%I A006918 M1349
%S A006918 0,1,2,5,8,14,20,30,40,55,70,91,112,140,168,204,240,285,330,385,440,506,
               572,
%T A006918 650,728,819,910,1015,1120,1240,1360,1496,1632,1785,1938,2109,2280,2470,
               2660,
%U A006918 2870,3080,3311,3542,3795,4048,4324,4600,4900,5200,5525,5850,6201,6552,
               6930
%N A006918 C(n+3,3)/4, n odd; n(n+2)(n+4)/24, n even.
%C A006918 Maximal number of inconsistent triples in a tournament on n nodes [Kac]
%C A006918 a(n-4)=number of aperiodic necklaces (Lyndon words) with 4 black beads 
               and n-4 white beads.
%C A006918 Comment from Erich Friedman (erich.friedman(AT)stetson.edu): also the 
               maximum number of squares that can be formed from n lines.
%C A006918 Number of trees with diameter 4 where at most 2 vertices 1 away from 
               the graph center have degree > 2. - Jon Perry (perry(AT)globalnet.co.uk), 
               Jul 11 2003
%C A006918 a(n+1) is the number of partitions of n into parts of two kinds, with 
               at most two parts of each kind. Also a(n-3) is the number of partitions 
               of n with Durfee square of size 2. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jan 27 2006
%C A006918 Starting with offset 1 = partial sums of (1, 1, 3, 3, 6, 6, 10, 10,...). 
               [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 30 2009]
%C A006918 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 01 2009: 
               (Start)
%C A006918 Equals (1, 2, 3, 4,...) convolved with (1, 0, 2, 0, 3, 0, 4,...).
%C A006918 a(6) = 20 = (1, 2, 3, 4, 5, 6) dot (0, 3, 0, 2, 0, 1) = (0 + 6 + 0 + 
               8 + 0 + 6). (End)
%C A006918 (1 + 2x + 5x^2 + ...) = convolution square of (1 + x + 2x^2 + 2x^3 + 
               3x^4 + 3x^5 + ...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jul 27 2009]
%D A006918 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006918 S. J. Cyvin et al., Polygonal systems including the corannulene ... homologs 
               ..., Z. Naturforsch., 52a (1997), 867-873.
%D A006918 M. Kac, An example of "counting without counting", Philips Res. Reports, 
               30 (1975), 20*-22* [Special issue in honour of C. J. Bouwkamp]
%D A006918 E. V. McLaughlin, Numbers of factorizations in non-unique factorial domains, 
               Senior Thesis, Allegeny College, Meadville, PA, 2004.
%D A006918 K. B. Reid and L. W. Beineke "Tournaments", pp. 169-204 in L. W. Beineke 
               and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic 
               Press, NY, 1978, p. 186 Theorem 6.11.
%D A006918 W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, 
               p. 33.
%D A006918 Brian OSullivan and Thomas Busch, Spontaneous emission in ultra-cold 
               spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 
               2008. [Eq 10b, lambda=2]
%H A006918 T. D. Noe, <a href="b006918.txt">Table of n, a(n) for n=0..1000</a>
%H A006918 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A006918 D. J. Broadhurst, <a href="http://arXiv.org/abs/hep-th/9604128">On the 
               enumeration of irreducible k-fold Euler sums and their roles in knot 
               theory and field theory</a>
%H A006918 <a href="Sindx_Lu.html#Lyndon">Index entries for sequences related to 
               Lyndon words</a>
%F A006918 G.f.: x/((1-x)^2*(1-x^2)^2).
%F A006918 0, 0, 0, 1, 2, ... has a(n)=sum{k=0..n, floor(k(n-k)/2) }/2 - Paul Barry 
               (pbarry(AT)wit.ie), Sep 14 2003
%F A006918 a(0)=0, a(1)=1 a(n)=(2/(n-1))*a(n-1)+((n+3)/(n-1))*a(n-2) - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Jun 28 2004
%F A006918 a(n)=floor(C(n+4, 4)/(n+4))-floor((n+2)/8)(1+(-1)^n)/2 - Paul Barry (pbarry(AT)wit.ie), 
               Jan 01 2005
%F A006918 a(n+1) = a(n) + C([n/2]+2,2) Convolution of A008619 with itself, then 
               shifted right (or A004526 with itself, shifted left by 3). - Frank 
               Adams-Watters (FrankTAW(AT)Netscape.net), Jan 27 2006
%F A006918 Contribution from Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Sep 12 
               2008: (Start)
%F A006918 A006918 (n+1)= (A027656 (n) + A003451(n))/2 with a(1)=0
%F A006918 (End)
%F A006918 Linear recurrence: a(n)=2a(n-1)+a(n-2)-4a(n-3)+a(n-4)+2a(n-5)-a(n-6) 
               [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 05 2008]
%F A006918 Euler transform of length 2 sequence [ 2, 2]. - Michael Somos Aug 15 
               2009
%F A006918 a(-4 - n) = -a(n).
%e A006918 x + 2*x^2 + 5*x^3 + 8*x^4 + 14*x^5 + 20*x^6 + 30*x^7 + 40*x^8 + 55*x^9 
               + ...
%p A006918 with(combstruct):ZL:=[st,{st=Prod(left,right),left=Set(U,card=r),right=Set(U,
               card=r),U=Sequence(Z,card>=3)}, unlabeled]: subs(r=1,stack): seq(count(subs(r=2,
               ZL),size=m),m=11..58) ; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 09 2007
%o A006918 (PARI) { parttrees(n)=local(pt,k,nk); if (n%2==0, pt=(n/2+1)^2, pt=ceil(n/
               2)*(ceil(n/2)+1)); pt+=floor(n/2); for (x=1,floor(n/2),pt+=floor(x/
               2)+floor((n-x)/2)); if (n%2==0 && n>2, pt-=floor(n/4)); k=1; while 
               (3*k<=n, for (x=k,floor((n-k)/2), pt+=floor(k/2); if (x!=k, pt+=floor(x/
               2)); if ((n-x-k)!=k && (n-x-k)!=x, pt+=floor((n-x-k)/2))); k++); 
               pt }
%o A006918 (PARI) {a(n) = n += 2; (n^3 - n * (2-n%2)^2) / 24} /* Michael Somos Aug 
               15 2009 */
%Y A006918 Cf. A000031, A001037, A028723, A051168. a(n) = T(n, 4), array T as in 
               A051168.
%Y A006918 Cf. A000094.
%Y A006918 Sequence in context: A111711 A095348 A022907 this_sequence A165189 A011842 
               A000094
%Y A006918 Adjacent sequences: A006915 A006916 A006917 this_sequence A006919 A006920 
               A006921
%K A006918 nonn,nice,easy
%O A006918 0,3
%A A006918 N. J. A. Sloane (njas(AT)research.att.com).