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 1-3 of 3 results.

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).

A178974 Number of ways to place 4 nonattacking amazons (superqueens) on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 98, 3328, 17496, 99600, 316052, 1041408, 2501538, 6157536, 12531150, 25938944, 47168268, 86938272, 145818008, 247240000, 390084786, 620964256, 933865918, 1414946304, 2047225000, 2980849040, 4177648224, 5886858432, 8032809818, 11012886000, 14689386642, 19674427392, 25732782504, 33779841296, 43433208000, 56027023488, 70963952198, 90145026976, 112667956362, 141187744000
Offset: 1

Views

Author

Vaclav Kotesovec, Jan 02 2011

Keywords

Comments

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

Links

Crossrefs

Cf. A173214, A172519, A178972, A178973.

Programs

  • Mathematica
    CoefficientList[Series[2 x^6 (162 x^30 - 350 x^29 - 1488 x^28 - 718 x^27 + 2389 x^26 + 6635 x^25 + 6157 x^24 - 3372 x^23 - 15873 x^22 - 22215 x^21 - 8561 x^20 + 23622 x^19 + 55919 x^18 + 38469 x^17 - 91949 x^16 - 461696 x^15 - 1076702 x^14 - 1978832 x^13 - 2858196 x^12 - 3576618 x^11 - 3727323 x^10 - 3419559 x^9 - 2634463 x^8 - 1782420 x^7 - 988307 x^6 - 472291 x^5 - 171451 x^4 - 53262 x^3 - 10265 x^2 - 1713 x - 49) / ((x - 1)^9 (x + 1)^7 (x^2 + 1)^3 (x^2 + x + 1)^3), {x, 0, 40}], x] (* _Vincenzo Librandi Jun 01 2013 *)

Formula

a(n)= (1/4)*n^2*(n^6/6 -4*n^5 +197*n^4/6 -66*n^3 -1941*n^2/4 +2638*n -18907/6 +(n^4/2 -10*n^3 +289*n^2/4 -210*n +357/2)*(-1)^n +18*cos(Pi*n/2) +32/3*cos(4*Pi*n/3)), n>=10.
G.f.: 2*x^7*(162*x^30 -350*x^29 -1488*x^28 -718*x^27 +2389*x^26 +6635*x^25 +6157*x^24 -3372*x^23 -15873*x^22 -22215*x^21 -8561*x^20 +23622*x^19 +55919*x^18 +38469*x^17 -91949*x^16 -461696*x^15 -1076702*x^14 -1978832*x^13 -2858196*x^12 -3576618*x^11 -3727323*x^10 -3419559*x^9 -2634463*x^8 -1782420*x^7 -988307*x^6 -472291*x^5 -171451*x^4 -53262*x^3 -10265*x^2 -1713*x -49)/((x-1)^9*(x+1)^7*(x^2+1)^3*(x^2+x+1)^3).

A178975 Number of ways to place 5 nonattacking amazons (superqueens) on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 640, 7290, 123640, 640574, 3869280, 12950132, 47022360, 123467040, 340840960, 759697190, 1758672648, 3494388306, 7150739360, 13041285516, 24354594440, 41566378136, 72345297024, 117101090250, 192694385416, 298703838186, 469581881888, 702148696580, 1062719841960, 1541332566284, 2259300468736, 3192255589842, 4552716843720, 6288527141890, 8758324830240, 11859789616944, 16178716174856, 21527161542900, 28834708173440
Offset: 1

Views

Author

Vaclav Kotesovec, Jan 02 2011

Keywords

Comments

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

Links

Crossrefs

Cf. A178967, A173775, A178972, A178973, A178974.

Programs

  • Mathematica
    CoefficientList[Series[- 2 x^7 * (1176 x^64 + 5556 x^63 + 15132 x^62 + 28428 x^61 + 39340 x^60 + 30066 x^59 - 16046 x^58 - 97562 x^57 - 191158 x^56 - 227584 x^55 - 150082 x^54 + 56017 x^53 + 289119 x^52 + 339896 x^51 + 45336 x^50 - 611255 x^49 - 1380704 x^48 - 2278261 x^47 - 3764650 x^46 - 7542849 x^45 - 7704482 x^44 + 18495516 x^43 + 165924351 x^42 + 637466559 x^41 + 1903273538 x^40 + 4724140916 x^39 + 10422040024 x^38 + 20690172375 x^37 + 37875420877 x^36 + 64238796480 x^35 + 102190978070 x^34 + 152823563437 x^33 + 216401077492 x^32 + 290462738417 x^31 + 371272897408 x^30 + 452086367452 x^29 + 526060962825 x^28 + 584865148004 x^27 + 622627590675 x^26 + 634259897550 x^25 + 619201117902 x^24 + 578669435625 x^23 + 518210895306 x^22 + 443951015905 x^21 + 364069798686 x^20 + 285127462600 x^19 + 213313173667 x^18 + 151952471981 x^17 + 103062047860 x^16 + 66251579160 x^15 + 40354587182 x^14 + 23135311545 x^13 + 12479773177 x^12 + 6269223018 x^11 + 2933204824 x^10 + 1256492269 x^9 + 493760966 x^8 + 172473531 x^7 + 54013568 x^6 + 14176791 x^5 + 3222186 x^4 + 525572 x^3 + 74355 x^2 + 4605 x + 320) / ((x - 1)^11 (x + 1)^9 (x^2 + 1)^5 (x^2 - x + 1)^3 (x^2 + x + 1)^5 (x^4 + x^3 + x^2 + x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 01 2013 *)

Formula

a(n) = n^2/5*(1/24*n^8 -5/3*n^7 +635/24*n^6 -1145/6*n^5 +4139/16*n^4 +10985/2*n^3 -916385/24*n^2 +273775/3*n -128930/3 +(5/24*n^6 -15/2*n^5 +5315/48*n^4 -1675/2*n^3 +25855/8*n^2 -4935*n -2290/3)*(-1)^n +(45/2*n^2 -390*n +1630)*cos(n*Pi/2) +400/3*cos(n*Pi/3) +(40/3*n^2 -640/3*n+2480/3)*cos(2*n*Pi/3) +16*cos(4*n*Pi/5) +16*cos(8*n*Pi/5)), n>=15.
G.f.: -2*x^8*(1176*x^64 + 5556*x^63 + 15132*x^62 + 28428*x^61 + 39340*x^60 + 30066*x^59 - 16046*x^58 - 97562*x^57 - 191158*x^56 - 227584*x^55 - 150082*x^54 + 56017*x^53 + 289119*x^52 + 339896*x^51 + 45336*x^50 - 611255*x^49 - 1380704*x^48 - 2278261*x^47 - 3764650*x^46 - 7542849*x^45 - 7704482*x^44 + 18495516*x^43 + 165924351*x^42 + 637466559*x^41 + 1903273538*x^40 + 4724140916*x^39 + 10422040024*x^38 + 20690172375*x^37 + 37875420877*x^36 + 64238796480*x^35 + 102190978070*x^34 + 152823563437*x^33 + 216401077492*x^32 + 290462738417*x^31 + 371272897408*x^30 + 452086367452*x^29 + 526060962825*x^28 + 584865148004*x^27 + 622627590675*x^26 + 634259897550*x^25 + 619201117902*x^24 + 578669435625*x^23 + 518210895306*x^22 + 443951015905*x^21 + 364069798686*x^20 + 285127462600*x^19 + 213313173667*x^18 + 151952471981*x^17 + 103062047860*x^16 + 66251579160*x^15 + 40354587182*x^14 + 23135311545*x^13 + 12479773177*x^12 + 6269223018*x^11 + 2933204824*x^10 + 1256492269*x^9 + 493760966*x^8 + 172473531*x^7 + 54013568*x^6 + 14176791*x^5 + 3222186*x^4 + 525572*x^3 + 74355*x^2 + 4605*x + 320)/((x-1)^11*(x+1)^9*(x^2+1)^5*(x^2-x+1)^3*(x^2+x+1)^5*(x^4+x^3+x^2+x+1)^3).
Showing 1-3 of 3 results.

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