A379726 -id:A379726 - cp's OEIS frontend

A379759 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 three kings.

3, 5, 12, 12, 16, 27, 27, 33, 48, 48, 56, 75, 75, 85, 108, 108, 120, 147, 147, 161, 192, 192, 208, 243, 243, 261, 300, 300, 320, 363, 363, 385, 432, 432, 456, 507, 507, 533, 588, 588, 616, 675, 675, 705, 768, 768, 800, 867, 867, 901, 972, 972, 1008, 1083, 1083

Offset: 2

Matthew Scroggs, Jan 02 2025

nonn

At most one king can be placed on each square.

			For a 3 by 3 chessboard, the five kings could be placed like this:
   oko
   kkk
   oko
For a 4 by 4 chessboard, the kings could be placed like this:
   okko
   kkkk
   kkkk
   okko
where o is an empty square and k is a king.

Dominic McCarty, Table of n, a(n) for n = 2..100
Matthew Scroggs, Python code to compute A379759
Dominic McCarty, Illustration of a(n) for n = 2...100
Dominic McCarty, Java program for A379759

Cf. A075561, A379726, A379766.

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

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

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.

Original entry on oeis.org

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

Matthew Scroggs, Jan 02 2025

nonn

At most one king can be placed on each square.

			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.

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).

Cf. A379726, A379759.

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

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

cp's OEIS Frontend

A379759 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 three kings.

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