0132_0213_0231_2103
Counting sequence:
1, 1, 2, 6, 20, 65, 203, 622, 1897, 5789, 17689, 54090, 165433, 505970, 1547419, 4732374, 14472590, 44260190, 135357068, 413951115, 1265952294, 3871556852, 11840060489, 36209471356, 110736410227, 338655938921, 1035683249076, 3167343816148, 9686423783477, 29623183074295, 90594113476003, 277056431629857, 847298608710263, 2591222763155878, 7924520752517684, 24234901780777380, 74115581581119353, 226661510032034811, 693180017399865210, 2119894711962948067, 6483097430686518729, 19826716892394541825, 60634396896536585212, 185433125764610939181, 567094683723953229788, 1734298437681547714621, 5303860461525276660178, 16220354688745056588706, 49605359744519340207420, 151703940055675790555018, 463943524388184487170213, 1418839838587810605362656, 4339119702594459535820382, 13269968379364889838760653, 40582439033441401714725796, 124109893167794552469154083, 379554949110587646220745227, 1160761287575772521839390574, 3549859565503899311551353824, 10856239839905005564484815603, 33200734081662420299919482213, 101535039739036002985253272885, 310516155138320723462947580965, 949625694240173444251508321325, 2904161165976777310932878515446, 8881554205118718311272472885766, 27161717477181070596926892148084, 83066418249748453505633891768449, 254035108296776598608154573694607, 776894413014496862728919687929746, 2375911475464346865150902956168617, 7266052174760374747142845821520505, 22221162173570311347704954335835253, 67957129465616077571570396565718678, 207827628867194193700338164514143681, 635581927315131946854357582111830342, 1943747270425524571003238141906920147, 5944400381626028962664958031727792986, 18179264575553567213545379839985305886, 55596130692928847026396273290093847910, 170025015873399528096126054108781930268, 519973344591525189642438225851481560403, 1590191170968723513304312105851129836394, 4863149210491460877653719462823216791564, 14872564176731226998908433325971366636529, 45483524279658206734777262879710814966576, 139098473962876449327113136336437932523355, 425393277351075444467015951629534966913957, 1300944828940285052984547053918851499777688, 3978571214113684454391139402789001729458564, 12167332967277284004572516190304788698162149, 37210340991614684566470809397884888235447667, 113797286589920436701632186843304756167036543, 348016763354754421359715279367338210182779837, 1064310680907280321447285890458981006819852067, 3254892708540683407130046506233803345464962322, 9954167269166260873179895714541971751552525156, 30442000672570682720449387669082313791499214124, 93098235130060612329753754152594185217252352393, 284714578307646462319024942285414412789017614463, 870718880842958179280643558374511292522481211470
Generating function in Maple syntax:
-(x^3+2*x-1)*(x-1)^2/(x^6-3*x^5+3*x^4-7*x^3+8*x^2-5*x+1)
Generating function in latex syntax:
-\frac{\left(x^{3}+2 x -1\right) \left(x -1\right)^{2}}{x^{6}-3 x^{5}+3 x^{4}-7 x^{3}+8 x^{2}-5 x +1}
Generating function in sympy syntax:
(x - 1)**2*(-x**3 - 2*x + 1)/(x**6 - 3*x**5 + 3*x**4 - 7*x**3 + 8*x**2 - 5*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^6-3*x^5+3*x^4-7*x^3+8*x^2-5*x+1)*F(x)+(x^3+2*x-1)*(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{6}-3 x^{5}+3 x^{4}-7 x^{3}+8 x^{2}-5 x +1\right) F \! \left(x \right)+\left(x^{3}+2 x -1\right) \left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
173099/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n+4)+173099/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n+4)+173099/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n+4)+173099/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n+4)+173099/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n+4)+173099/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 6)^(-n+4)-469881/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n+3)-469881/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n+3)-469881/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n+3)-469881/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n+3)-469881/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n+3)-469881/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 6)^(-n+3)+379076/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n+2)+379076/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n+2)+379076/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n+2)+379076/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n+2)+379076/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n+2)+379076/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 6)^(-n+2)-1017769/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n+1)-1017769/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n+1)-1017769/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n+1)-1017769/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n+1)-1017769/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n+1)-1017769/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 6)^(-n+1)-46698/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n-1)-46698/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n-1)-46698/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n-1)-46698/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n-1)-46698/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n-1)-46698/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 6)^(-n-1)+1409581/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 1)^(-n)+1409581/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 2)^(-n)+1409581/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 3)^(-n)+1409581/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 4)^(-n)+1409581/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 5)^(-n)+1409581/3883673*RootOf(_Z^6-3*_Z^5+3*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1,index = 6)^(-n)
Explicit closed form in latex syntax:
\frac{173099 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +4}}{3883673}+\frac{173099 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +4}}{3883673}+\frac{173099 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +4}}{3883673}+\frac{173099 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +4}}{3883673}+\frac{173099 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +4}}{3883673}+\frac{173099 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +4}}{3883673}-\frac{469881 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{3883673}-\frac{469881 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{3883673}-\frac{469881 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{3883673}-\frac{469881 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{3883673}-\frac{469881 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{3883673}-\frac{469881 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +3}}{3883673}+\frac{379076 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{3883673}+\frac{379076 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{3883673}+\frac{379076 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{3883673}+\frac{379076 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{3883673}+\frac{379076 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{3883673}+\frac{379076 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +2}}{3883673}-\frac{1017769 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{3883673}-\frac{1017769 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{3883673}-\frac{1017769 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{3883673}-\frac{1017769 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{3883673}-\frac{1017769 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{3883673}-\frac{1017769 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n +1}}{3883673}-\frac{46698 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{3883673}-\frac{46698 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{3883673}-\frac{46698 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{3883673}-\frac{46698 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{3883673}-\frac{46698 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{3883673}-\frac{46698 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n -1}}{3883673}+\frac{1409581 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{3883673}+\frac{1409581 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{3883673}+\frac{1409581 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{3883673}+\frac{1409581 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{3883673}+\frac{1409581 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{3883673}+\frac{1409581 \mathit{RootOf}\! \left(\textit{\_Z}^{6}-3 \textit{\_Z}^{5}+3 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =6\right)^{-n}}{3883673}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(n+6) = -a(n)+3*a(n+1)-3*a(n+2)+7*a(n+3)-8*a(n+4)+5*a(n+5), 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) = 20
a \! \left(5\right) = 65
a \! \left(n +6\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-3 a \! \left(n +2\right)+7 a \! \left(n +3\right)-8 a \! \left(n +4\right)+5 a \! \left(n +5\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15246/
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[14,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[7,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]+F[17,x]+F[47,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[29,x]
F[19,x] = F[20,x]+F[7,x]
F[20,x] = F[16,x]+F[21,x]+F[26,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]
F[23,x] = F[11,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[23,x]*F[4,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[13,x]+F[28,x]
F[28,x] = F[20,x]
F[29,x] = F[30,x]+F[37,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[2,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[33,x]*F[4,x]
F[36,x] = F[30,x]
F[37,x] = 2*F[16,x]+F[38,x]+F[43,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[34,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[15,x]
F[46,x] = F[37,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[60,x]
F[49,x] = F[2,x]+F[50,x]
F[50,x] = F[16,x]+F[51,x]+F[59,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[11,x]+F[54,x]
F[54,x] = F[16,x]+F[21,x]+F[55,x]
F[55,x] = F[13,x]*F[4,x]
F[56,x] = F[34,x]+F[57,x]
F[57,x] = 2*F[16,x]+F[38,x]+F[58,x]
F[58,x] = F[4,x]*F[45,x]
F[59,x] = F[4,x]*F[49,x]
F[60,x] = F[15,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_{14}\! \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_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{47}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{29}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{26}\! \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_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{20}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \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_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \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_{30}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{43}\! \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_{40}\! \left(x \right) = F_{34}\! \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_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right)
F_{46}\! \left(x \right) = F_{37}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{60}\! \left(x \right)
F_{49}\! \left(x \right) = F_{2}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{16}\! \left(x \right)+F_{51}\! \left(x \right)+F_{59}\! \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_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{11}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{56}\! \left(x \right) = F_{34}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{60}\! \left(x \right) = F_{15}\! \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_14(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_7(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_47(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_29(x))
Eq(F_19(x), F_20(x) + F_7(x))
Eq(F_20(x), F_16(x) + F_21(x) + F_26(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_11(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_23(x)*F_4(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_13(x) + F_28(x))
Eq(F_28(x), F_20(x))
Eq(F_29(x), F_30(x) + F_37(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) + F_36(x))
Eq(F_33(x), F_2(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_30(x))
Eq(F_37(x), 2*F_16(x) + F_38(x) + F_43(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_34(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_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_15(x))
Eq(F_46(x), F_37(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_60(x))
Eq(F_49(x), F_2(x) + F_50(x))
Eq(F_50(x), F_16(x) + F_51(x) + F_59(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_11(x) + F_54(x))
Eq(F_54(x), F_16(x) + F_21(x) + F_55(x))
Eq(F_55(x), F_13(x)*F_4(x))
Eq(F_56(x), F_34(x) + F_57(x))
Eq(F_57(x), 2*F_16(x) + F_38(x) + F_58(x))
Eq(F_58(x), F_4(x)*F_45(x))
Eq(F_59(x), F_4(x)*F_49(x))
Eq(F_60(x), F_15(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, 1, 3], "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, 1, 3], "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": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "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": []}, "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, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [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, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [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": [2, 1, 0, 3], "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, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [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, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 2], [1, 0], [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": [2, 1, 0, 3], "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": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [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, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [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": [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": [{"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [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, 0]]}, {"patt": [0, 1, 3, 2], "pos": 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