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

2,1

Comments

At most one king can be placed on each square.

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).

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

Example

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.

Crossrefs

Cf. A379726, A379759.

Sequence in context: A313306 A313307 A313308 * A248928 A080819 A313309

Adjacent sequences: A379763 A379764 A379765 * A379767 A379768 A379769

Keyword

nonn

Author

Matthew Scroggs, Jan 02 2025

Extensions

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

Status

approved