0,2

Alkane (or paraffin) numbers l(6,n).

Also multidigraphs with loops on 2 nodes with n arcs - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 27 1999

Euler transform of finite sequence [2,3,0,-1]. - Michael Somos Mar 17 2004

a(n-2) is the number of plane partitions with trace 2. - Michael Somos Mar 17 2004

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]

Equals (1,2,3,4...) convolved with (1,0,3,0,5,...) [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 16 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

L. Smith, Polynomial Invariants of Finite Groups, A K Peters, 1995, p. 96.

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)

Winston C. Yang (paper in preparation).

T. D. Noe, Table of n, a(n) for n=0..1000

Index entries for sequences related to linear recurrences with constant coefficients

Dragomir Z. Djokovic, Poincare series of some pure and mixed trace algebras of two generic matrices. See Table 8.

N. J. A. Sloane, Classic Sequences

Washington Bomfim, The 19 bracelets with 8 beads - one blue, three reds and four blacks. [From Washington Bomfim (webonfim(AT)bol.com.br), Aug 05 2008]

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.

Dimension of the space of homogeneous degree n polynomials in (x1, y1, x2, y2) invariant under permutation of variables x1<->y1, x2<->y2.

G.f.: (1+x^2)/((1-x)^2*(1-x^2)^2) = (1/(1-x)^4 +1/(1-x^2)^2)/2.

a(2n)=(n+1)(2n^2+4n+3)/3, a(2n+1)=(n+1)(n+2)(2n+3)/3. a(-4-n)=-a(n).

Contribution from Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Sep 12 2008: (Start)

a(n+1)= a(n) + A008794(n+3) with a(1)=1

a(n)= A027656 (n) + 2*A006918 (n)

a(n+2)= a(n) + A000982 (n+2) with a(1)=1, a(2)=2

(End)

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]

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.

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

(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]

(PARI) a(n)=polcoeff((1+x^2)/(1-x)^2/(1-x^2)^2+x*O(x^n), n)

(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]

Sequence in context: A006553 A054273 A127567 this_sequence A028247 A065054 A128165

Adjacent sequences: A005990 A005991 A005992 this_sequence A005994 A005995 A005996

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Winston C. Yang (yang(AT)math.wisc.edu)