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 1-10 of 23 results. Next

A194644 Number of ways to place 2n nonattacking kings on a 4 X 2n cylindrical chessboard.

Original entry on oeis.org

12, 32, 90, 256, 732, 2102, 6060, 17536, 50922, 148352, 433500, 1270246, 3731532, 10987232, 32418810, 95835136, 283784412, 841611542, 2499330540, 7431221056, 22118855562, 65898914432, 196498594140, 586358987206, 1750864725132, 5231094261152, 15636995277210
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 31 2011

Keywords

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 4, number of rows = 2n).

Links

Crossrefs

Cf. A061593, A137432, A195590.

Programs

  • Mathematica
    Table[2*3^n+2*LucasL[2n], {n,25}]
Drop[CoefficientList[Series[-2*(3 - 12*x + 10*x^2)/((-1 + 3*x)*(1 - 3*x + x^2)), {x, 0, 27}],x], 1] (* or *) LinearRecurrence[{6,-10,3},{12,32,90},27] (* Indranil Ghosh, Mar 05 2017 *)
  • PARI
    print(Vec(-2*(3 - 12*x + 10*x^2)/((-1 + 3*x)*(1 - 3*x + x^2)) + O(x^27))); \\ Indranil Ghosh, Mar 05 2017

Formula

a(n) = 2*3^n + 2*((3+sqrt(5))/2)^n + 2*((3-sqrt(5))/2)^n.
Recurrence: a(n) = 3*a(n-3) - 10*a(n-2) + 6*a(n-1).
G.f.: -2*(3-12*x+10*x^2)/((-1+3*x)*(1-3*x+x^2)).

A194645 Number of ways to place 3n nonattacking kings on a 6 X 2n cylindrical chessboard.

Original entry on oeis.org

32, 100, 344, 1220, 4392, 15988, 58776, 218052, 815816, 3076180, 11682296, 44653028, 171670440, 663421684, 2575592664, 10039703172, 39273896840, 154109956756, 606353229752, 2391296071460, 9449664931176, 37407140524084, 148300497571992, 588693691298244
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 31 2011

Keywords

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 6, number of rows = 2n).

Links

Crossrefs

Cf. A061594, A194644, A137432, A195591.

Programs

  • Mathematica
    Table[FullSimplify[2*4^n+2*3^n+4*(2+Sqrt[2])^n+4*(2-Sqrt[2])^n+2], {n,25}]
LinearRecurrence[{12,-53,104,-86,24},{32,100,344,1220,4392},30] (* Harvey P. Dale, Jul 25 2016 *)

Formula

a(n) = 2*4^n + 2*3^n + 4*(2+sqrt(2))^n + 4*(2-sqrt(2))^n + 2.
Recurrence: a(n) = 24*a(n-5) - 86*a(n-4) + 104*a(n-3) - 53*a(n-2) + 12*a(n-1).
G.f.: -2*(7-68*x+229*x^2-308*x^3+134*x^4)/((-1+x)*(-1+3*x)*(-1+4*x)*(1-4*x+2*x^2)).

A195653 Number of ways to place 9n nonattacking kings on an 18 X 2n cylindrical chessboard.

Original entry on oeis.org

5120, 21508, 109796, 626780, 3877300, 25603228, 178909300, 1314748124, 10105541204, 80812754568, 668845118276, 5700499630916, 49800720887968, 444140848321356, 4029482453905756, 37080781799409148, 345278411878468044, 3246772078088155432, 30781946900321278256
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 22 2011

Keywords

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 18, number of rows = 2n).

Links

Crossrefs

Cf. A137432, A195648, A195658.

Formula

Recurrence order is 400.

A194646 Number of ways to place 4n nonattacking kings on an 8 X 2n cylindrical chessboard.

Original entry on oeis.org

80, 276, 1082, 4460, 18890, 81606, 358564, 1599820, 7238864, 33175486, 153802520, 720390254, 3404944506, 16221905696, 77820675992, 375564803020, 1821845982082, 8876847931644, 43416046650306, 213033152875350, 1048198981050148, 5169676077206180
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 31 2011

Keywords

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 8, number of rows = 2n).

Links

Crossrefs

Cf. A173782, A194644, A194645, A137432, A195592.

Programs

  • Mathematica
    Table[FullSimplify[4+2*5^n+2*4^n + 2*(2+Sqrt[3])^n + 2*(2-Sqrt[3])^n+ 4*((5+Sqrt[5])/2)^n + 4*((5-Sqrt[5])/2)^n + 4*((5+Sqrt[13])/2)^n+4*((5-Sqrt[13])/2)^n+ 2*(2*Cos[Pi/7])^(2n) + 2*(2*Cos[2*Pi/7])^(2n) + 2*(2*Cos[3*Pi/7])^(2n)], {n,10}]

Formula

a(n) = 4 + 2*5^n + 2*4^n + 2*(2+sqrt(3))^n+2*(2-sqrt(3))^n + 4*((5+sqrt(5))/2)^n+4*((5-sqrt(5))/2)^n + 4*((5+sqrt(13))/2)^n+4*((5-sqrt(13))/2)^n + 2*(2*cos(Pi/7))^(2n)+2*(2*cos(2*Pi/7))^(2n)+2*(2*cos(3*Pi/7))^(2n).
Recurrence: a(n) = -300*a(n-12) + 4235*a(n-11) - 23320*a(n-10) + 66422*a(n-9) - 111545*a(n-8) + 118727*a(n-7) - 83449*a(n-6) + 39539*a(n-5) - 12676*a(n-4) + 2708*a(n-3) - 369*a(n-2) + 29*a(n-1).
G.f.: -2*(-17 + 453*x - 5251*x^2 + 34737*x^3 - 144635*x^4 + 394423*x^5 - 711101*x^6 + 836705*x^7 - 620007*x^8 + 270365*x^9 - 61055*x^10 + 5335*x^11)/((-1+x)*(-1+4*x)*(-1+5*x)*(1-4*x+x^2)*(1-5*x+3*x^2)*(1-5*x+5*x^2)*(-1+5*x-6*x^2+x^3)).

A195654 Number of ways to place 10n nonattacking kings on a 20 X 2n cylindrical chessboard.

Original entry on oeis.org

11264, 48132, 251126, 1473764, 9434784, 64923594, 476033636, 3694894500, 30179587994, 257860425672, 2290966142762, 21042965606234, 198765197377402, 1921681775292272, 18940490557328616, 189679152583318596, 1924757095250611458, 19746535064318923758
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 22 2011

Keywords

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 20, number of rows = 2n).

Links

Crossrefs

Cf. A137432, A195653, A195649, A195659.

Formula

Recurrence order is 916.

A194647 Number of ways to place 5n nonattacking kings on a 10 X 2n cylindrical chessboard.

Original entry on oeis.org

192, 708, 3036, 13932, 66532, 327192, 1649420, 8500668, 44693472, 239238888, 1301236304, 7177627944, 40078823652, 226167613792, 1287874058656, 7390391650172, 42688584938548, 247956702607932, 1447080255512308, 8479116559291112, 49852445684576540
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 31 2011

Keywords

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 10, number of rows = 2n).

Links

Crossrefs

Cf. A173783, A194644, A194645, A194646, A137432, A195593.

Formula

G.f.: -2*(7089408*x^21 - 132938496*x^20 + 1125112128*x^19 - 5717239392*x^18 + 19578445344*x^17 - 48082847384*x^16 + 88003026752*x^15 - 123138008952*x^14 + 134072006560*x^13 - 114991853490*x^12 + 78336556962*x^11 - 42596878318*x^10 + 18524447581*x^9 - 6435525481*x^8 + 1778018953*x^7 - 387290192*x^6 + 65568715*x^5 - 8436954*x^4 + 796245*x^3 - 51918*x^2 + 2088*x - 39)/((x-1)*(2*x-1)*(4*x-1)*(6*x-1)*(x^2-4*x+1)*(2*x^2-5*x+1)*(2*x^2-4*x+1)*(4*x^2-6*x+1)*(6*x^2-6*x+1)*(7*x^2-6*x+1)*(2*x^3-8*x^2+6*x-1)*(3*x^3-9*x^2+6*x-1)).
Asymptotic: a(n) ~ 2*6^n.

A195655 Number of ways to place 11n nonattacking kings on a 22 X 2n cylindrical chessboard.

Original entry on oeis.org

24576, 106500, 565512, 3392964, 22327496, 158877948, 1212120160, 9849731140, 84719304384, 766319864440, 7241521734020, 71028444904044, 718816489322444, 7466044767879028, 79230397598482712, 855840660674700612, 9381236750764316676, 104090420921618696952
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 22 2011

Keywords

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 22, number of rows = 2n).

Links

Crossrefs

Cf. A137432, A195653, A195654, A195650, A195660.

Formula

Recurrence order is 1829.

A194648 Number of ways to place 6n nonattacking kings on a 12 X 2n cylindrical chessboard.

Original entry on oeis.org

448, 1732, 7918, 39316, 205628, 1118398, 6286658, 36383284, 216134044, 1314160492, 8155899320, 51526819510, 330559583178, 2148524237842, 14120142260138, 93669254201140, 626289974615094, 4215364545901036, 28531464984810918, 194028126730583796
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 31 2011

Keywords

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 12, number of rows = 2n).

Links

  • Ray Chandler, Table of n, a(n) for n = 1..1182
  • V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
  • Vaclav Kotesovec, G.f., recurrence and explicit formula
  • Index entries for linear recurrences with constant coefficients, signature (123, -7321, 280953, -7815096, 167948132, -2902322885, 41450235787, -499005760654, 5139688443504, -45815780494341, 356686744049422, -2442977988253415, 14807521518627422, -79813193163925620, 384075945698236159, -1655461203049171846, 6408095608117333736, -22324060511037585983, 70110767534529096739, -198758740577568913437, 509105053427808565774, -1178957762144560119277, 2469102261987638350625, -4676788353519183205994, 8009846446628877143184, -12398045659209581154381, 17330242433059284917247, -21854027730161965985121, 24829391878777653306741, -25375425632156076733736, 23283124318667970185580, -19136737348463193623732, 14052187658858602811086, -9190302655918861018594, 5334155219056391326621, -2736099335631729732358, 1234232572662337908295, -486797186990900726158, 166723091105251092029, -49174878927647152193, 12365157710942649267, -2617610437109862140, 459103033290045078, -65327706620774553, 7328352058508736, -621736878459564, 37366066553760, -1412422401600, 25147584000).

Crossrefs

Cf. A174154, A194644, A194645, A194646, A194647, A137432, A195594.

Formula

Asymptotic: a(n) ~ 2*7^n.

A195656 Number of ways to place 12n nonattacking kings on a 24 X 2n cylindrical chessboard.

Original entry on oeis.org

53248, 233476, 1257754, 7682812, 51698178, 378088270, 2980927200, 25173962492, 226268016376, 2149806985106, 21437333168798, 222770819826574, 2396574908171782, 26535227773186536, 300870737118178194, 3479000496509382748, 40885324678195409820
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 22 2011

Keywords

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 24, number of rows = 2n).

Links

Crossrefs

Cf. A137432, A195653, A195654, A195655, A195651, A195661.

Formula

Recurrence order is 4248.

A195004 Number of ways to place 7n nonattacking kings on a 14 X 2n cylindrical chessboard.

Original entry on oeis.org

1024, 4100, 19648, 103508, 580664, 3419648, 20984924, 133538996, 877751236, 5937279840, 41180193352, 291859775552, 2106967145904, 15448890481568, 114765555945488, 861942483797204, 6533144250310688, 49899718750389380, 383593821097441412, 2964842429047018248
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 07 2011

Keywords

Comments

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 14, number of rows = 2n).

Links

  • Ray Chandler, Table of n, a(n) for n = 1..1106
  • V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
  • Vaclav Kotesovec, G.f., recurrence and explicit formula
  • Index entries for linear recurrences with constant coefficients, signature (234, -26876, 2019709, -111697218, 4847894553, -171967643248, 5126864261813, -131103713766739, 2920551204572253, -57369727964797769, 1003499987301575705, -15756273391572396056, 223560731166583964170, -2882674835842994926649, 33942752076287377374732, -366481987107253117428678, 3641474337928260994018833, -33403183417194915064568472, 283651397179433118312259120, -2235256807858321803493302712, 16381469343464643012631705641, -111864817301276143786034674593, 713000590137978010047099019123, -4248167585872182220244662296078, 23692825139854478095829187332583, -123839650151492238580849239743842, 607291338747966869409975798808119, -2796687736808231684915622029956501, 12105097441367664211274616723997585, -49282739018956175502285900368718340, 188846441297378176522518597171831038, -681488797315874299501618146366584502, 2317175918409451996565304916085049225, -7426636629919740985496045649451822465, 22444540561400607122897712881328419731, -63979187901854273527092491251827548655, 172057350932125535425088191059644660790, -436601690854837262203616552560815387262, 1045493932598927036221255467761014916672, -2362673225064751662372547547704555633600, 5038838587531054046651836530047833987628, -10140921814167342121211938118088849001792, 19257391087913181405263225040629151186272, -34500132785018636006710664752520219397318, 58298198399790056471455253509791714097825, -92893574441178572445869535970403724406464, 139532260661304247096543190046679153931782, -197497210344518691909307839617377332496726, 263305424445938364650937820234766261392812, -330492584893065300696048626280270629928420, 390329890315638397879899046383015257779768, -433519474585554675321027469844536795515096, 452483527667626510122157361959823577545116, -443501901490900685964486690632625788788496, 407883969421282557544660851181757217890384, -351675933152655222446863110197897961028664, 283984050472933807032016387053214808490768, -214551731812442749852409765446269884375456, 151480235225815576544104717761138879307344, -99820208079286562792944814404190179497760, 61308554282442277738368818159312699696768, -35043595589343382666629911544555460702048, 18610707623818189757146652551568358579392, -9166308055333364648206086359099895891840, 4178629157017923157800455107231658228096, -1759220243521242971272733964440808344960, 682319955535117483948474380530632894464, -243136873223743567485233920640254062592, 79356174222862273290100936536508525568, -23641977426444551429474710014044615168, 6404251284485474979502046840644695040, -1570379926552348612740435108569370624, 346795140719354500498066916678473728, -68563667046969861210140030991157248, 12051369611781880146558356044689408, -1867616610183259872317123737116672, 252622266308749887603294277730304, -29456297112310112325526951329792, 2914402636658640861712637558784, -239667516215588644643093741568, 15925650465805208711732920320, -820834685159380646929367040, 30762599549292146117836800, -745091435273014738944000, 8746516217730170880000).

Crossrefs

Cf. A174155, A194644, A194645, A194646, A194647, A194648, A137432, A195595.
Showing 1-10 of 23 results. Next

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