A172226 Number of ways to place 3 nonattacking wazirs on an n X n board.

0, 0, 22, 276, 1474, 5248, 14690, 35012, 74326, 144544, 262398, 450580, 739002, 1166176, 1780714, 2642948, 3826670, 5420992, 7532326, 10286484, 13830898, 18336960, 24002482, 31054276, 39750854, 50385248, 63287950

Offset: 1

Author

Vaclav Kotesovec, Jan 29 2010

Keywords

• nonn
• easy

Comments

A wazir is a (fairy chess) leaper [0,1].

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
• Eric Weisstein's World of Mathematics, Grid Graph
• Wikipedia, Wazir (chess)
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A172225, A047659, A061996, A172124, A172134, A172138.

Programs

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]]; // Vincenzo Librandi, Apr 30 2013

Magma

[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

Maple

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

Mathematica

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 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,22,276,1474,5248,14690,35012},30] (* Harvey P. Dale, Apr 08 2022 *)

Formula

a(n) = (n-2)*(n^5 + 2*n^4 - 11*n^3 - 10*n^2 + 42*n - 12)/6, n>=2.

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

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

a(n) = A232833(n,3). - R. J. Mathar, Apr 11 2024