0123_0132_1302_2031
Counting sequence:
1, 1, 2, 6, 20, 64, 197, 597, 1805, 5466, 16577, 50308, 152705, 463538, 1407099, 4271445, 12966923, 39364896, 119505712, 362805074, 1101441936, 3343893121, 10151845139, 30820444337, 93569381598, 284072576689, 862432983698, 2618314205909, 7949108477794, 24133219925649, 73267648164179, 222438177521825, 675315121587458, 2050235130811402, 6224448862920620, 18897230558469848, 57371398825715509, 174177771466218649, 528798275338511805, 1605415077730995402, 4873990181152021709, 14797282531359706256, 44924089606041753845, 136388139609504586606, 414070153623739305583, 1257104122111368600345, 3816529065372959830651, 11586863703466132833472, 35177358352805229560804, 106797367528163756471498, 324233491371854001007832, 984362811497475681374057, 2988494929321223345772903, 9072977808319296197813209, 27545278900217231816839926, 83626611437320316430289133, 253887795663785413268955246, 770795464290666436678723233, 2340111096072136901718401718, 7104504626432892055372754293, 21569055448934726855881972383, 65482982618366242085948845345, 198804301967991402061036400626, 603563687860205581122167939854, 1832400615574718332194802316660, 5563111372510501249495483170336, 16889433391355467044161840400773, 51275795356364635902920973062333, 155671722579911102005177470907325, 472614516120030857286992260122794, 1434842996182892598792885080048153, 4356138784305060574802391887248492, 13225102090343417181999379655675785, 40150999304754615062839774754214122, 121897187194327224802423213745047811, 370076075394992861582495363685394525, 1123539473978494968579920936553113387, 3411030956920467479291559582099926704, 10355784072160686392769056414821793368, 31439838894350147690657166382153327346, 95450374671290105690005738232190008784, 289784373752851998728463177522118568369, 879776360863169508319444172821667774331, 2670973714385906062157503053522124812913, 8108993262720828137079984640555712685662, 24618651760101746808736276881451369282601, 74741462330652398213009136351361632573242, 226912758901678262854549487518852943574221, 688900090348577743024788282730950237495146, 2091479283841932606458041043449005473735801, 6349666164982813318263119564314403690918091, 19277389318752965827559321349643763426907905, 58525555405758401816735465512812431011786322, 177681768984166206788048010020809440305570258, 539436333589007251817704834128622661994562860, 1637711959193078255384751161966991232600760152, 4972042656895826805052544527545168804094291381, 15094967123627876522790515683156560390696351361, 45827851486226799832572104259105746393109214605, 139131934150175646640376828939039630312264639418, 422400232011457398194680430538527486671302180901
Generating function in Maple syntax:
(2*x-1)*(x^3-2*x^2+3*x-1)*(x-1)^2/(5*x^6-20*x^5+39*x^4-41*x^3+25*x^2-8*x+1)
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{2}}{5 x^{6}-20 x^{5}+39 x^{4}-41 x^{3}+25 x^{2}-8 x +1}
Generating function in sympy syntax:
(x - 1)**2*(2*x - 1)*(x**3 - 2*x**2 + 3*x - 1)/(5*x**6 - 20*x**5 + 39*x**4 - 41*x**3 + 25*x**2 - 8*x + 1)
Implicit equation for the generating function in Maple syntax:
(5*x^6-20*x^5+39*x^4-41*x^3+25*x^2-8*x+1)*F(x)-(2*x-1)*(x^3-2*x^2+3*x-1)*(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(5 x^{6}-20 x^{5}+39 x^{4}-41 x^{3}+25 x^{2}-8 x +1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,2/5,0)-1751/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 1)^(-n+4)-1751/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 2)^(-n+4)-1751/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 3)^(-n+4)-1751/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 4)^(-n+4)-1751/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 5)^(-n+4)-1751/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 6)^(-n+4)+6203/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 1)^(-n+3)+6203/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 2)^(-n+3)+6203/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 3)^(-n+3)+6203/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 4)^(-n+3)+6203/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 5)^(-n+3)+6203/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 6)^(-n+3)-52534/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 1)^(-n+2)-52534/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 2)^(-n+2)-52534/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 3)^(-n+2)-52534/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 4)^(-n+2)-52534/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 5)^(-n+2)-52534/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 6)^(-n+2)+43742/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 1)^(-n+1)+43742/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 2)^(-n+1)+43742/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 3)^(-n+1)+43742/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 4)^(-n+1)+43742/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 5)^(-n+1)+43742/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 6)^(-n+1)+3392/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 1)^(-n-1)+3392/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 2)^(-n-1)+3392/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 3)^(-n-1)+3392/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 4)^(-n-1)+3392/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 5)^(-n-1)+3392/4145*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 6)^(-n-1)-3796/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 1)^(-n)-3796/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 2)^(-n)-3796/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 3)^(-n)-3796/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 4)^(-n)-3796/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 5)^(-n)-3796/829*RootOf(5*_Z^6-20*_Z^5+39*_Z^4-41*_Z^3+25*_Z^2-8*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
-\frac{1751 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{829}-\frac{1751 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{829}-\frac{1751 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{829}-\frac{1751 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{829}-\frac{1751 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{829}-\frac{1751 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{829}+\frac{6203 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{829}+\frac{6203 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{829}+\frac{6203 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{829}+\frac{6203 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{829}+\frac{6203 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{829}+\frac{6203 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{829}-\frac{52534 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{4145}-\frac{52534 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{4145}-\frac{52534 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{4145}-\frac{52534 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{4145}-\frac{52534 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{4145}-\frac{52534 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{4145}+\frac{43742 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{4145}+\frac{43742 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{4145}+\frac{43742 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{4145}+\frac{43742 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{4145}+\frac{43742 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{4145}+\frac{43742 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{4145}+\frac{3392 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{4145}+\frac{3392 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{4145}+\frac{3392 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{4145}+\frac{3392 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{4145}+\frac{3392 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{4145}+\frac{3392 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{4145}-\frac{3796 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{829}-\frac{3796 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{829}-\frac{3796 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{829}-\frac{3796 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{829}-\frac{3796 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{829}-\frac{3796 \mathit{RootOf}\! \left(5 \textit{\_Z}^{6}-20 \textit{\_Z}^{5}+39 \textit{\_Z}^{4}-41 \textit{\_Z}^{3}+25 \textit{\_Z}^{2}-8 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{829}+\left(\left\{\begin{array}{cc}\frac{2}{5} & 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) = 20
a(5) = 64
a(6) = 197
a(n+6) = -5*a(n)+20*a(n+1)-39*a(n+2)+41*a(n+3)-25*a(n+4)+8*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) = 20
a \! \left(5\right) = 64
a \! \left(6\right) = 197
a \! \left(n +6\right) = -5 a \! \left(n \right)+20 a \! \left(n +1\right)-39 a \! \left(n +2\right)+41 a \! \left(n +3\right)-25 a \! \left(n +4\right)+8 a \! \left(n +5\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14857/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[12,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[54,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[29,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[16,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[25,x]
F[29,x] = F[20,x]+F[30,x]
F[30,x] = 2*F[13,x]+F[31,x]+F[35,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[24,x]
F[34,x] = F[30,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[49,x]
F[37,x] = F[38,x]
F[38,x] = F[13,x]+F[39,x]+F[48,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[28,x]+F[4,x]
F[42,x] = F[38,x]+F[43,x]
F[43,x] = 2*F[13,x]+F[35,x]+F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[28,x]
F[47,x] = F[43,x]
F[48,x] = F[2,x]*F[4,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[38,x]
F[53,x] = F[50,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[2,x]+F[38,x]
F[57,x] = F[58,x]+F[70,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[63,x]
F[61,x] = F[2,x]+F[62,x]
F[62,x] = F[48,x]
F[63,x] = F[58,x]+F[64,x]
F[64,x] = 2*F[13,x]+F[65,x]+F[69,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[68,x]
F[67,x] = F[62,x]
F[68,x] = F[64,x]
F[69,x] = F[4,x]*F[58,x]
F[70,x] = 2*F[13,x]+F[69,x]+F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[52,x]+F[73,x]
F[73,x] = F[70,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{54}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{29}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{20}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \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_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{24}\! \left(x \right)
F_{34}\! \left(x \right) = F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{49}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{13}\! \left(x \right)+F_{39}\! \left(x \right)+F_{48}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{28}\! \left(x \right)+F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{35}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{28}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \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_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{38}\! \left(x \right)
F_{53}\! \left(x \right) = F_{50}\! \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_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{2}\! \left(x \right)+F_{38}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{70}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{63}\! \left(x \right)
F_{61}\! \left(x \right) = F_{2}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{48}\! \left(x \right)
F_{63}\! \left(x \right) = F_{58}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{65}\! \left(x \right)+F_{69}\! \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_{68}\! \left(x \right)
F_{67}\! \left(x \right) = F_{62}\! \left(x \right)
F_{68}\! \left(x \right) = F_{64}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{70}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{69}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{52}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{70}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_54(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_29(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_16(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_25(x))
Eq(F_29(x), F_20(x) + F_30(x))
Eq(F_30(x), 2*F_13(x) + F_31(x) + F_35(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_24(x))
Eq(F_34(x), F_30(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_49(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_13(x) + F_39(x) + F_48(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_28(x) + F_4(x))
Eq(F_42(x), F_38(x) + F_43(x))
Eq(F_43(x), 2*F_13(x) + F_35(x) + F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_28(x))
Eq(F_47(x), F_43(x))
Eq(F_48(x), F_2(x)*F_4(x))
Eq(F_49(x), F_50(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_38(x))
Eq(F_53(x), F_50(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_2(x) + F_38(x))
Eq(F_57(x), F_58(x) + F_70(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_63(x))
Eq(F_61(x), F_2(x) + F_62(x))
Eq(F_62(x), F_48(x))
Eq(F_63(x), F_58(x) + F_64(x))
Eq(F_64(x), 2*F_13(x) + F_65(x) + F_69(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_68(x))
Eq(F_67(x), F_62(x))
Eq(F_68(x), F_64(x))
Eq(F_69(x), F_4(x)*F_58(x))
Eq(F_70(x), 2*F_13(x) + F_69(x) + F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_52(x) + F_73(x))
Eq(F_73(x), F_70(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "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": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "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": [], 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