%I A005993 M1576
%S A005993 1,2,6,10,19,28,44,60,85,110,146,182,231,280,344,408,489,570,670,
%T A005993 770,891,1012,1156,1300,1469,1638,1834,2030,2255,2480,2736,2992,
%U A005993 3281,3570,3894,4218,4579,4940,5340,5740,6181,6622,7106,7590,8119
%N A005993 G.f.: (1+x^2)/((1-x)^2*(1-x^2)^2).
%C A005993 Alkane (or paraffin) numbers l(6,n).
%C A005993 Also multidigraphs with loops on 2 nodes with n arcs - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Dec 27 1999
%C A005993 Euler transform of finite sequence [2,3,0,-1]. - Michael Somos Mar 17 
               2004
%C A005993 a(n-2) is the number of plane partitions with trace 2. - Michael Somos 
               Mar 17 2004
%C A005993 With offset 4, a(n) is the number of bracelets with n beads, 3 of which 
               are red, 1 of which is blue. For odd n, a(n) = C(n-1,3)/2. For even 
               n, a(n) = C(n-1,3)/2 +(n-2)/4. For n >= 6, with K = (n-1)(n-2)/((n-5)(n-4)), 
               for odd n, a(n) = K*a(n-2). For even n, a(n) = K*a(n-2) -(n-2)/(n-5). 
               [From Washington Bomfim (webonfim(AT)bol.com.br), Aug 05 2008]
%C A005993 Equals (1,2,3,4...) convolved with (1,0,3,0,5,...) [From Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Feb 16 2009]
%D A005993 S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, 
               Chem. Ber. 30 (1897), 1917-1926.
%D A005993 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005993 L. Smith, Polynomial Invariants of Finite Groups, A K Peters, 1995, p. 
               96.
%D A005993 L. Smith, Polynomial invariants of finite groups. A survey of recent 
               developments. Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 3, 211-250. 
               See page 218. MR1433171 (98i:13009)
%D A005993 Winston C. Yang (paper in preparation).
%H A005993 T. D. Noe, <a href="b005993.txt">Table of n, a(n) for n=0..1000</a>
%H A005993 Dragomir Z. Djokovic, <a href="http://arXiv.org/abs/math.AC/0609262">
               Poincare series of some pure and mixed trace algebras of two generic 
               matrices</a>. See Table 8.
%H A005993 N. J. A. Sloane, <a href="classic.html#LOSS">Classic Sequences</a>
%H A005993 Washington Bomfim, <a href="http://commons.wikimedia.org/wiki/Image:Comentario1Blue3Reds.PNG">
               The 19 bracelets with 8 beads - one blue, three reds and four blacks.</
               a> [From Washington Bomfim (webonfim(AT)bol.com.br), Aug 05 2008]
%H A005993 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A005993 l(c, r) = 1/2 C(c+r-3, r) + 1/2 d(c, r), where d(c, r) is C((c + r - 
               3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, 
               C((c + r - 4)/2, r/2) if c is even and r is even, C((c + r - 4)/2, 
               (r - 1)/2) if c is odd and r is odd.
%F A005993 Dimension of the space of homogeneous degree n polynomials in (x1, y1, 
               x2, y2) invariant under permutation of variables x1<->y1, x2<->y2.
%F A005993 G.f.: (1+x^2)/((1-x)^2*(1-x^2)^2) = (1/(1-x)^4 +1/(1-x^2)^2)/2.
%F A005993 a(2n)=(n+1)(2n^2+4n+3)/3, a(2n+1)=(n+1)(n+2)(2n+3)/3. a(-4-n)=-a(n).
%F A005993 Contribution from Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Sep 12 
               2008: (Start)
%F A005993 a(n+1)= a(n) + A008794(n+3) with a(1)=1
%F A005993 a(n)= A027656 (n) + 2*A006918 (n)
%F A005993 a(n+2)= a(n) + A000982 (n+2) with a(1)=1, a(2)=2
%F A005993 (End)
%F A005993 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]
%e A005993 a(2)=6 since ( x1*y1, x2*y2, x1*x1+y1*y1, x2*x2+y2*y2, x1*x2+y1*y2, x1*y2+x2*y1 
               ) are a basis for homogenous quadratic invariant polynomials.
%p A005993 g := proc(n) local i; add(floor(i/2)^2,i=1..n+1) end: # Joseph S. Riel 
               (joer(AT)k-online.com), Mar 22 2002
%p A005993 (Maple) a := n -> (Matrix([[1, 0$3, -1, -2]]).Matrix(6, (i,j)-> if (i=j-1) 
               then 1 elif j=1 then [2, 1, -4, 1, 2, -1][i] else 0 fi)^n)[1,1]; 
               seq (a(n), n=0..44); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), 
               Jul 31 2008]
%o A005993 (PARI) a(n)=polcoeff((1+x^2)/(1-x)^2/(1-x^2)^2+x*O(x^n),n)
%o A005993 (PARI) a(n) = (binomial(n+3, n) + (1-n%2)*binomial((n+2)/2, n>>1))/2. 
               [From Washington Bomfim (webonfim(AT)bol.com.br), Aug 05 2008]
%Y A005993 Sequence in context: A006553 A054273 A127567 this_sequence A028247 A065054 
               A128165
%Y A005993 Adjacent sequences: A005990 A005991 A005992 this_sequence A005994 A005995 
               A005996
%K A005993 nonn,easy,nice
%O A005993 0,2
%A A005993 N. J. A. Sloane (njas(AT)research.att.com), Winston C. Yang (yang(AT)math.wisc.edu)