A355509 Peaceable coexisting armies of knights: a(n) is the maximum number m such that m white knights and m black knights can coexist on an n X n chessboard without attacking each other.

0, 2, 3, 6, 10, 14, 18, 24, 32, 40, 50, 60, 72, 84, 98, 112, 128, 144, 162, 180, 200, 220, 242, 264, 288, 312, 338, 364, 392, 420, 450, 480, 512, 544, 578, 612, 648, 684, 722, 760, 800, 840, 882, 924, 968, 1012, 1058, 1104, 1152, 1200, 1250, 1300, 1352, 1404

Offset: 1

Author

Aaron Khan, Jul 04 2022

Keywords

• nonn
• easy

Comments

After the first 7 terms, the first differences are terms of A052928: for n >= 8, a(n) - a(n-1) = A052928(n-1).

The increase in differences going from an even n to an odd n, but not from an odd n to an even n, is due to the differing optimal layouts for odd vs. even n values. See example section for a(7) and a(8).

Examples

			Examples for n=2 to n=6 have been included as they do not follow the general formula.
.
A solution illustrating a(2)=2:
  +-----+
  | B B |
  | W W |
  +-----+
.
A solution illustrating a(3)=3:
  +-------+
  | . . . |
  | B B W |
  | W W B |
  +-------+
.
A solution illustrating a(4)=6:
  +---------+
  | B B . W |
  | W W . B |
  | B B . W |
  | W W . B |
  +---------+
.
A solution illustrating a(5)=10:
  +-----------+
  | W B W B W |
  | W B W B W |
  | . . . . . |
  | B W B W B |
  | B W B W B |
  +-----------+
.
A solution illustrating a(6)=14:
  +-------------+
  | B B W W B B |
  | W W B B W W |
  | B . . . . B |
  | W . . . . W |
  | B B W W B B |
  | W W B B W W |
  +-------------+
.
Examples for n=7 and n=8 are provided, as while both follow the same formula, the layout for even values of n differs from the layout for odd values of n (related to the fact that, for even values of n, the floor function rounds down a non-integer value).
.
A solution illustrating a(7)=18:
  +---------------+
  | B B B B B B B |
  | B B B B B B B |
  | B . B . B . B |
  | . . . . . . . |
  | W . W . W . W |
  | W W W W W W W |
  | W W W W W W W |
  +---------------+
.
A solution illustrating a(8)=24:
  +-----------------+
  | B B B B B B B B |
  | B B B B B B B B |
  | B B B B B B B B |
  | . . . . . . . . |
  | . . . . . . . . |
  | W W W W W W W W |
  | W W W W W W W W |
  | W W W W W W W W |
  +-----------------+
		

Links

• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A007590, A052928, A176222 (peaceable kings), A250000 (peaceable queens), A002620 (peaceable rooks).

Formula

For n > 6, a(n) = floor(((n-1)^2)/2).

G.f.: x^2*(2 - x + 2*x^3 - 2*x^4 - x^5 + 2*x^6 + 2*x^7 - 2*x^8)/((1 - x)^3*(1 + x)). - Stefano Spezia, Jul 05 2022