A172207 -id:A172207 - cp's OEIS frontend

A172208 Number of ways to place 4 nonattacking bishops on a 4 X n board.

1, 9, 61, 260, 927, 2578, 5965, 12066, 22135, 37678, 60457, 92488, 136043, 193650, 268093, 362412, 479903, 624118, 798865, 1008208, 1256467, 1548218, 1888293, 2281780, 2734023, 3250622, 3837433, 4500568, 5246395, 6081538, 7012877

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. 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 (5, -10, 10, -5, 1).

Cf. A172127, A061990, A172207.

CoefficientList[Series[-1 (2 x^12 - 2 x^11 + 4 x^10 - 24 x^9 + 50 x^8 - 10 x^7 + 41 x^6 - 23 x^5 + 152 * x^4 + 35 x^3 + 26  x^2 + 4 x + 1) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,9,61,260,927,2578,5965,12066,22135,37678,60457,92488,136043},40] (* Harvey P. Dale, Dec 13 2021 *)

a(n) = (32*n^4 -336*n^3 +1702*n^2 -4701*n +5844) / 3, n>=9.

G.f.: -x * (2*x^12 -2*x^11 +4*x^10 -24*x^9 +50*x^8 -10*x^7 +41*x^6 -23*x^5 +152*x^4 +35*x^3 +26*x^2 +4*x+1) / (x-1)^5. - Vaclav Kotesovec, Mar 25 2010

A172210 Number of ways to place 5 nonattacking bishops on a 5 X n board.

Original entry on oeis.org

1, 12, 143, 770, 3368, 12632, 38566, 98968, 222351, 450682, 843169, 1479116, 2460912, 3917228, 6006056, 8917888, 12878847, 18153806, 25049515, 33917724, 45158308, 59222392, 76615476, 97900560, 123701269, 154704978, 191665937, 235408396, 286829730, 346903564

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. 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 (6,-15,20,-15,6,-1).

Cf. A172129, A061991, A172207, A172208.

CoefficientList[Series[(2 x^20 - 4 x^19 + 8 x^18 - 12 x^17 - 48 x^16 + 140 x^15 - 158 x^14 + 208 x^13 + 134 x^12 - 932 x^11 + 1048*x^10 -182*x^9+ 436 * x^8 + 396 x^7 - 32 x^6 + 1288 * x^5 + 668 * x^4 + 72 * x^3 + 86 * x^2 + 6 * x + 1) / (x - 1)^6, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)

a(n) = (625n^5-11250n^4+98875n^3-515250n^2+1566016n-2194944)/24, n>=16.

G.f.: x*(2*x^20 -4*x^19 +8*x^18 -12*x^17 -48*x^16 +140*x^15 -158*x^14 +208*x^13 +134*x^12 -932*x^11 +1048*x^10 -182*x^9 +436*x^8 +396*x^7 -32*x^6 +1288*x^5 +668*x^4 +72*x^3 +86*x^2 +6*x+1)/(x-1)^6. - Vaclav Kotesovec, Mar 25 2010

A172211 Number of ways to place 6 nonattacking bishops on a 6 X n board.

Original entry on oeis.org

1, 16, 313, 2320, 12160, 53744, 209428, 683524, 1905625, 4664384, 10297579, 20907590, 39664250, 71114916, 121559433, 199459466, 315906248, 485124352, 725031335, 1057839684, 1510706686, 2116429956, 2914190277, 3950340692

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Vaclav Kotesovec and Vincenzo Librandi, Table of n, a(n) for n = 1..1000 (first 32 terms from V. Kotesovec)
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Cf. A061992, A172207, A172208, A172210.

CoefficientList[Series[-(2 x^30 - 6 x^29 + 14 x^28 - 26 x^27 + 44 x^26 - 220 x^25 + 596 x^24 - 1060 x^23 + 1654 x^22 - 2266 x^21 + 5622 x^20 - 13570 x^19 + 19848 x^18 - 22392 x^17 + 24048 x^16 - 30525 x^15 + 57673 x^14 - 80154 x^13 + 61962 x^12 - 30874 x^11 + 25832 x^10 - 9360 x^9 + 16960 x^8 - 4710 x^7 + 18006 x^6 + 6928 x^5 + 1968 x^4 + 430 x^3 + 222 x^2 + 9 x + 1) / (x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)

a(n) = (648n^6-17820n^5+240930n^4-2011545n^3+10806047n^2-35094560n+53430940)/10, n>=25.
For any fixed value of k > 1, a(n) = 1/k!*(kn)^k - (2k-1)/2/(k-2)!*(kn)^(k-1) + ...
G.f.: -x*(2*x^30-6*x^29+14*x^28-26*x^27+44*x^26-220*x^25+596*x^24-1060*x^23+1654*x^22
-2266*x^21+5622*x^20-13570*x^19+19848*x^18-22392*x^17+24048*x^16-30525*x^15+57673*x^14
-80154*x^13+61962*x^12-30874*x^11+25832*x^10-9360*x^9+16960*x^8-4710*x^7+18006*x^6+6928*x^5
+1968*x^4+430*x^3+222*x^2+9*x+1)/(x-1)^7. - Vaclav Kotesovec, Mar 25 2010

cp's OEIS Frontend

A172208 Number of ways to place 4 nonattacking bishops on a 4 X n board.

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A172210 Number of ways to place 5 nonattacking bishops on a 5 X n board.

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A172211 Number of ways to place 6 nonattacking bishops on a 6 X n board.

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