1,2

Number of odd parts in all compositions (ordered partitions) of n: a(3)=6 because in 3=2+1=1+2=1+1+1 we have 6 odd parts. Number of even parts in all compositions (ordered partitions) of n+1: a(3)=6 because in 4=3+1=1+3=2+2=2+1+1=1+2+1=1+1+2=1+1+1+1 we have 6 even parts.

S. Heubach and T. Mansour, Counting rises, levels, and drops in compositions

a(n) = (3*n+4)*2^n/18 - 2*(-1)^n/9; G.f.: z*(1-z)/((1+z)*(1-2*z)^2).

a(n)=sum{j=0..n, sum{k=0..n, binomial(n-k, k+j)2^k}} - Paul Barry (pbarry(AT)wit.ie), Aug 29 2004

a(n)=sum{k=0..n+1, (-1)^(k+1)*C(n+1, k+j)A001045(k)} - Paul Barry (pbarry(AT)wit.ie), Jan 30 2005

Convolution of "Expansion of (1-x)/(1-x-2*x^2)" (A078008) with "Powers of 2" (A000079), treating the result as if offset=1. - Graeme McRae (g_m(AT)mcraefamily.com), Jul 12 2006

Convolution of "Difference sequence of A045623" (A045891) with "Positive integers repeated" (A008619), treating the result as if offset=1. - Graeme McRae (g_m(AT)mcraefamily.com), Jul 12 2006

a(n)=3*a(n-1)-4*a(n-3) ; a(1)=1,a(2)=2,a(3)=6 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 30 2006

a(n)=3*a(n-1)-4*a(n-3) ; a(1)=1,a(2)=2,a(3)=6 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 30 2006

Equals row sums of A128255. (1, 2, 6, 14, 34,...) - (0, 0, 1, 2, 6, 14, 34,...) = A045623: (1, 2, 5, 12, 28, 64,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 20 2007

a(3)=6 because in the 231-avoiding involutions of {1,2,3}, i.e. in 123, 132, 213, 321, we have alltogether 6 fixed points (3+1+1+1).

Cf. A027934.

Cf. A128255, A045623.

Sequence in context: A124613 A124614 A070933 this_sequence A018016 A099425 A105635

Adjacent sequences: A059567 A059568 A059569 this_sequence A059571 A059572 A059573

nonn

Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 16 2001

More terms from Eugene McDonnell (eemcd(AT)mac.com), Jan 13 2005