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Number of ways of placing k non-attacking wazirs on a 3 X n board, where k is chosen so as to maximize this number.
3

%I #13 Apr 14 2024 17:36:07

%S 1,3,8,24,84,276,880,3063,10692,36257,121580,436847,1530534,5259906,

%T 18389910,65748491,230935493,799429185,2860613606,10203350814,

%U 35899202776,125660232367,453360413253,1614905346286,5688690345179,20241845359246,72805688610204

%N Number of ways of placing k non-attacking wazirs on a 3 X n board, where k is chosen so as to maximize this number.

%H Alois P. Heinz, <a href="/A371979/b371979.txt">Table of n, a(n) for n = 0..1788</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Wazir_(chess)">Wazir (chess)</a>

%e a(3) = 24 = A371967(3,2):

%e +-------+ +-------+ +-------+ +-------+ +-------+ +-------+

%e | W . W | | W . . | | W . . | | W . . | | W . . | | W . . |

%e | . . . | | . W . | | . . W | | . . . | | . . . | | . . . |

%e | . . . | | . . . | | . . . | | W . . | | . W . | | . . W |

%e +-------+ +-------+ +-------+ +-------+ +-------+ +-------+

%e +-------+ +-------+ +-------+ +-------+ +-------+ +-------+

%e | . W . | | . W . | | . W . | | . W . | | . W . | | . . W |

%e | W . . | | . . W | | . . . | | . . . | | . . . | | W . . |

%e | . . . | | . . . | | W . . | | . W . | | . . W | | . . . |

%e +-------+ +-------+ +-------+ +-------+ +-------+ +-------+

%e +-------+ +-------+ +-------+ +-------+ +-------+ +-------+

%e | . . W | | . . W | | . . W | | . . W | | . . . | | . . . |

%e | . W . | | . . . | | . . . | | . . . | | W . W | | W . . |

%e | . . . | | W . . | | . W . | | . . W | | . . . | | . W . |

%e +-------+ +-------+ +-------+ +-------+ +-------+ +-------+

%e +-------+ +-------+ +-------+ +-------+ +-------+ +-------+

%e | . . . | | . . . | | . . . | | . . . | | . . . | | . . . |

%e | W . . | | . W . | | . W . | | . . W | | . . W | | . . . |

%e | . . W | | W . . | | . . W | | W . . | | . W . | | W . W |

%e +-------+ +-------+ +-------+ +-------+ +-------+ +-------+ .

%p b:= proc(n, l) option remember; `if`(n=0, 1,

%p add(`if`(Bits[And](j, l)>0, 0, expand(b(n-1, j)*

%p x^add(i, i=Bits[Split](j)))), j=[0, 1, 2, 4, 5]))

%p end:

%p a:= n-> max(coeffs(b(n, 0))):

%p seq(a(n), n=0..30);

%Y Row maxima of A371967.

%Y Cf. A371978.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Apr 14 2024