A278684 Number of non-equivalent ways to place 5 non-attacking ferses on an n X n board.

0, 0, 2, 99, 2102, 19987, 124676, 571418, 2122841, 6704061, 18711691, 47235845, 109938296, 238950999, 490309398, 957267228, 1790325363, 3224010105, 5615368229, 9493358359, 15627413290, 25112609019, 39484650296, 60859027054, 92114682749, 137111560949, 200972392655

Offset

1,3

Comments

A fers is a leaper [1, 1].

Rotations and reflections of placements are not counted. If they are to be counted, see A201246.

Links

Heinrich Ludwig, Table of n, a(n) for n = 1..1000

Wikipedia, Fairy chess piece

Index entries for linear recurrences with constant coefficients, signature (5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1).

Formula

a(n) = (n^10 - 50*n^8 + 80*n^7 + 955*n^6 - 2828*n^5 - 7090*n^4 + 36860*n^3 - 10856*n^2 - 133712*n + 161280 + ((1-(-1)^n)/2)*(52*n^5 - 145*n^4 - 580*n^3 + 2320*n^2 - 1152*n - 15))/960 for n >= 4.

a(n) = 5*a(n-1) - 4*a(n-2) - 20*a(n-3) + 40*a(n-4) + 16*a(n-5) - 100*a(n-6) + 44*a(n-7) + 110*a(n-8) - 110*a(n-9) - 44*a(n-10) + 100*a(n-11) - 16*a(n-12) - 40*a(n-13) + 20*a(n-14) + 4*a(n-15) - 5*a(n-16) + a(n-17) for n >= 21.

G.f.: x^3*(2 +89*x +1615*x^2 +9913*x^3 +35049*x^4 +66034*x^5 +78731*x^6 +45748*x^7 +9902*x^8 -5540*x^9 -1343*x^10 +1685*x^11 +409*x^12 -334*x^13 -83*x^14 +38*x^15 +6*x^16 -x^17) / ((1 -x)^11*(1 +x)^6). - Colin Barker, Dec 10 2016

Example

There are 2 ways to place 5 non-attacking ferses on a 3 X 3 board, rotations and reflections being ignored:
   XXX   XXX
   ...   ...
   X.X   XX.

Maple

A278684:=n->(n^10 - 50*n^8 + 80*n^7 + 955*n^6 - 2828*n^5 - 7090*n^4 + 36860*n^3 - 10856*n^2 - 133712*n + 161280 + ((1-(-1)^n)/2)*(52*n^5 - 145*n^4 - 580*n^3 + 2320*n^2 - 1152*n - 15))/960: 0, 0, 2, seq(A278684(n), n=4..30); # Wesley Ivan Hurt, Nov 27 2016

Mathematica

Join[{0, 0, 2}, Table[(n^10 - 50*n^8 + 80*n^7 + 955*n^6 - 2828*n^5 - 7090*n^4 + 36860*n^3 - 10856*n^2 - 133712*n + 161280 + ((1-(-1)^n)/2)*(52*n^5 - 145*n^4 - 580*n^3 + 2320*n^2 - 1152*n - 15))/960, {n, 4, 30}]] (* Wesley Ivan Hurt, Nov 27 2016 *)

Prog

(Magma) [0, 0, 2] cat [(n^10 - 50*n^8 + 80*n^7 + 955*n^6 - 2828*n^5 - 7090*n^4 + 36860*n^3 - 10856*n^2 - 133712*n + 161280 + ((1-(-1)^n)/2)*(52*n^5 - 145*n^4 - 580*n^3 + 2320*n^2 - 1152*n - 15))/960 : n in [4..30]]; // Wesley Ivan Hurt, Nov 27 2016
(PARI) concat(vector(2), Vec(x^3*(2 +89*x +1615*x^2 +9913*x^3 +35049*x^4 +66034*x^5 +78731*x^6 +45748*x^7 +9902*x^8 -5540*x^9 -1343*x^10 +1685*x^11 +409*x^12 -334*x^13 -83*x^14 +38*x^15 +6*x^16 -x^17) / ((1 -x)^11*(1 +x)^6) + O(x^40))) \\ Colin Barker, Dec 10 2016

Crossrefs

Cf. A201246, A232567 (2 ferses), A278682 (3 ferses), A278683 (4 ferses), A278685 (6 ferses), A278686 (7 ferses), A278687, A278688.

Sequence in context: A258399 A212838 A024241 * A024242 A276386 A333023

Adjacent sequences: A278681 A278682 A278683 * A278685 A278686 A278687

Keyword

nonn,easy

Author

Heinrich Ludwig, Nov 26 2016

Status

approved