A172213 Number of ways to place 4 nonattacking knights on a 4 X n board.

1, 16, 84, 412, 1416, 3640, 7928, 15384, 27352, 45432, 71480, 107608, 156184, 219832, 301432, 404120, 531288, 686584, 873912, 1097432, 1361560, 1670968, 2030584, 2445592, 2921432

Offset: 1

Author

Vaclav Kotesovec, Jan 29 2010

Keywords

• nonn
• easy

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
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A172135, A061990, A172212.

Programs

Mathematica

CoefficientList[Series[-(16 x^9 - 20 x^8 - 40 x^7 + 172 x^6 - 81 x^5 + 41 x^4 + 142 x^3 + 14 x^2 + 11 x + 1) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,16,84,412,1416,3640,7928,15384,27352,45432},30] (* Harvey P. Dale, Apr 16 2022 *)

Formula

Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = 8*(4*n^4-36*n^3+170*n^2-450*n+537)/3, n>=6.

G.f.: -x*(16*x^9-20*x^8-40*x^7+172*x^6-81*x^5+41*x^4+142*x^3+14*x^2+11*x+1)/(x-1)^5. - Vaclav Kotesovec, Mar 25 2010

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Jun 26 2025