A321251 a(n) is the number of ways to place non-attacking knights on a 3 X n chessboard.

1, 8, 36, 94, 278, 1062, 3650, 11856, 39444, 135704, 456980, 1534668, 5166204, 17480600, 58888528, 198548648, 669291696, 2258436248, 7613387344, 25676313144, 86575342536, 291991130840, 984557555352, 3320284572360, 11196209499736, 37757232570616

Offset

0,2

Comments

For n = 3, a(3) = 94 is the same as A141243(3). In both cases these are 3 X 3 chessboards.

Links

Dimitrios Noulas, Table of n, a(n) for n = 0..1000

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 297-301.

Formula

G.f.: -(36*x^15 - 72*x^14 - 60*x^13 + 72*x^12 - 120*x^11 + 250*x^10 + 270*x^9 - 256*x^8 - 30*x^7 - 78*x^6 - 98*x^5 + 92*x^4 + 36*x^3 - 8*x^2 - 5*x - 1)/(36*x^16 - 108*x^15 + 48*x^14 + 24*x^13 - 144*x^12 + 376*x^11 - 70*x^10 - 174*x^9 + 108*x^8 - 168*x^7 + 26*x^6 + 78*x^5 - 24*x^4 + 10*x^3 - 4*x^2 - 3*x + 1).

Mathematica

CoefficientList[Series[-(36 x^15 - 72 x^14 - 60 x^13 + 72 x^12 - 120 x^11 + 250 x^10 + 270 x^9 - 256 x^8 - 30 x^7 - 78 x^6 - 98 x^5 + 92 x^4 + 36 x^3 - 8 x^2 - 5 x - 1)/(36 x^16 - 108 x^15 + 48 x^14 + 24 x^13 - 144 x^12 + 376 x^11 - 70 x^10 - 174 x^9 + 108 x^8 - 168 x^7 + 26 x^6 + 78 x^5 - 24 x^4 + 10 x^3 - 4 x^2 - 3 x + 1), {x, 0, 25}], x] (* Michael De Vlieger, Nov 05 2018 *)

Prog

(Sage) G(x)=-(36*x^15 - 72*x^14 - 60*x^13 + 72*x^12 - 120*x^11 + 250*x^10 + 270*x^9 - 256*x^8 - 30*x^7 - 78*x^6 - 98*x^5 + 92*x^4 + 36*x^3 - 8*x^2 - 5*x - 1)/(36*x^16 - 108*x^15 + 48*x^14 + 24*x^13 - 144*x^12 + 376*x^11 - 70*x^10 - 174*x^9 + 108*x^8 - 168*x^7 + 26*x^6 + 78*x^5 - 24*x^4 + 10*x^3 - 4*x^2 - 3*x + 1)
G.series(x, 1001)

Crossrefs

Cf. A141243, A189145.

Sequence in context: A009923 A187287 A224159 * A035006 A245360 A032768

Adjacent sequences: A321248 A321249 A321250 * A321252 A321253 A321254

Keyword

nonn,easy

Author

Dimitrios Noulas, Nov 01 2018

Status

approved