01234_01243_01324_01342_01423_01432_02134_02143_02314_02341_02413_02431_03124_03142_03214_03241_03412_03421_04123_04132_04213_04231_04312_04321_12034_12043_12304_12403_13024_13042_13204_13402_14023_14032_14203_14302_23014_23104_24013_24103
Counting sequence:
1, 1, 2, 6, 24, 80, 240, 692, 2040, 6156, 18712, 56620, 170504, 512748, 1543336, 4649852, 14013080, 42223036, 127198040, 383167532, 1154284776, 3477397004, 10476109640, 31560402844, 95078347000, 286430877340, 862896499704, 2599550549068, 7831374595784, 23592697205356, 71075030120296, 214119626558012, 645053787599448, 1943280127944572, 5854298952494488, 17636580177062764, 53131717206572968, 160063875001436108, 482206213063682696, 1452687762980411868, 4376347049814201016, 13184122547360558428, 39718305078758797880, 119654816047293593484, 360470442444040636296, 1085948265030481955372, 3271512710871119769768, 9855713906273767673468, 29691187283958904410648, 89447259803890063534140, 269467576690107770493400, 811794291370560666710188, 2445600244738400295329384, 7367581443522281581283724, 22195473869318137485593032, 66865777332755544124364828, 201438915187794011341746296, 606852087430443812666935324, 1828194198104050548059645048, 5507592534011467212652594892, 16592097027851653205337228360, 49985121826201666226516735468, 150584486083116222663525430760, 453648738273756099224273888956, 1366655909186951946304305933784, 4117168673769155967372919586044, 12403325353746538693405882569240, 37366086264824573159670380603884, 112568554232802545767275144050984, 339122468225744821197201295998796, 1021635653396435475188171930285320, 3077765427196560646226670130604636, 9272033521299460693699034419939896, 27932799836018947738896606545639132, 84149966119810855738898142741422776, 253509023067361357382283325330442764, 763717773636013981384158279408434440, 2300765592917642215167360322651887020, 6931254576348971393332796567584605992, 20881001589229812482658679636629017340, 62905816337666375220566757336683251096, 189509191510682293255216482010127377852, 570912767020690315212292769786460984920, 1719923898935784799208348087074977683244, 5181418929493420301832485846885289725928, 15609470942013526224883016766462319864076, 47024876082233035986144709984698799303752, 141666490732716880488096730758257840574620, 426782508930535752327686583594061009854456, 1285719078569496263765476785133461237414300, 3873339498237621356898722184881367932939512, 11668768954801453180300370820473202032574796, 35153171825627832870386938551833729266577864, 105901958826053468747534996926078627629787500, 319038775187246454352808510299919642383508584, 961131797761776306439961543876400643127202620, 2895492349250066529432807623583905635961297240, 8722920169834684324924525900429078082400821884, 26278548554623506826395651372589931074794088088, 79166391608831225290336227082024648302112372844, 238495575481855793670067583513352074537052431016
Generating function in Maple syntax:
(x+1)*(x-1)^3/(4*x^6+6*x^5+10*x^4-x^3-x^2+3*x-1)
Generating function in latex syntax:
\frac{\left(x +1\right) \left(x -1\right)^{3}}{4 x^{6}+6 x^{5}+10 x^{4}-x^{3}-x^{2}+3 x -1}
Generating function in sympy syntax:
(x - 1)**3*(x + 1)/(4*x**6 + 6*x**5 + 10*x**4 - x**3 - x**2 + 3*x - 1)
Implicit equation for the generating function in Maple syntax:
(4*x^6+6*x^5+10*x^4-x^3-x^2+3*x-1)*F(x)-(x+1)*(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(4 x^{6}+6 x^{5}+10 x^{4}-x^{3}-x^{2}+3 x -1\right) F \! \left(x \right)-\left(x +1\right) \left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
160980/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 1)^(-n+4)+160980/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 2)^(-n+4)+160980/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 3)^(-n+4)+160980/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 4)^(-n+4)+160980/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 5)^(-n+4)+160980/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 6)^(-n+4)+348600/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 1)^(-n+3)+348600/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 2)^(-n+3)+348600/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 3)^(-n+3)+348600/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 4)^(-n+3)+348600/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 5)^(-n+3)+348600/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 6)^(-n+3)+1031875/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 1)^(-n+2)+1031875/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 2)^(-n+2)+1031875/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 3)^(-n+2)+1031875/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 4)^(-n+2)+1031875/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 5)^(-n+2)+1031875/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 6)^(-n+2)+1783122/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 1)^(-n+1)+1783122/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 2)^(-n+1)+1783122/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 3)^(-n+1)+1783122/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 4)^(-n+1)+1783122/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 5)^(-n+1)+1783122/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 6)^(-n+1)+1828222/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 1)^(-n-1)+1828222/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 2)^(-n-1)+1828222/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 3)^(-n-1)+1828222/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 4)^(-n-1)+1828222/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 5)^(-n-1)+1828222/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 6)^(-n-1)+6758201/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 1)^(-n)+6758201/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 2)^(-n)+6758201/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 3)^(-n)+6758201/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 4)^(-n)+6758201/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 5)^(-n)+6758201/42652879*RootOf(4*_Z^6+6*_Z^5+10*_Z^4-_Z^3-_Z^2+3*_Z-1,index = 6)^(-n)
Explicit closed form in latex syntax:
\frac{160980 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{42652879}+\frac{160980 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{42652879}+\frac{160980 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{42652879}+\frac{160980 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{42652879}+\frac{160980 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{42652879}+\frac{160980 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{42652879}+\frac{348600 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{42652879}+\frac{348600 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{42652879}+\frac{348600 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{42652879}+\frac{348600 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{42652879}+\frac{348600 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{42652879}+\frac{348600 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{42652879}+\frac{1031875 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{42652879}+\frac{1031875 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{42652879}+\frac{1031875 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{42652879}+\frac{1031875 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{42652879}+\frac{1031875 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{42652879}+\frac{1031875 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{42652879}+\frac{1783122 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{42652879}+\frac{1783122 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{42652879}+\frac{1783122 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{42652879}+\frac{1783122 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{42652879}+\frac{1783122 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{42652879}+\frac{1783122 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{42652879}+\frac{1828222 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{42652879}+\frac{1828222 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{42652879}+\frac{1828222 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{42652879}+\frac{1828222 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{42652879}+\frac{1828222 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{42652879}+\frac{1828222 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{42652879}+\frac{6758201 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{42652879}+\frac{6758201 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{42652879}+\frac{6758201 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{42652879}+\frac{6758201 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{42652879}+\frac{6758201 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{42652879}+\frac{6758201 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}+6 \textit{\_Z}^{5}+10 \textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{42652879}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 80
a(n+6) = 4*a(n)+6*a(n+1)+10*a(n+2)-a(n+3)-a(n+4)+3*a(n+5), n >= 6
Recurrence in latex format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 80
a{\left(n + 6 \right)} = 4 a{\left(n \right)} + 6 a{\left(n + 1 \right)} + 10 a{\left(n + 2 \right)} - a{\left(n + 3 \right)} - a{\left(n + 4 \right)} + 3 a{\left(n + 5 \right)}, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/25003/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[23,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[16,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[15,x]
F[14,x] = F[1,x]+F[4,x]
F[15,x] = F[4,x]
F[16,x] = F[11,x]+F[17,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[4,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[15,x]
F[23,x] = F[2,x]+F[24,x]
F[24,x] = F[18,x]+F[25,x]+F[89,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[36,x]
F[27,x] = F[28,x]+F[7,x]
F[28,x] = F[18,x]+F[29,x]+F[33,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[11,x]+F[17,x]
F[32,x] = F[17,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[16,x]+F[35,x]
F[35,x] = F[17,x]
F[36,x] = F[24,x]+F[37,x]
F[37,x] = F[18,x]+F[38,x]+F[71,x]+F[86,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[48,x]
F[40,x] = F[28,x]+F[41,x]
F[41,x] = F[18,x]+F[42,x]+F[44,x]+F[46,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[17,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[32,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[35,x]
F[48,x] = F[37,x]+F[49,x]
F[49,x] = F[18,x]+F[50,x]+F[54,x]+F[65,x]+F[68,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[41,x]
F[53,x] = F[49,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[18,x]+F[57,x]+F[58,x]+F[62,x]
F[57,x] = 0
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[2,x]*F[4,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[60,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[56,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]
F[70,x] = F[56,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[67,x]+F[73,x]
F[73,x] = F[56,x]+F[74,x]
F[74,x] = F[18,x]+F[75,x]+F[83,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[77,x]+F[80,x]
F[77,x] = F[4,x]+F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[11,x]*F[4,x]
F[80,x] = F[60,x]+F[81,x]
F[81,x] = F[82,x]
F[82,x] = F[4,x]*F[74,x]
F[83,x] = F[4,x]*F[84,x]
F[84,x] = F[64,x]+F[85,x]
F[85,x] = F[2,x]+F[60,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[70,x]+F[88,x]
F[88,x] = F[56,x]+F[74,x]
F[89,x] = F[4,x]*F[90,x]
F[90,x] = F[88,x]+F[91,x]
F[91,x] = F[2,x]+F[74,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{23}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{16}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)
F_{23}\! \left(x \right) = F_{2}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \left(x \right)+F_{89}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{36}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{11}\! \left(x \right)+F_{17}\! \left(x \right)
F_{32}\! \left(x \right) = F_{17}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{16}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{17}\! \left(x \right)
F_{36}\! \left(x \right) = F_{24}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)+F_{71}\! \left(x \right)+F_{86}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{48}\! \left(x \right)
F_{40}\! \left(x \right) = F_{28}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{17}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{32}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{35}\! \left(x \right)
F_{48}\! \left(x \right) = F_{37}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)+F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{41}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{18}\! \left(x \right)+F_{57}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)
F_{57}\! \left(x \right) = 0
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{60}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{56}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{56}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{67}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{56}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{18}\! \left(x \right)+F_{75}\! \left(x \right)+F_{83}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{80}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{80}\! \left(x \right) = F_{60}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{64}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{2}\! \left(x \right)+F_{60}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{70}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{56}\! \left(x \right)+F_{74}\! \left(x \right)
F_{89}\! \left(x \right) = F_{4}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{88}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{2}\! \left(x \right)+F_{74}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_23(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_16(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_15(x))
Eq(F_14(x), F_1(x) + F_4(x))
Eq(F_15(x), F_4(x))
Eq(F_16(x), F_11(x) + F_17(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_4(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_15(x))
Eq(F_23(x), F_2(x) + F_24(x))
Eq(F_24(x), F_18(x) + F_25(x) + F_89(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_36(x))
Eq(F_27(x), F_28(x) + F_7(x))
Eq(F_28(x), F_18(x) + F_29(x) + F_33(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_11(x) + F_17(x))
Eq(F_32(x), F_17(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_16(x) + F_35(x))
Eq(F_35(x), F_17(x))
Eq(F_36(x), F_24(x) + F_37(x))
Eq(F_37(x), F_18(x) + F_38(x) + F_71(x) + F_86(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_48(x))
Eq(F_40(x), F_28(x) + F_41(x))
Eq(F_41(x), F_18(x) + F_42(x) + F_44(x) + F_46(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_17(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_32(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_35(x))
Eq(F_48(x), F_37(x) + F_49(x))
Eq(F_49(x), F_18(x) + F_50(x) + F_54(x) + F_65(x) + F_68(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_41(x))
Eq(F_53(x), F_49(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_18(x) + F_57(x) + F_58(x) + F_62(x))
Eq(F_57(x), 0)
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_2(x)*F_4(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_60(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_56(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_56(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_67(x) + F_73(x))
Eq(F_73(x), F_56(x) + F_74(x))
Eq(F_74(x), F_18(x) + F_75(x) + F_83(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_77(x) + F_80(x))
Eq(F_77(x), F_4(x) + F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_11(x)*F_4(x))
Eq(F_80(x), F_60(x) + F_81(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_4(x)*F_74(x))
Eq(F_83(x), F_4(x)*F_84(x))
Eq(F_84(x), F_64(x) + F_85(x))
Eq(F_85(x), F_2(x) + F_60(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_70(x) + F_88(x))
Eq(F_88(x), F_56(x) + F_74(x))
Eq(F_89(x), F_4(x)*F_90(x))
Eq(F_90(x), F_88(x) + F_91(x))
Eq(F_91(x), F_2(x) + F_74(x))
Pack JSON:
{"name": "point_placements", "initial_strats": [{"class_module": "tilings.strategies.factor", "strategy_class": "FactorFactory", "interleaving": null, "unions": false, "ignore_parent": true, "workable": true, "tracked": false}, {"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "PointCorroborationFactory", "ignore_parent": true}], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "strategy_class": "RowColumnSeparationStrategy", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "strategy_class": "InsertionEncodingVerificationStrategy", "ignore_parent": false}, {"class_module": "tilings.strategies.verification", "strategy_class": "OneByOneVerificationStrategy", "ignore_parent": false, "basis": [], "symmetry": false}, {"class_module": "tilings.strategies.verification", "strategy_class": "LocallyFactorableVerificationStrategy", "ignore_parent": false, "basis": [], "symmetry": false}], "expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "CellInsertionFactory", "ignore_parent": false, "maxreqlen": 1, "extra_basis": [], "one_cell_only": false}, {"class_module": "tilings.strategies.requirement_placement", "strategy_class": "PatternPlacementFactory", "partial": false, "ignore_parent": false, "dirs": [0, 1, 2, 3], "point_only": false}]], "symmetries": [], "iterative": false}
Specification JSON:
{"root": {"class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, 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