cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A173214 Number of ways to place 4 nonattacking amazons (superqueens) on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 2, 112, 1754, 13074, 63400, 234014, 712248, 1882132, 4457246, 9679760, 19584514, 37367934, 67849336, 118085614, 198107620, 321870956, 508359070, 782972820, 1179105738, 1740089734, 2521359260, 3593085246, 5043058972
Offset: 1

Views

Author

Vaclav Kotesovec, Feb 12 2010

Keywords

Comments

A amazon (superqueen) moves like a queen and a knight.

Links

Crossrefs

Cf. A172200, A172201, A061994.

Programs

  • Mathematica
    CoefficientList[Series[2 x^4 (28 x^17 - 18 x^16 - 162 x^15 - 139 x^14 + 261 x^13 + 1268 x^12 + 2387 x^11 + 1220 x^10 - 5937 x^9 - 18637 x^8 - 30086 x^7 - 31557 x^6 - 23251 x^5 - 11716 x^4 - 3859 x^3 - 708 x^2 - 53 x - 1) / ((x + 1)^4 (x - 1)^9 (x^2 + x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2013 *)

Formula

a(n) = n^8/24-5n^7/6+47n^6/9+43n^5/10-5053n^4/24+112585n^3/108-15433n^2/8+55669n/270+119917/54 + (n^3/4-21n^2/8+7n-3/2)*(-1)^n + 32/27*(n-1)*cos(2*Pi*n/3) + 40*sqrt(3)*sin(2*Pi*n/3)/81, n>=6.
Recurrence: a(n) = 3a(n-1)+a(n-2)-9a(n-3)+12a(n-5)+7a(n-6)-15a(n-7)-16a(n-8)+16a(n-9)+15a(n-10)-7a(n-11)-12a(n-12)+9a(n-14)-a(n-15)-3a(n-16)+a(n-17), n>=23. - Vaclav Kotesovec, Feb 18 2010
G.f.: 2x^5*(28x^17-18x^16-162x^15-139x^14+261x^13+1268x^12+2387x^11+1220x^10-5937x^9-18637x^8-30086x^7-31557x^6-23251x^5-11716x^4-3859x^3-708x^2-53x-1)/((x+1)^4*(x-1)^9*(x^2+x+1)^2). - Vaclav Kotesovec, Mar 24 2010

A174642 Number of ways to place 4 nonattacking amazons (superqueens) on a 4 X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 12, 60, 180, 432, 900, 1692, 2940, 4800, 7452, 11100, 15972, 22320, 30420, 40572, 53100, 68352, 86700, 108540, 134292, 164400, 199332, 239580, 285660, 338112, 397500, 464412, 539460, 623280, 716532, 819900, 934092, 1059840
Offset: 1

Views

Author

Vaclav Kotesovec, Mar 25 2010

Keywords

Comments

An amazon (superqueen) moves like a queen and a knight

Links

Crossrefs

Cf. A173214, A172201, A172200, A051223, A051224, A036464.

Programs

  • Mathematica
    CoefficientList[Series[- 12 x^7 (x^3 + 1) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2013 *)

Formula

G.f.: -12*x^8*(x^3+1)/(x-1)^5.
Explicit formula: a(n) = (n-7)(n^3-21n^2+158n-420), n>=7.

Extensions

More terms from Vincenzo Librandi, May 30 2013

A178973 Number of ways to place 3 nonattacking amazons (superqueens) on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 588, 3328, 9720, 27600, 59048, 124992, 226460, 408464, 666900, 1086464, 1650768, 2505168, 3610000, 5198400, 7191828, 9945232, 13320220, 17835264, 23265000, 30341584, 38718648, 49401408, 61880780, 77504400, 95550308, 117788672, 143225280, 174144464, 209210400, 251325504, 298732228, 355068048, 418062060, 492217600
Offset: 1

Views

Author

Vaclav Kotesovec, Jan 02 2011

Keywords

Comments

An amazon (superqueen) moves like a queen and a knight.

Links

Crossrefs

Cf. A172201, A172518, A178972.

Programs

  • Mathematica
    CoefficientList[Series[- 4 x^6 (36 x^11 - 47 x^10 - 178 x^9 + 228 x^8 + 354 x^7 - 419 x^6 - 356 x^5 + 297 x^4 + 182 x^3 + 178 x^2 + 538 x + 147) / ((x - 1)^7 (x + 1)^5), {x, 0, 50}], x] (* Vincenzo Librandi, May 31 2013 *)

Formula

a(n) = 1/3*n^2*(n^4/2 -6*n^3 +61*n^2/4 +42*n -285/2 +(3*n^2/4 -6*n +21/2)*(-1)^n), n>=7.
G.f.: -4*x^7 * (36*x^11 -47*x^10 -178*x^9 +228*x^8 +354*x^7 -419*x^6 -356*x^5 +297*x^4 +182*x^3 +178*x^2 +538*x +147)/((x-1)^7*(x+1)^5).

A178967 Number of ways to place 5 nonattacking amazons (superqueens) on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 248, 7320, 82758, 562384, 2756122, 10771928, 35504296, 102677536, 267284836, 638673432, 1420555842, 2974232240, 5911536526, 11232560320, 20516606128, 36191817440, 61893239340, 102950022616, 167010533830, 264869097472, 411497661102, 627378473416, 940130628920, 1386570370640, 2015178519904, 2889176379864, 4090150245318, 5722507236712, 7918655437366, 10845295301648, 14710646654420, 19773136732920, 26351274869008, 34835414789584
Offset: 1

Views

Author

Vaclav Kotesovec, Jan 01 2011

Keywords

Comments

An amazon (superqueen) moves like a queen and a knight.

Links

Crossrefs

Cf. A173214, A172200, A172201, A108792.

Programs

  • Mathematica
    Flatten[{{0, 0, 0, 0, 0, 0, 248, 7320, 82758},FullSimplify[Table[1/120*n^10-5/18*n^9+253/72*n^8-689/45*n^7-34217/360*n^6+28391/18*n^5-6828569/810*n^4+29655659/1620*n^3+14328773/1296*n^2-779503661/6480*n+9261910451/64800 +(1/8*n^5-143/48*n^4+79/3*n^3-4711/48*n^2+5171/48*n+2549/32)*(-1)^n +1/2*(29*n-35)*Cos[Pi*n/2] +(2*n+15)*Sin[Pi*n/2] +1/81*(96*n^3-1328*n^2+4744*n-2248)*Cos[4*Pi*n/3] -1/243*(120*n^2-1496*n+5224)*Sqrt[3]*Sin[4*Pi*n/3] +8/25*((5-Sqrt[5])*n+2*Sqrt[5]-8)*Cos[4*Pi*n/5] +8/25*((5+Sqrt[5])*n-2*Sqrt[5]-8)*Cos[8*Pi*n/5] +8/25*Sqrt[50-22*Sqrt[5]]*Sin[4*Pi*n/5] -8/25*Sqrt[50+22*Sqrt[5]]*Sin[8*Pi*n/5], {n, 10, 20}]]}]

Formula

a(n) = 1/120*n^10-5/18*n^9+253/72*n^8-689/45*n^7-34217/360*n^6+28391/18*n^5-6828569/810*n^4+29655659/1620*n^3+14328773/1296*n^2-779503661/6480*n+9261910451/64800 +(1/8*n^5-143/48*n^4+79/3*n^3-4711/48*n^2+5171/48*n+2549/32)*(-1)^n +1/2*(29*n-35)*cos(Pi*n/2) +(2*n+15)*sin(Pi*n/2) +1/81*(96*n^3-1328*n^2+4744*n-2248)*cos(4*Pi*n/3) -1/243*(120*n^2-1496*n+5224)*sqrt(3)*sin(4*Pi*n/3) +8/25*((5-sqrt(5))*n+2*sqrt(5)-8)*cos(4*Pi*n/5) +8/25*((5+sqrt(5))*n-2*sqrt(5)-8)*cos(8*Pi*n/5) +8/25*sqrt(50-22*sqrt(5))*sin(4*Pi*n/5) -8/25*sqrt(50+22*sqrt(5))*sin(8*Pi*n/5), n>=10.
a(n) = n^10/120 - 5*n^9/18 + 253*n^8/72 - 689*n^7/45 - 34307*n^6/360 + 57001*n^5/36 - 55000657*n^4/6480 + 60118543*n^3/3240 + 34387307*n^2/3240 - 155720509*n/1296 + 142960 + (n^5/2 - 143*n^4/12 + 316*n^3/3 - 4711*n^2/12 + 5123*n/12 + 2309/8)*floor[n/2] + (32*n^3/9 - 1328*n^2/27 + 4744*n/27 - 2248/27)*floor[n/3] + (16*n^3/9 - 724*n^2/27 + 1040*n/9 - 3736/27)*floor[(n+1)/3] + (33*n - 5)*floor[n/4] + (25*n - 65)*floor[(n+1)/4] + (32*n/5 - 48/5)*floor[n/5] + (24*n/5 - 64/5)*floor[(n+1)/5] + (16*n/5 - 56/5)*floor[(n+2)/5] + (8*n/5 - 32/5)*floor[(n+3)/5], n>=10.
G.f.: (2*x^7*(-124 - 3784*x - 44667*x^2 - 310723*x^3 - 1509124*x^4 - 5621180*x^5 - 16954312*x^6 - 42976662*x^7 - 93896850*x^8 - 180088868*x^9 - 307206501*x^10 - 470650261*x^11 - 652017897*x^12 - 820670989*x^13 - 941074901*x^14 - 984212615*x^15 - 938015444*x^16 - 812413066*x^17 - 635893628*x^18 - 445615046*x^19 - 275100707*x^20 - 145295581*x^21 - 61597137*x^22 - 17181649*x^23 + 704005*x^24 + 4589289*x^25 + 3324134*x^26 + 1424132*x^27 + 316332*x^28 - 58210*x^29 - 91844*x^30 - 47684*x^31 - 15863*x^32 - 3119*x^33 + 490*x^34 + 982*x^35 + 632*x^36 + 260*x^37 + 126*x^38 + 54*x^39))/((-1+x)^11*(1+x)^6*(1+x^2)^2*(1+x+x^2)^4*(1+x+x^2+x^3+x^4)^2).
Recurrence: a(n) = a(n-37) + a(n-36) - 3a(n-35) - 7a(n-34) - 3a(n-33) + 11a(n-32) + 21a(n-31) + 13a(n-30) - 13a(n-29) - 41a(n-28) - 44a(n-27) - 8a(n-26) + 49a(n-25) + 81a(n-24) + 57a(n-23) - 15a(n-22) - 88a(n-21) - 106a(n-20) - 48a(n-19) + 48a(n-18) + 106a(n-17) + 88a(n-16) + 15a(n-15) - 57a(n-14) - 81a(n-13) - 49a(n-12) + 8a(n-11) + 44a(n-10) + 41a(n-9) + 13a(n-8) - 13a(n-7) - 21a(n-6) - 11a(n-5) + 3a(n-4) + 7a(n-3) + 3a(n-2) - a(n-1), n>=47.
Showing 1-4 of 4 results.

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