A193580 Triangle read by rows: T(n,k) = number of ways to place k nonattacking kings on an n X n board.

1, 1, 1, 1, 4, 1, 9, 16, 8, 1, 1, 16, 78, 140, 79, 1, 25, 228, 964, 1987, 1974, 978, 242, 27, 1, 1, 36, 520, 3920, 16834, 42368, 62266, 51504, 21792, 3600, 1, 49, 1020, 11860, 85275, 397014, 1220298, 2484382, 3324193, 2882737, 1601292, 569818, 129657, 18389, 1520, 64, 1

Offset: 0

Author

Andrew Woods, Aug 27 2011

Keywords

• nonn
• tabf

Comments

Rows 2n and 2n-1 both contain 1 + n^2 entries. Cf. A008794.

Row n sums to A063443(n+1).

Number of walks of length n-1 on a graph in which each node represents a 11-avoiding n-bit binary sequence B and adjacency of B and B' is determined by B'&(B|(B<<1)|(B>>1))=0 and the total number of nonzero bits in the walk is k.

Row n gives the coefficients of the independence polynomial of the n X n king graph. - Eric W. Weisstein, Jun 20 2017

Examples

			The table begins with T(0,0):
  1;
  1,   1;
  1,   4;
  1,   9,  16,   8,   1;
  1,  16,  78, 140,  79;
  ...
T(4,3) = 140 because there are 140 ways to place 3 kings on a 4 X 4 chessboard so that no king threatens any other.
		

References

• Norman Biggs, Algebraic Graph Theory, Cambridge University Press, New York, NY, second edition, 1993.

Links

• Liang Kai, Rows n = 0..26, flattened (Rows n = 0..20 from Andrew Woods, row n = 21 from Alois P. Heinz)
• Kai Liang, Independent Set Enumeration in King Graphs by Tensor Network Contractions, arXiv:2505.12776 [math.CO], 2025. See p. 1.
• R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares arXiv:1609.03964 [math.CO], 2016, Section 4.1.
• Johan Nilsson, On Counting the Number of Tilings of a Rectangle with Squares of Size 1 and 2, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.2.
• Eric Weisstein's World of Mathematics, Independence Polynomial
• Eric Weisstein's World of Mathematics, King Graph

Crossrefs

Columns 2 to 10: A061995, A061996, A061997, A061998, A172158, A194788, A201369, A201771, A220467.

Diagonal: A201513.

Cf. A179403, etc., for extension to toroidal boards.

Cf. A166540, etc., for extension into three dimensions.

Cf. A098487 for a clipped version.

Row n sums to A063443(n+1).

Formula

T(n, 0) = 1;

T(n, 1) = n^2;

T(2n-1, n^2-1) = n^3;

T(2n-1, n^2) = 1.