A196810 Number of ways to place 2 nonattacking nightriders on an n X n cylindrical board.

0, 4, 18, 80, 200, 420, 756, 1472, 2358, 3860, 5500, 8304, 11232, 15484, 21090, 27392, 34816, 44604, 55404, 69840, 84294, 102124, 122452, 147264, 173800, 203476, 237762, 276752, 318304, 368340, 418500, 478208, 541398, 611524, 689780, 774576, 863136, 963148

Offset

1,2

Comments

A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.

Links

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

S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-queens problem I. General theory, arXiv:1303.1879 [math.CO], 2013-2014. See also. [N. J. A. Sloane, Feb 16 2013]

V. Kotesovec, Non-attacking chess pieces

Formula

G.f.: -(2*x^2*(2 + 17*x + 96*x^2 + 384*x^3 + 1203*x^4 + 3100*x^5 + 6917*x^6 + 13670*x^7 + 24466*x^8 + 39974*x^9 + 60206*x^10 + 83709*x^11 + 107667*x^12 + 128088*x^13 + 141070*x^14 + 143882*x^15 + 136037*x^16 + 119239*x^17 + 96892*x^18 + 72808*x^19 + 50428*x^20 + 31926*x^21 + 18321*x^22 + 9388*x^23 + 4223*x^24 + 1622*x^25 + 514*x^26 + 127*x^27 + 22*x^28 + 2*x^29))/((-1+x)^5*(1+x)^3*(1+x^2)^3*(1+x+x^2)^3*(1+x+x^2+x^3+x^4)^3).

Recurrence: a(n) = a(n-32) + 4*a(n-31) + 10*a(n-30) + 17*a(n-29) + 20*a(n-28) + 11*a(n-27) - 15*a(n-26) - 54*a(n-25) - 90*a(n-24) - 99*a(n-23) - 63*a(n-22) + 18*a(n-21) + 116*a(n-20) + 188*a(n-19) + 194*a(n-18) + 123*a(n-17) - 123*a(n-15) - 194*a(n-14) - 188*a(n-13) - 116*a(n-12) - 18*a(n-11) + 63*a(n-10) + 99*a(n-9) + 90*a(n-8) + 54*a(n-7) + 15*a(n-6) - 11*a(n-5) - 20*a(n-4) - 17*a(n-3) - 10*a(n-2) - 4*a(n-1).

Explicit formula: a(n) = -n/4+(572*n^2)/225-(3*n^3)/2+n^4/2+(-1)^n*(n/4+n^2/2)+1/2*n^2*cos((n*Pi)/2)+16/25*n^2*cos((4*n*Pi)/5)+4/9*n^2*cos((4*n*Pi)/3)+16/25*n^2*cos((8*n*Pi)/5).

Chaiken et al. give a 4th degree quasi-polynomial formula. - N. J. A. Sloane, Feb 16 2013

Note that cited formula is for normal chessboard (not cylindrical), see sequence A172141. - Vaclav Kotesovec, Dec 09 2013

Mathematica

Table[(143*n^2)/30-(79*n^3)/15+n^4/2+16/5*n^2*Floor[n/5]+n^2*Floor[n/4]+4/3*n^2*Floor[n/3]+(n+2*n^2)*Floor[n/2]+8/5*n^2*Floor[(1+n)/5]+n^2*Floor[(1+n)/4]+2/3*n^2*Floor[(1+n)/3]+8/5*n^2*Floor[(2+n)/5]+8/5*n^2*Floor[(3+n)/5], {n, 1, 100}]

Crossrefs

Cf. A172141, A196812.

Sequence in context: A240342 A208309 A112619 * A177755 A037965 A045902

Adjacent sequences: A196807 A196808 A196809 * A196811 A196812 A196813

Keyword

nonn,easy

Author

Vaclav Kotesovec, Oct 06 2011

Status

approved