A194644 -id:A194644 - cp's OEIS frontend

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

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

Vaclav Kotesovec, Aug 31 2011

nonn

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).

Ray Chandler, Table of n, a(n) for n = 1..1660
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
Index entries for linear recurrences with constant coefficients, signature (12,-53,104,-86,24).

Cf. A061594, A194644, A137432, A195591.

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 *)

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)).

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

Vaclav Kotesovec, Aug 31 2011

nonn

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).

Ray Chandler, Table of n, a(n) for n = 1..1430
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
Index entries for linear recurrences with constant coefficients, signature (29, -369, 2708, -12676, 39539, -83449, 118727, -111545, 66422, -23320, 4235, -300).

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

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}]

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)).

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

Vaclav Kotesovec, Aug 31 2011

nonn

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).

Ray Chandler, Table of n, a(n) for n = 1..1284
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
Vaclav Kotesovec, Explicit formula and recurrence
Index entries for linear recurrences with constant coefficients, signature (56, -1460, 23541, -263028, 2162701, -13565686, 66416673, -257594833, 798883747, -1991743054, 4000492482, -6469161690, 8395031504, -8690399936, 7111450512, -4541105512, 2222092032, -811893408, 213097152, -37748736, 4020480, -193536).

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

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.

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

Vaclav Kotesovec, Aug 31 2011

nonn

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).

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).

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

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

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

Vaclav Kotesovec, Sep 07 2011

nonn

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).

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).

Cf. A174155, A194644, A194645, A194646, A194647, A194648, A137432, A195595.

A195590 Number of ways to place 2n nonattacking kings on a vertical cylinder 4 X 2n.

Original entry on oeis.org

8, 32, 100, 276, 708, 1732, 4100, 9476, 21508, 48132, 106500, 233476, 507908, 1097732, 2359300, 5046276, 10747908, 22806532, 48234500, 101711876, 213909508, 448790532, 939524100, 1962934276, 4093640708, 8522825732, 17716740100, 36775657476, 76235669508

Offset: 1

Vaclav Kotesovec, Sep 21 2011

nonn

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

Ray Chandler, Table of n, a(n) for n = 1..3307
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (5, -8, 4).

Cf. A194644, A061593, A137432.

Recurrence: a(n) = 4*a(n-3) - 8*a(n-2) + 5*a(n-1).
G.f.: -(1+3*x)/((x-1)*(2*x-1)^2).
Explicit formula: a(n) = (5*n-3)*2^n + 4.

A195642 Number of lower triangles of an n X n 0..6 array with all row sums equal to the length of the row and all column sums equal to the length of the column.

Original entry on oeis.org

1, 1, 3, 19, 391, 25532, 5539434, 4116036800, 10694128575819, 98630958757213465, 3267417937114919996252

Offset: 1

R. H. Hardin Sep 21 2011

nonn

Column 6 of A194644

			Some solutions for n=6
..1............1............1............1............1............1
..1.1..........1.1..........0.2..........2.0..........0.2..........2.0
..2.1.0........2.1.0........2.1.0........2.1.0........2.1.0........2.1.0
..2.0.0.2......1.2.1.0......0.0.3.1......0.2.1.1......0.1.3.0......0.3.1.0
..0.0.2.1.2....1.0.2.0.2....1.0.1.1.2....0.1.2.2.0....1.0.1.1.2....1.0.2.1.1
..0.3.2.0.0.1..0.1.1.3.0.1..2.2.0.1.0.1..1.1.1.0.2.1..2.1.0.2.0.1..0.1.1.2.1.1

A195643 Number of lower triangles of an n X n 0..7 array with all row sums equal to the length of the row and all column sums equal to the length of the column.

Original entry on oeis.org

1, 1, 3, 19, 391, 25532, 5539434, 4116290300, 10699674783969, 98827025484631957, 3284291571314714878300

Offset: 1

R. H. Hardin Sep 21 2011

nonn

Column 7 of A194644

			Some solutions for n=7
..1..............1..............1..............1..............1
..1.1............0.2............2.0............2.0............2.0
..0.3.0..........0.3.0..........3.0.0..........2.1.0..........2.1.0
..4.0.0.0........2.1.1.0........0.2.2.0........1.3.0.0........2.2.0.0
..0.2.2.0.1......2.0.1.1.1......1.1.2.0.1......1.2.2.0.0......0.1.0.3.1
..1.0.1.0.2.2....2.0.2.0.2.0....0.3.0.1.2.0....0.0.2.4.0.0....0.1.4.0.1.0
..0.0.2.4.0.0.1..0.0.1.3.0.2.1..0.0.1.3.0.2.1..0.0.1.0.3.2.1..0.1.1.1.1.2.1

cp's OEIS Frontend

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

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

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

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

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

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A195590 Number of ways to place 2n nonattacking kings on a vertical cylinder 4 X 2n.

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A195642 Number of lower triangles of an n X n 0..6 array with all row sums equal to the length of the row and all column sums equal to the length of the column.

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A195643 Number of lower triangles of an n X n 0..7 array with all row sums equal to the length of the row and all column sums equal to the length of the column.

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