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%I #27 Apr 11 2024 13:56:53
%S 0,0,22,276,1474,5248,14690,35012,74326,144544,262398,450580,739002,
%T 1166176,1780714,2642948,3826670,5420992,7532326,10286484,13830898,
%U 18336960,24002482,31054276,39750854,50385248,63287950
%N Number of ways to place 3 nonattacking wazirs on an n X n board.
%C A wazir is a (fairy chess) leaper [0,1].
%H Vincenzo Librandi, <a href="/A172226/b172226.txt">Table of n, a(n) for n = 1..1000</a>
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Wazir_(chess)">Wazir (chess)</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = (n-2)*(n^5 + 2*n^4 - 11*n^3 - 10*n^2 + 42*n - 12)/6, n>=2.
%F G.f.: 2*x^3*(x^5-9*x^4+22*x^3-2*x^2-61*x-11)/(x-1)^7. - _Vaclav Kotesovec_, Mar 25 2010
%F a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - _Vincenzo Librandi_, Apr 30 2013
%F a(n) = A232833(n,3). - _R. J. Mathar_, Apr 11 2024
%p A172226:=n->`if`(n=1, 0, (n-2)*(n^5 + 2*n^4 - 11*n^3 - 10*n^2 + 42*n - 12)/6); seq(A172226(n), n=1..60); # _Wesley Ivan Hurt_, Feb 06 2014
%t CoefficientList[Series[2 x^2 (x^5 - 9 x^4 + 22 x^3 - 2 x^2 - 61 x - 11) / (x-1)^7, {x, 0, 60}], x] (* _Vincenzo Librandi_, Apr 30 2013 *)
%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,22,276,1474,5248,14690,35012},30] (* _Harvey P. Dale_, Apr 08 2022 *)
%o (Magma) I:=[0, 0, 22, 276, 1474, 5248, 14690, 35012]; [n le 8 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..40]]; /* or */ [0] cat [(n-2)*(n^5+2*n^4-11*n^3-10*n^2+42*n-12)/6: n in [2..30]]; // _Vincenzo Librandi_, Apr 30 2013
%Y Cf. A172225, A047659, A061996, A172124, A172134, A172138.
%K nonn,easy
%O 1,3
%A _Vaclav Kotesovec_, Jan 29 2010
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