0,6

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

S. M. Jacob, The enumeration of the Latin rectangle of depth three..., Proc. London Math. Soc., 31 (1928), 329-336.

R. P. Stanley, Space Programs Summary. Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, Vol. 37-40-4 (1966), pp. 208-214.

R. P. Stanley, Enumerative Combinatorics I, Chap. 2, Exercise 8, p. 88.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 182 and p. 183, Table 5.6.

a(n)=(-1)^n + sum((-1)^k*binomial(n, k)*(n-k-1)!, k=0..n-1); e.g.f.: (1-ln(1-x))/e^x; a(n) = (n-3)*a(n-1) + (n-2)*(2*a(n-2) + a(n-3)).

a(n)=(n-2)*a(n-1)+(n-1)*a(n-2)-(-1)^n, n>0.

a(n)=sum(((-1)^(n-j))*D(j-1),j=3..n), n>=3, with the derangements numbers (subfactorials) D(n)=A000166(n).

a(4)=1 because from the 4!/4=6 circular permutations of n=4 elements (1,2,3,4), (1,4,3,2), (1,3,4,2),(1,2,4,3), (1,4,2,3) and (1,3,2,4) only (1,4,3,2) has no successor pair (i,i+1). Note that (4,1) is also a successor pair. W. Lang, Jan 21 2008.

(PARI) a(n)=if(n<0, 0, (-1)^n + sum(k=0, n-1, (-1)^k*binomial(n, k)*(n-k-1)!))

A000757(n)=(-1)^n+A002741(n).

Sequence in context: A001555 A032770 A032794 this_sequence A126756 A058823 A050536

Adjacent sequences: A000754 A000755 A000756 this_sequence A000758 A000759 A000760

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Better description from Len Smiley (smiley(AT)math.uaa.alaska.edu)

Additional comments from Michael Somos, Jun 21, 2002