0132_0321_1023_1302_2031

Counting sequence:
1, 1, 2, 6, 19, 55, 157, 453, 1318, 3843, 11209, 32704, 95454, 278673, 813676, 2375948, 6938080, 20260598, 59165921, 172780506, 504568372, 1473487926, 4303025252, 12566133094, 36696925428, 107166206725, 312958051567, 913933199551, 2668964579132, 7794193531430, 22761431465141, 66470349757436, 194113777189434, 566871677802979, 1655438911834804, 4834388633425122, 14117895451694031, 41228578673295621, 120400077041730682, 351605100668101314, 1026794582439722736, 2998554664322022409, 8756698008638287338, 25572240162529786124, 74678773470912820021, 218084890949164295325, 636874676029479347841, 1859869114286771709638, 5431387449489761770421, 15861314863449611966488, 46319897362742251158665, 135268287034633709354942, 395024827753740890185527, 1153593484198852580304365, 3368846293417695249452430, 9838062978100480146255647, 28730157071928332367868114, 83900858046454412940538334, 245016202428262211244090874, 715522354004154904653027963, 2089544421983881516321462438, 6102109692325012064112360542, 17820029239586143714131820208, 52039943250956512284426695783, 151972572948801185159839912352, 443806458767692773730166504197, 1296050787468578805990733256008, 3784865250411271990617704791754, 11052965749706840640361306523499, 32278045262223654946756434329392, 94261778199918214502700071869362, 275273262591566986459334907844536, 803882236733271586106574235895193, 2347582342183802405059257134041889, 6855659450479668588228683481562987, 20020625328622169560463582129354725, 58466357823684354664442614458498695, 170739671766403665430742795802506110, 498612135252419543070920676271161156, 1456100148541440194036247748514647194, 4252258404238499077836077028775414310, 12417896910819761911183851163211200406, 36264062300174841513633223465122406348, 105902168777478412230878060537437458600, 309266768266043123128089352270582957055, 903153684743639289172144138742582483108, 2637485374970284997966848411894083149267, 7702265096949366943781039767710329590911, 22492973112449133374817295227471435719549, 65686370576591120943825335598497345859138, 191824320331277558064707644481504665030454, 560185766812178013502500721402226752417922, 1635914011304751571830491082440164867908009, 4777369956421077669053194185623539594279449, 13951383472968508920079108328256097756091597, 40742312733852503715810115075490062711226476, 118980031630500841654127456011920548731543691, 347458133250070829230647628445521437902336887, 1014684167647130250221468655177986940890843233, 2963188544309430542958427254916626447001382163, 8653418106923859581034898220588138468756383394

Generating function in Maple syntax:
(x^3-2*x^2+3*x-1)*(x-1)^2/(x^8-3*x^7+4*x^6-10*x^4+15*x^3-13*x^2+6*x-1)

Generating function in latex syntax:
\frac{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{2}}{x^{8}-3 x^{7}+4 x^{6}-10 x^{4}+15 x^{3}-13 x^{2}+6 x -1}

Generating function in sympy syntax:
(x - 1)**2*(x**3 - 2*x**2 + 3*x - 1)/(x**8 - 3*x**7 + 4*x**6 - 10*x**4 + 15*x**3 - 13*x**2 + 6*x - 1)

Implicit equation for the generating function in Maple syntax:
(x^8-3*x^7+4*x^6-10*x^4+15*x^3-13*x^2+6*x-1)*F(x)-(x^3-2*x^2+3*x-1)*(x-1)^2 = 0

Implicit equation for the generating function in latex syntax:
\left(x^{8}-3 x^{7}+4 x^{6}-10 x^{4}+15 x^{3}-13 x^{2}+6 x -1\right) F \! \left(x \right)-\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{2} = 0

Explicit closed form in Maple syntax:
51047379/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 1)^(-n+6)+51047379/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 2)^(-n+6)+51047379/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 3)^(-n+6)+51047379/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 4)^(-n+6)+51047379/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 5)^(-n+6)+51047379/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 6)^(-n+6)+51047379/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 7)^(-n+6)+51047379/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 8)^(-n+6)-140963359/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 1)^(-n+5)-140963359/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 2)^(-n+5)-140963359/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 3)^(-n+5)-140963359/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 4)^(-n+5)-140963359/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 5)^(-n+5)-140963359/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 6)^(-n+5)-140963359/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 7)^(-n+5)-140963359/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 8)^(-n+5)+141821286/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 1)^(-n+4)+141821286/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 2)^(-n+4)+141821286/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 3)^(-n+4)+141821286/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 4)^(-n+4)+141821286/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 5)^(-n+4)+141821286/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 6)^(-n+4)+141821286/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 7)^(-n+4)+141821286/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 8)^(-n+4)+89105829/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 1)^(-n+3)+89105829/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 2)^(-n+3)+89105829/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 3)^(-n+3)+89105829/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 4)^(-n+3)+89105829/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 5)^(-n+3)+89105829/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 6)^(-n+3)+89105829/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 7)^(-n+3)+89105829/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 8)^(-n+3)-514631101/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 1)^(-n+2)-514631101/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 2)^(-n+2)-514631101/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 3)^(-n+2)-514631101/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 4)^(-n+2)-514631101/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 5)^(-n+2)-514631101/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 6)^(-n+2)-514631101/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 7)^(-n+2)-514631101/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 8)^(-n+2)+538637734/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 1)^(-n+1)+538637734/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 2)^(-n+1)+538637734/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 3)^(-n+1)+538637734/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 4)^(-n+1)+538637734/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 5)^(-n+1)+538637734/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 6)^(-n+1)+538637734/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 7)^(-n+1)+538637734/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 8)^(-n+1)+93344121/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 3)^(-n-1)+93344121/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 4)^(-n-1)+93344121/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 5)^(-n-1)+93344121/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 6)^(-n-1)+93344121/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 7)^(-n-1)+93344121/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 8)^(-n-1)+93344121/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 1)^(-n-1)+93344121/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 2)^(-n-1)-275394486/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 1)^(-n)-275394486/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 2)^(-n)-275394486/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 3)^(-n)-275394486/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 4)^(-n)-275394486/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 5)^(-n)-275394486/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 6)^(-n)-275394486/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 7)^(-n)-275394486/509179709*RootOf(_Z^8-3*_Z^7+4*_Z^6-10*_Z^4+15*_Z^3-13*_Z^2+6*_Z-1,index = 8)^(-n)

Explicit closed form in latex syntax:
\frac{51047379 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +6}}{509179709}+\frac{51047379 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +6}}{509179709}+\frac{51047379 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +6}}{509179709}+\frac{51047379 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +6}}{509179709}+\frac{51047379 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +6}}{509179709}+\frac{51047379 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +6}}{509179709}+\frac{51047379 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +6}}{509179709}+\frac{51047379 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +6}}{509179709}-\frac{140963359 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{509179709}-\frac{140963359 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{509179709}-\frac{140963359 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{509179709}-\frac{140963359 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{509179709}-\frac{140963359 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{509179709}-\frac{140963359 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{509179709}-\frac{140963359 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{509179709}-\frac{140963359 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +5}}{509179709}+\frac{141821286 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{509179709}+\frac{141821286 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{509179709}+\frac{141821286 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{509179709}+\frac{141821286 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{509179709}+\frac{141821286 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{509179709}+\frac{141821286 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{509179709}+\frac{141821286 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{509179709}+\frac{141821286 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +4}}{509179709}+\frac{89105829 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{509179709}+\frac{89105829 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{509179709}+\frac{89105829 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{509179709}+\frac{89105829 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{509179709}+\frac{89105829 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{509179709}+\frac{89105829 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{509179709}+\frac{89105829 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{509179709}+\frac{89105829 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +3}}{509179709}-\frac{514631101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{509179709}-\frac{514631101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{509179709}-\frac{514631101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{509179709}-\frac{514631101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{509179709}-\frac{514631101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{509179709}-\frac{514631101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{509179709}-\frac{514631101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{509179709}-\frac{514631101 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +2}}{509179709}+\frac{538637734 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{509179709}+\frac{538637734 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{509179709}+\frac{538637734 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{509179709}+\frac{538637734 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{509179709}+\frac{538637734 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{509179709}+\frac{538637734 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{509179709}+\frac{538637734 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{509179709}+\frac{538637734 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =8\right)^{-n +1}}{509179709}+\frac{93344121 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{509179709}+\frac{93344121 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{509179709}+\frac{93344121 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{509179709}+\frac{93344121 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{509179709}+\frac{93344121 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{509179709}+\frac{93344121 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =8\right)^{-n -1}}{509179709}+\frac{93344121 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{509179709}+\frac{93344121 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{509179709}-\frac{275394486 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{509179709}-\frac{275394486 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{509179709}-\frac{275394486 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{509179709}-\frac{275394486 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{509179709}-\frac{275394486 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{509179709}-\frac{275394486 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{509179709}-\frac{275394486 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{509179709}-\frac{275394486 \mathit{RootOf}\! \left(\textit{\_Z}^{8}-3 \textit{\_Z}^{7}+4 \textit{\_Z}^{6}-10 \textit{\_Z}^{4}+15 \textit{\_Z}^{3}-13 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =8\right)^{-n}}{509179709}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 157
a(7) = 453
a(n+2) = -1/4*a(n)+3/4*a(n+1)+5/2*a(n+4)-15/4*a(n+5)+13/4*a(n+6)-3/2*a(n+7)+1/4*a(n+8), n >= 8

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) = 19
a \! \left(5\right) = 55
a \! \left(6\right) = 157
a \! \left(7\right) = 453
a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{5 a \! \left(n +4\right)}{2}-\frac{15 a \! \left(n +5\right)}{4}+\frac{13 a \! \left(n +6\right)}{4}-\frac{3 a \! \left(n +7\right)}{2}+\frac{a \! \left(n +8\right)}{4}, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19320/
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[11,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[16,x]
F[19,x] = F[13,x]*F[4,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[15,x]+F[22,x]+F[59,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[34,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[1,x]+F[4,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[17,x]+F[31,x]
F[31,x] = F[18,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[31,x]*F[4,x]
F[34,x] = F[35,x]+F[40,x]
F[35,x] = F[15,x]+F[22,x]+F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[2,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[2,x]*F[4,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[53,x]
F[43,x] = F[44,x]+F[51,x]
F[44,x] = F[15,x]+F[39,x]+F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[49,x]
F[47,x] = F[4,x]+F[48,x]
F[48,x] = F[29,x]
F[49,x] = F[44,x]+F[50,x]
F[50,x] = F[41,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[43,x]
F[53,x] = F[54,x]+F[57,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[44,x]+F[54,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[53,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[64,x]
F[61,x] = F[2,x]+F[62,x]
F[62,x] = F[15,x]+F[45,x]+F[63,x]
F[63,x] = F[4,x]*F[61,x]
F[64,x] = F[65,x]+F[68,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[2,x]+F[65,x]
F[68,x] = 2*F[15,x]+F[55,x]+F[69,x]
F[69,x] = F[4,x]*F[64,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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \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_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{59}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{34}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{17}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{18}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{53}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{51}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right)+F_{39}\! \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_{49}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{29}\! \left(x \right)
F_{49}\! \left(x \right) = F_{44}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{41}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{57}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{44}\! \left(x \right)+F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \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_{64}\! \left(x \right)
F_{61}\! \left(x \right) = F_{2}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{2}\! \left(x \right)+F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \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_11(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_13(x)*F_4(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_15(x) + F_22(x) + F_59(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_34(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_1(x) + F_4(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_17(x) + F_31(x))
Eq(F_31(x), F_18(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_31(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_40(x))
Eq(F_35(x), F_15(x) + F_22(x) + F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_2(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_2(x)*F_4(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_53(x))
Eq(F_43(x), F_44(x) + F_51(x))
Eq(F_44(x), F_15(x) + F_39(x) + F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_49(x))
Eq(F_47(x), F_4(x) + F_48(x))
Eq(F_48(x), F_29(x))
Eq(F_49(x), F_44(x) + F_50(x))
Eq(F_50(x), F_41(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_43(x))
Eq(F_53(x), F_54(x) + F_57(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_44(x) + F_54(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_53(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_64(x))
Eq(F_61(x), F_2(x) + F_62(x))
Eq(F_62(x), F_15(x) + F_45(x) + F_63(x))
Eq(F_63(x), F_4(x)*F_61(x))
Eq(F_64(x), F_65(x) + F_68(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_2(x) + F_65(x))
Eq(F_68(x), 2*F_15(x) + F_55(x) + F_69(x))
Eq(F_69(x), F_4(x)*F_64(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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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, 0], [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": [2, 1, 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, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [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, 0], [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": [2, 1, 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, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 2], [1, 2], [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": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [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": [2, 1, 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, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [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": [[{"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], "pos": [[0, 0], [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": [2, 1, 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, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [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": []}, "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], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 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, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 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, 3, 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