A379759 -id:A379759 - cp's OEIS frontend

A379726 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 two kings.

2, 3, 8, 8, 10, 18, 18, 21, 32, 32, 36, 50, 50, 55, 72, 72, 78, 98, 98, 105, 128, 128, 136, 162, 162, 171, 200, 200, 210, 242, 242, 253, 288, 288, 300, 338, 338, 351, 392, 392, 406, 450, 450, 465, 512, 512, 528, 578, 578, 595, 648, 648, 666, 722, 722, 741, 800, 800, 820, 882, 882, 903, 968, 968, 990, 1058, 1058, 1081, 1152, 1152, 1176, 1250, 1250, 1275, 1352, 1352, 1378, 1458, 1458, 1485, 1568, 1568, 1596, 1682, 1682, 1711, 1800, 1800, 1830, 1922, 1922, 1953, 2048, 2048, 2080, 2178, 2178, 2211, 2312

Offset: 2

Matthew Scroggs, Dec 31 2024

nonn

At most one king can be placed on each square.
Every third term is conjectured to be A014105. Other terms are A001105. A093353 is conjectured to be this sequence with repeated terms removed.
The above conjectures are true (see Beveridge link). - Colin Beveridge, Jan 13 2025

			For a 3 by 3 chessboard, the three kings could be placed like this (where o is an empty square and k is a king):
   ooo
   kkk
   ooo
For a 4 by 4 chessboard, the kings could be placed like this:
   oooo
   kkkk
   okko
   okko

Rob Pratt, Table of n, a(n) for n = 2..100
Colin Beveridge, Proof of the case where n is a multiple of 3
Matthew Scroggs, December 23
Matthew Scroggs, Friendly squares
Matthew Scroggs, Python code to compute A379726
Puzzling StackExchange, Minimum Number of Squares to Color
Dominic McCarty, Illustration of a(n) for n = 2..100

Cf. A001105, A014105, A075561, A093353, A379759, A379766.

If n is not a multiple of 3, a(n) = 2*floor((n+2)/3)^2.
If n is a multiple of 3, it is conjectured that a(n)=2*(n/3)^2+n/3.
The above conjectures are true (see Beveridge link). - Colin Beveridge, Jan 13 2025

a(15)-a(100) via integer linear programming by Rob Pratt, Jan 02 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

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A379726 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 two kings.

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