A379766 Minimum number of kings that must be placed on an n X n chessboard such that each square is attacked or occupied by at least four kings.

4, 9, 16, 16, 24, 36, 36, 47, 64, 64, 78, 100, 100, 117, 144, 144, 164, 196, 196, 219, 256, 256, 282, 324, 324, 353, 400, 400, 432, 484, 484, 519, 576, 576, 614, 676, 676, 717, 784, 784, 828, 900, 900, 947, 1024, 1024, 1074, 1156, 1156, 1209, 1296, 1296, 1352

Offset: 2

Author

Matthew Scroggs, Jan 02 2025

Keywords

• nonn

Comments

At most one king can be placed on each square.

Examples

			For a 5 by 5 chessboard, the sixteen kings could be placed like this:
  kkokk
  kkokk
  ooooo
  kkokk
  kkokk
For a 6 by 6 chessboard, the kings could be placed like this:
  kkookk
  kkkkkk
  okooko
  okooko
  kkkkkk
  kkookk
where o is an empty square and k is a king.
		

Links

• Dominic McCarty, Table of n, a(n) for n = 2..100
• Matthew Scroggs, Python code to compute A379766
• Dominic McCarty, Java program for A379766
• Dominic McCarty, Illustration of a(n) for n = 2..100
• Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Crossrefs

Cf. A379726, A379759.

Formula

It appears that a(3n+1) = a(3n+2) - Dominic McCarty, Jan 17 2025

For n >= 2 we have a(n) = 4*floor(n/3)^2+3*floor(n/3)+2 if 3 divides n, a(n) = 4*(floor(n/3)+1)^2 otherwise. - Benoit Cloitre, Jan 17 2025

G.f.: -x^2*(4+5*x+7*x^2-2*x^4-2*x^5-8*x^3+4*x^6)/(1+x+x^2)^2/(x-1)^3 . - R. J. Mathar, Jan 27 2025

Extensions

a(9)-a(100) from Dominic McCarty, Jan 17 2025