1,2

Number of n X n binary matrices with n ones under action of dihedral group of the square D_4.

Mario Velucchi, Title?

Mario Velucchi, Different Dispositions in the ChessBoard.

See Velucchi link or the PARI program. Note that the polynomial whose coefficient of a^k is divided by 8 differs based upon whether the term's index is even or odd.

Let A(n) = C(n^2, n); B(n) = C((n^2-(n mod 2))/2, n/2); C(n) = C((n^2-(n mod 2))/4, n/4); D(n) = Sum(p = 0 to [n/2], C((n^2-n)/2, p)*C(n, n-2p)). Then a(n) = (A(n) + 3B(n) + 2C(n) + 2D(n))/8 if n == 0 (mod 4), (A(n) + B(n) + 2C(n) + 4D(n))/8 if n == 1 (mod 4), (A(n) + 3B(n) + 2D(n))/8 if n == 2 (mod 4), (A(n) + B(n) + 4D(n))/8 if n == 3 (mod 4). - David W. Wilson (davidwwilson(AT)comcast.net), May 29 2003

For n=3 the 16 solutions are

111 110 110 110 110 110 110 101 101 101 100 100 100 010 010 010

000 100 010 001 000 000 000 010 000 000 011 010 001 110 101 010

000 000 000 000 100 010 001 000 100 010 000 001 010 000 000 010

(PARI) {p(a, b, N) = if(N%2==0, (a+b)^(N^2) + 2*(a+b)^N*(a^2+b^2)^((N^2-N)/2) + 3*(a^2+b^2)^(N^2/2) + 2*(a^4+b^4)^(N^2/4), (a+b)^(N^2) + 2*(a+b)*(a^4+b^4)^((N^2-1)/4) + (a+b)*(a^2+b^2)^((N^2-1)/2) + 4*(a+b)^N*(a^2+b^2)^((N^2-N)/2))} for(k=1, 20, print1(polcoeff(p(a, 1, k), k)/8, ", "))

Cf. A054252 and A014409.

Sequence in context: A138764 A009833 A009044 this_sequence A090727 A108242 A140307

Adjacent sequences: A019315 A019316 A019317 this_sequence A019319 A019320 A019321

nonn,nice

Mario Velucchi (mathchess(AT)velucchi.it)

More terms from Rick L. Shepherd (rshepherd2(AT)hotmail.com) and David W. Wilson (davidwwilson(AT)comcast.net), May 28 2003