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%I #35 Jul 16 2022 07:12:45
%S 0,2,3,6,10,14,18,24,32,40,50,60,72,84,98,112,128,144,162,180,200,220,
%T 242,264,288,312,338,364,392,420,450,480,512,544,578,612,648,684,722,
%U 760,800,840,882,924,968,1012,1058,1104,1152,1200,1250,1300,1352,1404
%N 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.
%C After the first 7 terms, the first differences are terms of A052928: for n >= 8, a(n) - a(n-1) = A052928(n-1).
%C 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).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F For n > 6, a(n) = floor(((n-1)^2)/2).
%F 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
%e Examples for n=2 to n=6 have been included as they do not follow the general formula.
%e .
%e A solution illustrating a(2)=2:
%e +-----+
%e | B B |
%e | W W |
%e +-----+
%e .
%e A solution illustrating a(3)=3:
%e +-------+
%e | . . . |
%e | B B W |
%e | W W B |
%e +-------+
%e .
%e A solution illustrating a(4)=6:
%e +---------+
%e | B B . W |
%e | W W . B |
%e | B B . W |
%e | W W . B |
%e +---------+
%e .
%e A solution illustrating a(5)=10:
%e +-----------+
%e | W B W B W |
%e | W B W B W |
%e | . . . . . |
%e | B W B W B |
%e | B W B W B |
%e +-----------+
%e .
%e A solution illustrating a(6)=14:
%e +-------------+
%e | B B W W B B |
%e | W W B B W W |
%e | B . . . . B |
%e | W . . . . W |
%e | B B W W B B |
%e | W W B B W W |
%e +-------------+
%e .
%e 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).
%e .
%e A solution illustrating a(7)=18:
%e +---------------+
%e | B B B B B B B |
%e | B B B B B B B |
%e | B . B . B . B |
%e | . . . . . . . |
%e | W . W . W . W |
%e | W W W W W W W |
%e | W W W W W W W |
%e +---------------+
%e .
%e A solution illustrating a(8)=24:
%e +-----------------+
%e | B B B B B B B B |
%e | B B B B B B B B |
%e | B B B B B B B B |
%e | . . . . . . . . |
%e | . . . . . . . . |
%e | W W W W W W W W |
%e | W W W W W W W W |
%e | W W W W W W W W |
%e +-----------------+
%Y Cf. A007590, A052928, A176222 (peaceable kings), A250000 (peaceable queens), A002620 (peaceable rooks).
%K nonn,easy
%O 1,2
%A _Aaron Khan_, Jul 04 2022
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