0231_0321_3102
Counting sequence:
1, 1, 2, 6, 21, 74, 256, 880, 3025, 10406, 35805, 123197, 423881, 1458425, 5017929, 17264954, 59402739, 204384285, 703215622, 2419521666, 8324736993, 28642540011, 98549071193, 339073260558, 1166633785846, 4013983255478, 13810727728612, 47517936238039, 163492779576375, 562522093545950, 1935443917139269, 6659192944366294, 22911979147317873, 78832193755738596, 271234306402980414, 933223413747133573, 3210898914357552376, 11047592341072652884, 38010943287184480216, 130782505814409031161, 449977357780085025648, 1548216416667238586149, 5326877078133794805075, 18327941171577220564843, 63060104947358476648775, 216968005230107284508627, 746511845053689375999544, 2568489000092135732960063, 8837282070346561042641651, 30406030311232525577327676, 104616631214005718325399030, 359949635468334453732591717, 1238462169640588697073529717, 4261119874827057532571931419, 14661039337935935261837772672, 50443564317051465591568530982, 173558853663559203169603713286, 597155972081583722048420024452, 2054607111452549988809875611059, 7069192271018031707874653860193, 24322644989431200281233487925996, 83685806892999936458686092537245, 287933910081555336861185674729900, 990680972710893749085842123973509, 3408590497081131774163919310195526, 11727780685036303511645896151079226, 40351236064904402360277226560601357, 138834643628962228778901347807921950, 477681978330906122919191768587706572, 1643538431459103992019502952185412600, 5654847154003009766464657344948968113, 19456372740092890994562738096604668490, 66942647589238142207980013665097412737, 230326491279768522561810150998175073761, 792473774129270849562082927617687109873, 2726628097329307542364596267650240872857, 9381383995090921867056718999154742956513, 32278096800056074971334128523264878845306, 111057764353210727669337294149526776584447, 382111346264748066357761205884440260761441, 1314712949559181458522028426248298156958542, 4523472429266788197283446607425149997167486, 15563703715854892372268929866021512796266529, 53549320161144074437465595135587335905565607, 184244685074512925240266356628204251519247265, 633922221160859489716880721354365534880750222, 2181107055104449737957780304714475266408979882, 7504434813966310846402600088859434248293299310, 25820164005829906004932090687658245998463325836, 88838251755776680112631787566579472066317415815, 305661690345606614379375242526377403030813863979, 1051676131603504986553685863398206277165925085014, 3618453737313142495937258653929348716765076246961, 12449847491653210673969179802763326343784008081930, 42835618144593702523115196586607471489462925911213, 147382542883325781272482215131345581661220291515328, 507092342485475605261928833849728010948634023357754, 1744729319882693419371053404392173136990856547319861, 6003009993678845634459574099411472300447919388193852, 20654280623100308384476472257692564555844722032450216, 71064234193676981610857251969954481905423459413940200
Generating function in Maple syntax:
(x^5-3*x^3+4*x^2-4*x+1)/(x^5+x^4-6*x^3+7*x^2-5*x+1)
Generating function in latex syntax:
\frac{x^{5}-3 x^{3}+4 x^{2}-4 x +1}{x^{5}+x^{4}-6 x^{3}+7 x^{2}-5 x +1}
Generating function in sympy syntax:
(x**5 - 3*x**3 + 4*x**2 - 4*x + 1)/(x**5 + x**4 - 6*x**3 + 7*x**2 - 5*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^5+x^4-6*x^3+7*x^2-5*x+1)*F(x)-x^5+3*x^3-4*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}+x^{4}-6 x^{3}+7 x^{2}-5 x +1\right) F \! \left(x \right)-x^{5}+3 x^{3}-4 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/142319*(10609*(RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)-85/103)*(RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)-85/103)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 2)+(-8755*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)+7225)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)+7225*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)+7855)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n+1)+47674/142319*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n+2)-19364/142319*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n+3)-10609/142319*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n+4)+1/142319*(10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^3+10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^2-63654*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)+53160)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n+1)+1/142319*(10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^2+10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)-15980)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n+2)+1/142319*(10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)-8755)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n+3)+1/142319*((-10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)+8755)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 2)^2+(-10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^2-1854*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)+8755)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 2)+8755*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^2+8755*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)+630)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n+1)+1/142319*((-10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)+8755)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 2)+8755*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)-7225)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n+2)-21103/142319*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n+1)+24576/142319*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n)+1/142319*(10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^4+10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^3-63654*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^2+74263*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)-28469)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n)+1/142319*((-10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)+8755)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 2)^3+(-10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^2-1854*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)+8755)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 2)^2+(-10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^3-1854*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^2+72409*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)-52530)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 2)+8755*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^3+8755*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^2-52530*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)+32816)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n)+1/142319*(((10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)-8755)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)-8755*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)+7225)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 2)^2+((10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)-8755)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^2+(10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)^2-6901*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)-1530)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)-8755*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)^2-1530*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)+7225)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 2)+(-8755*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)+7225)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)^2+(-8755*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)^2-1530*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)+7225)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)+7225*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)^2+7225*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)-10534)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n)+1/142319*((((-10609*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 4)+8755)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)+8755*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 4)-7225)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)+(8755*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 4)-7225)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)-7225*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 4)-7855)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 2)+((8755*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 4)-7225)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)-7225*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 4)-7855)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 1)+(-7225*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 4)-7855)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 3)-7855*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 4)-18389)*RootOf(_Z^5+_Z^4-6*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(10609 \left(\mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)-\frac{85}{103}\right) \left(\mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)-\frac{85}{103}\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)+\left(-8755 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)+7225\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)+7225 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)+7855\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{142319}\\+\\\frac{47674 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{142319}\\-\\\frac{19364 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{142319}\\-\\\frac{10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{142319}\\+\\\frac{\left(10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{3}+10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{2}-63654 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)+53160\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{142319}\\+\\\frac{\left(10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{2}+10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)-15980\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{142319}\\+\\\frac{\left(10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)-8755\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{142319}\\+\\\frac{\left(\left(-10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)+8755\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{2}+\left(-10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{2}-1854 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)+8755\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)+8755 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{2}+8755 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)+630\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{142319}\\+\\\frac{\left(\left(-10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)+8755\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)+8755 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)-7225\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{142319}\\-\\\frac{21103 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{142319}\\+\\\frac{24576 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{142319}\\+\\\frac{\left(10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{4}+10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{3}-63654 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{2}+74263 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)-28469\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{142319}\\+\\\frac{\left(\left(-10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)+8755\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{3}+\left(-10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{2}-1854 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)+8755\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{2}+\left(-10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{3}-1854 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{2}+72409 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)-52530\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)+8755 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{3}+8755 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{2}-52530 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)+32816\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{142319}\\+\\\frac{\left(\left(\left(10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)-8755\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)-8755 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)+7225\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{2}+\left(\left(10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)-8755\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{2}+\left(10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{2}-6901 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)-1530\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)-8755 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{2}-1530 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)+7225\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)+\left(-8755 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)+7225\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{2}+\left(-8755 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{2}-1530 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)+7225\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)+7225 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{2}+7225 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)-10534\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{142319}\\+\\\frac{\left(\left(\left(\left(-10609 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)+8755\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)+8755 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)-7225\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)+\left(8755 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)-7225\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)-7225 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)-7855\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)+\left(\left(8755 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)-7225\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)-7225 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)-7855\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)+\left(-7225 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)-7855\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)-7855 \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)-18389\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+\textit{\_Z}^{4}-6 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{142319} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 74
a(n+5) = -a(n)-a(n+1)+6*a(n+2)-7*a(n+3)+5*a(4+n), n >= 6
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 21
a \! \left(5\right) = 74
a \! \left(n +5\right) = -a \! \left(n \right)-a \! \left(n +1\right)+6 a \! \left(n +2\right)-7 a \! \left(n +3\right)+5 a \! \left(4+n \right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14759/
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[17,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[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[14,x]+F[7,x]
F[17,x] = F[18,x]+F[34,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[26,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[21,x]+F[25,x]
F[25,x] = F[11,x]
F[26,x] = F[18,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[26,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[18,x]+F[31,x]
F[34,x] = F[35,x]+F[36,x]+F[72,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[52,x]
F[38,x] = F[39,x]+F[7,x]
F[39,x] = F[35,x]+F[40,x]+F[43,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[38,x]+F[42,x]
F[42,x] = F[14,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[10,x]+F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[11,x]+F[46,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[14,x]+F[49,x]
F[52,x] = F[34,x]+F[53,x]
F[53,x] = 2*F[35,x]+F[54,x]+F[60,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[52,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[34,x]+F[57,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[31,x]+F[57,x]
F[63,x] = F[64,x]+F[68,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[31,x]+F[67,x]
F[67,x] = F[65,x]
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[57,x]+F[71,x]
F[71,x] = F[69,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[17,x]+F[74,x]
F[74,x] = F[75,x]+F[90,x]
F[75,x] = F[32,x]+F[35,x]+F[76,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[78,x]+F[84,x]
F[78,x] = F[11,x]+F[79,x]
F[79,x] = F[35,x]+F[80,x]+F[82,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[78,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[11,x]
F[84,x] = F[31,x]+F[85,x]
F[85,x] = 2*F[35,x]+F[86,x]+F[88,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[84,x]
F[88,x] = F[4,x]*F[89,x]
F[89,x] = F[31,x]
F[90,x] = 2*F[35,x]+F[58,x]+F[91,x]
F[91,x] = F[4,x]*F[92,x]
F[92,x] = F[93,x]+F[99,x]
F[93,x] = F[14,x]+F[94,x]
F[94,x] = 2*F[35,x]+F[95,x]+F[97,x]
F[95,x] = F[4,x]*F[96,x]
F[96,x] = F[93,x]
F[97,x] = F[4,x]*F[98,x]
F[98,x] = F[14,x]
F[99,x] = F[100,x]+F[57,x]
F[100,x] = 3*F[35,x]+F[101,x]+F[103,x]
F[101,x] = F[102,x]*F[4,x]
F[102,x] = F[99,x]
F[103,x] = F[104,x]*F[4,x]
F[104,x] = F[57,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_{17}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \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_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)
F_{26}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{72}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{52}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{7}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{38}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{14}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{11}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{14}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{34}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{54}\! \left(x \right)+F_{60}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{52}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{34}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{31}\! \left(x \right)+F_{57}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{68}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{31}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{57}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{69}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{17}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{90}\! \left(x \right)
F_{75}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{84}\! \left(x \right)
F_{78}\! \left(x \right) = F_{11}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{35}\! \left(x \right)+F_{80}\! \left(x \right)+F_{82}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{78}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{11}\! \left(x \right)
F_{84}\! \left(x \right) = F_{31}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{86}\! \left(x \right)+F_{88}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{84}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{31}\! \left(x \right)
F_{90}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{58}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{4}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)+F_{99}\! \left(x \right)
F_{93}\! \left(x \right) = F_{14}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{95}\! \left(x \right)+F_{97}\! \left(x \right)
F_{95}\! \left(x \right) = F_{4}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{93}\! \left(x \right)
F_{97}\! \left(x \right) = F_{4}\! \left(x \right) F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{14}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{57}\! \left(x \right)
F_{100}\! \left(x \right) = 3 F_{35}\! \left(x \right)+F_{101}\! \left(x \right)+F_{103}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{4}\! \left(x \right)
F_{102}\! \left(x \right) = F_{99}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{4}\! \left(x \right)
F_{104}\! \left(x \right) = F_{57}\! \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_17(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_6(x))
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_14(x) + F_7(x))
Eq(F_17(x), F_18(x) + F_34(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_26(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_21(x) + F_25(x))
Eq(F_25(x), F_11(x))
Eq(F_26(x), F_18(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_26(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_18(x) + F_31(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_72(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_52(x))
Eq(F_38(x), F_39(x) + F_7(x))
Eq(F_39(x), F_35(x) + F_40(x) + F_43(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_38(x) + F_42(x))
Eq(F_42(x), F_14(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_10(x) + F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_11(x) + F_46(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_14(x) + F_49(x))
Eq(F_52(x), F_34(x) + F_53(x))
Eq(F_53(x), 2*F_35(x) + F_54(x) + F_60(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_52(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_34(x) + F_57(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_31(x) + F_57(x))
Eq(F_63(x), F_64(x) + F_68(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_31(x) + F_67(x))
Eq(F_67(x), F_65(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_57(x) + F_71(x))
Eq(F_71(x), F_69(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_17(x) + F_74(x))
Eq(F_74(x), F_75(x) + F_90(x))
Eq(F_75(x), F_32(x) + F_35(x) + F_76(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_84(x))
Eq(F_78(x), F_11(x) + F_79(x))
Eq(F_79(x), F_35(x) + F_80(x) + F_82(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_78(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_11(x))
Eq(F_84(x), F_31(x) + F_85(x))
Eq(F_85(x), 2*F_35(x) + F_86(x) + F_88(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_84(x))
Eq(F_88(x), F_4(x)*F_89(x))
Eq(F_89(x), F_31(x))
Eq(F_90(x), 2*F_35(x) + F_58(x) + F_91(x))
Eq(F_91(x), F_4(x)*F_92(x))
Eq(F_92(x), F_93(x) + F_99(x))
Eq(F_93(x), F_14(x) + F_94(x))
Eq(F_94(x), 2*F_35(x) + F_95(x) + F_97(x))
Eq(F_95(x), F_4(x)*F_96(x))
Eq(F_96(x), F_93(x))
Eq(F_97(x), F_4(x)*F_98(x))
Eq(F_98(x), F_14(x))
Eq(F_99(x), F_100(x) + F_57(x))
Eq(F_100(x), F_101(x) + F_103(x) + 3*F_35(x))
Eq(F_101(x), F_102(x)*F_4(x))
Eq(F_102(x), F_99(x))
Eq(F_103(x), F_104(x)*F_4(x))
Eq(F_104(x), F_57(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 0], [1, 0], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": 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