A132439 Square array a(m,n) read by antidiagonals, where a(m,n) is the number of ways to move a chess queen from the lower left corner to square (m,n), with the queen moving only up, right, or diagonally up-right.

1, 1, 1, 2, 3, 2, 4, 7, 7, 4, 8, 17, 22, 17, 8, 16, 40, 60, 60, 40, 16, 32, 92, 158, 188, 158, 92, 32, 64, 208, 401, 543, 543, 401, 208, 64, 128, 464, 990, 1498, 1712, 1498, 990, 464, 128, 256, 1024, 2392, 3985, 5079, 5079, 3985, 2392, 1024, 256

Offset: 1

Author

Martin J. Erickson (erickson(AT)truman.edu), Nov 13 2007

Keywords

• easy
• nonn
• tabl

Comments

a(m,n) is the sum of all the entries above it plus the sum of all the entries to the left of it plus the sum of all the entries on the northwest diagonal from it.

Examples

			The table begins
  1  1   2   4    8   16    32 ...
  1  3   7  17   40   92   208 ...
  2  7  22  60  158  401   990 ...
  4 17  60 188  543 1498  3985 ...
  8 40 158 543 1712 5079 14430 ...
a(3,4)=4+17+2+7+22+1+7=60.
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened)
• Peter Kagey, Parity bitmap of first 1024 rows and columns. (Even and odd entries and represented by black and white pixels respectively.)

Crossrefs

Cf. A035002.

Formula

a(1,1)=1; a(1,2)=1; a(1,3)=2; a(2,1)=1; a(2,2)=3; a(2,3)=7; a(3,1)=2; a(3,2)=7; a(3,3)=22; a(m,n) = 2*a(m-1,n)+2*a(m,n-1)-a(m-1,n-1)-3*a(m-2,n-1)-3*a(m-1,n-2)+4*a(m-2,n-2), where m >=3 or n >= 3 and a(m,n)=0 if m <= 0 or n <= 0.

G.f.: (xy-x^2y-xy^2+x^3y^2+x^2y^3-x^3y^3)/(1-2x-2y+xy+3x^2y+3xy^2-4x^2y^2).