A278685 Number of non-equivalent ways to place 6 non-attacking ferses on an n X n board.

0, 0, 1, 76, 3773, 66201, 651193, 4318451, 21754341, 89267490, 312974387, 968069337, 2704548145, 6942663519, 16594368633, 37311795887, 79570707969, 162013125016, 316669793867, 596873304925, 1089009784181, 1929545889877, 3329316638249, 5607471933963, 9238336533613

Offset

1,4

Comments

A fers is a leaper [1, 1].

Rotations and reflections of placements are not counted. If they are to be counted, see A201247.

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 (6,-8,-22,69,-8,-176,168,182,-364,0,364,-182,-168,176,8,-69,22,8,-6,1).

Formula

a(n) = n^12 - 75*n^10 + 120*n^9 + 2305*n^8 - 6960*n^7 - 32008*n^6 + 152880*n^5 + 138204*n^4 - 1543560*n^3 + 1178528*n^2 + 5238720*n - 7977600 + IF(MOD(n, 2) = 1, 122*n^6 - 600*n^5 - 1645*n^4 + 14520*n^3 - 19447*n^2 - 30480*n + 81855)/5760 for n>=5.

a(n) = 6*a(n-1)-8*a(n-2)-22*a(n-3)+69*a(n-4)-8*a(n-5)-176*a(n-6)+168*a(n-7)+182*a(n-8)-364*a(n-9)+364*a(n-11)-182*a(n-12)-168*a(n-13)+176*a(n-14)+8*a(n-15)-69*a(n-16)+22*a(n-17)+8*a(n-18)-6*a(n-19)+*a(n-20) for n>=25.

G.f.: x^3*(1 +70*x +3325*x^2 +44193*x^3 +285774*x^4 +1018671*x^5 +2250048*x^6 +3090821*x^7 +2658486*x^8 +1198906*x^9 +139256*x^10 -84845*x^11 +22114*x^12 +28024*x^13 -6172*x^14 -5377*x^15 +485*x^16 +592*x^17 +199*x^18 -56*x^19 -44*x^20 +9*x^21) / ((1 -x)^13*(1 +x)^7). - Colin Barker, Dec 10 2016

Example

There is 1 way to place 6 non-attacking ferses on a 3 X 3 board, rotations and reflections being ignored:
   XXX
   ...
   XXX

Prog

(PARI) concat(vector(2), Vec(x^3*(1 +70*x +3325*x^2 +44193*x^3 +285774*x^4 +1018671*x^5 +2250048*x^6 +3090821*x^7 +2658486*x^8 +1198906*x^9 +139256*x^10 -84845*x^11 +22114*x^12 +28024*x^13 -6172*x^14 -5377*x^15 +485*x^16 +592*x^17 +199*x^18 -56*x^19 -44*x^20 +9*x^21) / ((1 -x)^13*(1 +x)^7) + O(x^30))) \\ Colin Barker, Dec 10 2016

Crossrefs

Cf. A201247, A232567 (2 ferses), A278682 (3 ferses), A278683 (4 ferses), A278684 (5 ferses), A278686 (7 ferses), A278687, A278688.

Sequence in context: A035733 A035804 A017739 * A198530 A234176 A116264

Adjacent sequences: A278682 A278683 A278684 * A278686 A278687 A278688

Keyword

nonn,easy

Author

Heinrich Ludwig, Nov 26 2016

Status

approved