A172202 Number of ways to place 3 nonattacking kings on a 3 X n board.

0, 0, 8, 34, 105, 248, 490, 858, 1379, 2080, 2988, 4130, 5533, 7224, 9230, 11578, 14295, 17408, 20944, 24930, 29393, 34360, 39858, 45914, 52555, 59808, 67700, 76258, 85509, 95480, 106198, 117690, 129983, 143104, 157080, 171938, 187705

Offset

1,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Vaclav 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 (4,-6,4,-1).

Formula

a(n) = (n-2)*(9*n^2 - 45*n + 70)/2, n>=2.

G.f.: x^3*(8+2*x+17*x^2)/(1-x)^4. - Vaclav Kotesovec, Mar 24 2010

E.g.f.: 70 + 17*x + (1/2)*(-140 + 106*x - 36*x^2 + 9*x^3)*exp(x). - G. C. Greubel, Apr 29 2022

Mathematica

CoefficientList[Series[x^2*(8+2*x+17*x^2)/(1-x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 0, 8, 34, 105}, 40] (* Harvey P. Dale, Oct 07 2023 *)

Prog

(Magma) [0] cat [(n-2)*(9*n^2-45*n+70)/2: n in [2..50]]; // G. C. Greubel, Apr 29 2022
(SageMath) [(1/8)*(n-2)*(9*(2*n-5)^2+55) +17*bool(n==1) for n in (1..50)] # G. C. Greubel, Apr 29 2022

Crossrefs

Cf. A047659, A061989, A061996.

Sequence in context: A240785 A066804 A033455 * A053298 A196311 A196284

Adjacent sequences: A172199 A172200 A172201 * A172203 A172204 A172205

Keyword

nonn,easy

Author

Vaclav Kotesovec, Jan 29 2010

Extensions

More terms from Vincenzo Librandi, May 27 2013

Status

approved