005580 this_sequence A005582 A005583 
               A005584
%K A005581 nonn,easy,nice
%O A005581 0,3
%A A005581 N. J. A. Sloane (njas(AT)research.att.com).
%E A005581 More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000