0132_0231_2103
Counting sequence:
1, 1, 2, 6, 21, 73, 241, 768, 2415, 7587, 23905, 75507, 238759, 755088, 2387570, 7548085, 23860518, 75425046, 238427267, 753705647, 2382596754, 7531821990, 23809446690, 75265885442, 237928695207, 752134317626, 2377628496399, 7516101147253, 23759716349192, 75108639092895, 237431608201632, 750563038119049, 2372661655388950, 7500400417829820, 23710083693147114, 74951740915897677, 236935623687447145, 748995141289997853, 2367705256606568302, 7484732374118776323, 23660554266628908785, 74795169715866443671, 236440674625625574339, 747430520310940309881, 2362759214663251029932, 7469097066248795029388, 23611128310848530736933, 74638925584523064945294, 235946759471544236134878, 745869167718353374279949, 2357823504755342934559281, 7453494419916864234272852, 23561805604061397077021118, 74483007841276828459612276, 235453876086959982534878504, 744311076729244290325967709, 2352898105347982500105052620, 7437924366889389830881458997, 23512585930441283206907880481, 74327415803985782892536096268, 234962022316143890729832802677, 742756240530181806741144138324, 2347982994903869254300658917333, 7422386839082669776208576206908, 23463469074759285783311679318838, 74172148792264474266536856971970, 234471196008268050854860701292109, 741204652322021755981123106905477, 2343078151929742628138054663124477, 7406881768552797756876153846013800, 23414454822232860434217824028946211, 74017206127147553843075533641372611, 233981395017003500201969724137191533, 739656305319840639326260273519623651, 2338183554977266510855708673963923935, 7391409087497814104346754700132523920, 23365542958528099508197149464197586766, 73862587131087879887615859265639806089, 233492617200504669161108098013671351386, 738111192752888245130163786914150688310, 2333299182642909707114045439319067343983, 7375968728257395614560215170293610143539, 23316733269758831910561706649205691847190, 73708291127953681039203943438009013417434, 233004860421400217291565006905257001960998, 736569307864558183336400102870444592409986, 2328425013567852134094524643616389505232583, 7360560623312558792130277222882596641079918, 23268025542485686243606940228661112473490015, 73554317443025598322222445477489522713757277, 232518122546783687855736427135506202335304900, 735030643912358345675774452123479684232603899, 2323561026437891433579481222069402921948237808, 7345184705285364609164591325255466368021444601, 23219419563715157456170410359690981874375231366, 73400665402993734632194288399793140458594845604, 232032401448204180766511410059305780587765602302, 733495194167881421392970815372581290085003339101, 2318707199983349767300391408612701515766969967421, 7329840906938623869258184914340915775834447661465, 23170915120898675447240659091495130331733899051878
Generating function in Maple syntax:
(x^3+2*x-1)*(x-1)^3/(4*x^6-7*x^5+9*x^4-15*x^3+13*x^2-6*x+1)
Generating function in latex syntax:
\frac{\left(x^{3}+2 x -1\right) \left(x -1\right)^{3}}{4 x^{6}-7 x^{5}+9 x^{4}-15 x^{3}+13 x^{2}-6 x +1}
Generating function in sympy syntax:
(x - 1)**3*(x**3 + 2*x - 1)/(4*x**6 - 7*x**5 + 9*x**4 - 15*x**3 + 13*x**2 - 6*x + 1)
Implicit equation for the generating function in Maple syntax:
(4*x^6-7*x^5+9*x^4-15*x^3+13*x^2-6*x+1)*F(x)-(x^3+2*x-1)*(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(4 x^{6}-7 x^{5}+9 x^{4}-15 x^{3}+13 x^{2}-6 x +1\right) F \! \left(x \right)-\left(x^{3}+2 x -1\right) \left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1/4,0)+1367036/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+4)+1367036/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+4)+1367036/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+4)+1367036/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+4)+1367036/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+4)+1367036/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+4)-250363/2851559*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+3)-250363/2851559*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+3)-250363/2851559*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+3)-250363/2851559*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+3)-250363/2851559*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+3)-250363/2851559*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+3)+4575648/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+2)+4575648/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+2)+4575648/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+2)+4575648/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+2)+4575648/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+2)+4575648/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+2)-15453889/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n+1)-15453889/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n+1)-15453889/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n+1)-15453889/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n+1)-15453889/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n+1)-15453889/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n+1)+2944759/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n-1)+2944759/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n-1)+2944759/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n-1)+2944759/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n-1)+2944759/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n-1)+2944759/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n-1)+27636497/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 1)^(-n)+27636497/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 2)^(-n)+27636497/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 3)^(-n)+27636497/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 4)^(-n)+27636497/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 5)^(-n)+27636497/134023273*RootOf(4*_Z^6-7*_Z^5+9*_Z^4-15*_Z^3+13*_Z^2-6*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
\frac{1367036 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{134023273}+\frac{1367036 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{134023273}+\frac{1367036 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{134023273}+\frac{1367036 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{134023273}+\frac{1367036 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{134023273}+\frac{1367036 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{134023273}-\frac{250363 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{2851559}-\frac{250363 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{2851559}-\frac{250363 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{2851559}-\frac{250363 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{2851559}-\frac{250363 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{2851559}-\frac{250363 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{2851559}+\frac{4575648 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{134023273}+\frac{4575648 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{134023273}+\frac{4575648 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{134023273}+\frac{4575648 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{134023273}+\frac{4575648 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{134023273}+\frac{4575648 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{134023273}-\frac{15453889 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{134023273}-\frac{15453889 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{134023273}-\frac{15453889 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{134023273}-\frac{15453889 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{134023273}-\frac{15453889 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{134023273}-\frac{15453889 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{134023273}+\frac{2944759 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{134023273}+\frac{2944759 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{134023273}+\frac{2944759 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{134023273}+\frac{2944759 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{134023273}+\frac{2944759 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{134023273}+\frac{2944759 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{134023273}+\frac{27636497 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{134023273}+\frac{27636497 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{134023273}+\frac{27636497 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{134023273}+\frac{27636497 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{134023273}+\frac{27636497 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{134023273}+\frac{27636497 \mathit{RootOf}\! \left(4 \textit{\_Z}^{6}-7 \textit{\_Z}^{5}+9 \textit{\_Z}^{4}-15 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{134023273}+\left(\left\{\begin{array}{cc}\frac{1}{4} & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 73
a(6) = 241
a(n+6) = -4*a(n)+7*a(n+1)-9*a(n+2)+15*a(n+3)-13*a(n+4)+6*a(n+5), n >= 7
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) = 73
a \! \left(6\right) = 241
a \! \left(n +6\right) = -4 a \! \left(n \right)+7 a \! \left(n +1\right)-9 a \! \left(n +2\right)+15 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14693/
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[20,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[67,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[36,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[15,x]+F[22,x]+F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[13,x]+F[29,x]
F[29,x] = F[26,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[18,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[33,x]*F[4,x]
F[36,x] = F[37,x]+F[50,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = F[37,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[41,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[47,x]
F[50,x] = 2*F[22,x]+F[51,x]+F[56,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[41,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[53,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[60,x]
F[58,x] = F[21,x]+F[59,x]
F[59,x] = F[51,x]
F[60,x] = F[50,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[48,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[64,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]+F[80,x]
F[69,x] = F[2,x]+F[70,x]
F[70,x] = F[22,x]+F[71,x]+F[79,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]+F[76,x]
F[73,x] = F[11,x]+F[74,x]
F[74,x] = F[15,x]+F[22,x]+F[75,x]
F[75,x] = F[13,x]*F[4,x]
F[76,x] = F[41,x]+F[77,x]
F[77,x] = 2*F[22,x]+F[51,x]+F[78,x]
F[78,x] = F[4,x]*F[58,x]
F[79,x] = F[4,x]*F[69,x]
F[80,x] = F[21,x]+F[81,x]
F[81,x] = 2*F[22,x]+F[51,x]+F[82,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[84,x]+F[86,x]
F[84,x] = F[14,x]+F[85,x]
F[85,x] = F[31,x]
F[86,x] = F[44,x]+F[87,x]
F[87,x] = F[62,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_{20}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \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_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{67}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{36}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right)+F_{29}\! \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_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{50}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \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_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{37}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{41}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{41}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{51}\! \left(x \right)
F_{60}\! \left(x \right) = F_{50}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \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_{48}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{80}\! \left(x \right)
F_{69}\! \left(x \right) = F_{2}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{22}\! \left(x \right)+F_{71}\! \left(x \right)+F_{79}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{76}\! \left(x \right)
F_{73}\! \left(x \right) = F_{11}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{76}\! \left(x \right) = F_{41}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{51}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{80}\! \left(x \right) = F_{21}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{51}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{86}\! \left(x \right)
F_{84}\! \left(x \right) = F_{14}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{31}\! \left(x \right)
F_{86}\! \left(x \right) = F_{44}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{62}\! \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_20(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_67(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_36(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_15(x) + F_22(x) + F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_13(x) + F_29(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)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_18(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_33(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_50(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_40(x))
Eq(F_43(x), F_37(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_41(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_47(x))
Eq(F_50(x), 2*F_22(x) + F_51(x) + F_56(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_41(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_53(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_60(x))
Eq(F_58(x), F_21(x) + F_59(x))
Eq(F_59(x), F_51(x))
Eq(F_60(x), F_50(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_48(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_64(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_80(x))
Eq(F_69(x), F_2(x) + F_70(x))
Eq(F_70(x), F_22(x) + F_71(x) + F_79(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_76(x))
Eq(F_73(x), F_11(x) + F_74(x))
Eq(F_74(x), F_15(x) + F_22(x) + F_75(x))
Eq(F_75(x), F_13(x)*F_4(x))
Eq(F_76(x), F_41(x) + F_77(x))
Eq(F_77(x), 2*F_22(x) + F_51(x) + F_78(x))
Eq(F_78(x), F_4(x)*F_58(x))
Eq(F_79(x), F_4(x)*F_69(x))
Eq(F_80(x), F_21(x) + F_81(x))
Eq(F_81(x), 2*F_22(x) + F_51(x) + F_82(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_84(x) + F_86(x))
Eq(F_84(x), F_14(x) + F_85(x))
Eq(F_85(x), F_31(x))
Eq(F_86(x), F_44(x) + F_87(x))
Eq(F_87(x), F_62(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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": 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