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

1,4

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.

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.)

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).

Example

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.

Crossrefs

Cf. A035002.

Sequence in context: A229012 A207606 A303845 * A338902 A116217 A333907

Adjacent sequences: A132436 A132437 A132438 * A132440 A132441 A132442

Keyword

easy,nonn,tabl

Author

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

Status

approved