0132_0231_1302_2013

Counting sequence:
1, 1, 2, 6, 20, 66, 212, 669, 2094, 6535, 20376, 63513, 197946, 616879, 1922397, 5990864, 18670019, 58184746, 181334464, 565139687, 1761300712, 5489239972, 17107685667, 53317569762, 166168754129, 517878998018, 1614013517402, 5030208552621, 15677066062767, 48858887245209, 152272805316273, 474570921069856, 1479039927726952, 4609551518372685, 14366052482924810, 44773003040553282, 139538805458919232, 434884348032273561, 1355353412847121999, 4224072175057486466, 13164673930474989155, 41028806448880580492, 127869703989628257739, 398516618284737696678, 1242010344068297518292, 3870829029427428532862, 12063762147052338400040, 37597722873846145201787, 117176445296796976955381, 365190183949713648014655, 1138145726433527011492255, 3547126268808259755345211, 11054915442416417086000587, 34453567811681501227355241, 107377423295306674364212348, 334650712999030686583940623, 1042966913098347817667243363, 3250493543161976889661646059, 10130434763985088913743288104, 31572346520499965807116996091, 98397856363906128101039260419, 306665142254322895118621659662, 955747543178803444259735708243, 2978667088073463997342094043979, 9283264900752319885531778771697, 28932070845580346658278966141741, 90169216580886769467476228743666, 281019898720899059627203322143729, 875821998589307390926516128618260, 2729572449155264254195077984133260, 8506940642263092235776552821696214, 26512591418258096078924895298162503, 82628706754969958114416666719823684, 257519269704318923059108567929051041, 802580324362355572564301490329256163, 2501308650778535981147704592110276244, 7795537439109687962861171089365458366, 24295443885202233927118382445547908229, 75718781185973492239732603542645806314, 235983909220994640132216776622424587223, 735463573752539807992981593321892106673, 2292133688701242345786205483945215133315, 7143626189496268460200681327919088271753, 22263707996967723829182967441129591072131, 69386706502513235512938980467986065463370, 216249469312193666596544825021512866732320, 673959542612266389874206343223573854332484, 2100451235893613884144404363854703073838275, 6546231806832957167743689172461108569398108, 20401878480439980875341699430274139031242600, 63584159225185418361143243249130132398883728, 198165345816065412595622985543639930442424644, 617598860485463619127791049325675112189082196, 1924798460105028486268595203830259306429910668, 5998795252164960414176082025611365782480479429, 18695746709727357650784053332634505826042913569, 58266856984014562034685560259801835188430482944, 181593529025999338145553001271258968517069774460, 565951408588306818489921701545144530929593608087, 1763834860201599308660125529342959107416597391349, 5497138741685737042737550269671524800788128969766

Generating function in Maple syntax:
-(2*x-1)^2*(x-1)^4/(3*x^7-16*x^6+48*x^5-74*x^4+65*x^3-33*x^2+9*x-1)

Generating function in latex syntax:
-\frac{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}{3 x^{7}-16 x^{6}+48 x^{5}-74 x^{4}+65 x^{3}-33 x^{2}+9 x -1}

Generating function in sympy syntax:
-(x - 1)**4*(2*x - 1)**2/(3*x**7 - 16*x**6 + 48*x**5 - 74*x**4 + 65*x**3 - 33*x**2 + 9*x - 1)

Implicit equation for the generating function in Maple syntax:
(3*x^7-16*x^6+48*x^5-74*x^4+65*x^3-33*x^2+9*x-1)*F(x)+(2*x-1)^2*(x-1)^4 = 0

Implicit equation for the generating function in latex syntax:
\left(3 x^{7}-16 x^{6}+48 x^{5}-74 x^{4}+65 x^{3}-33 x^{2}+9 x -1\right) F \! \left(x \right)+\left(2 x -1\right)^{2} \left(x -1\right)^{4} = 0

Explicit closed form in Maple syntax:
1668192/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+5)+1668192/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+5)+1668192/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+5)+1668192/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+5)+1668192/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+5)+1668192/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+5)+1668192/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+5)-10629386/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+4)-10629386/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+4)-10629386/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+4)-10629386/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+4)-10629386/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+4)-10629386/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+4)-10629386/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+4)+33513512/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+3)+33513512/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+3)+33513512/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+3)+33513512/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+3)+33513512/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+3)+33513512/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+3)+33513512/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+3)-58205329/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+2)-58205329/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+2)-58205329/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+2)-58205329/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+2)-58205329/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+2)-58205329/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+2)-58205329/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+2)+49899222/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+1)+49899222/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+1)+49899222/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+1)+49899222/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+1)+49899222/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+1)+49899222/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+1)+49899222/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+1)+3944364/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n-1)+3944364/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n-1)+3944364/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n-1)+3944364/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n-1)+3944364/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n-1)+3944364/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n-1)+3944364/7727983*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n-1)-405479/145811*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n)-405479/145811*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n)-405479/145811*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n)-405479/145811*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n)-405479/145811*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n)-405479/145811*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n)-405479/145811*RootOf(3*_Z^7-16*_Z^6+48*_Z^5-74*_Z^4+65*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n)

Explicit closed form in latex syntax:
\frac{1668192 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{7727983}+\frac{1668192 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{7727983}+\frac{1668192 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{7727983}+\frac{1668192 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{7727983}+\frac{1668192 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{7727983}+\frac{1668192 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{7727983}+\frac{1668192 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{7727983}-\frac{10629386 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{7727983}-\frac{10629386 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{7727983}-\frac{10629386 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{7727983}-\frac{10629386 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{7727983}-\frac{10629386 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{7727983}-\frac{10629386 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{7727983}-\frac{10629386 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{7727983}+\frac{33513512 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{7727983}+\frac{33513512 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{7727983}+\frac{33513512 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{7727983}+\frac{33513512 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{7727983}+\frac{33513512 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{7727983}+\frac{33513512 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{7727983}+\frac{33513512 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{7727983}-\frac{58205329 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{7727983}-\frac{58205329 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{7727983}-\frac{58205329 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{7727983}-\frac{58205329 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{7727983}-\frac{58205329 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{7727983}-\frac{58205329 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{7727983}-\frac{58205329 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{7727983}+\frac{49899222 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{7727983}+\frac{49899222 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{7727983}+\frac{49899222 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{7727983}+\frac{49899222 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{7727983}+\frac{49899222 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{7727983}+\frac{49899222 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{7727983}+\frac{49899222 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{7727983}+\frac{3944364 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{7727983}+\frac{3944364 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{7727983}+\frac{3944364 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{7727983}+\frac{3944364 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{7727983}+\frac{3944364 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{7727983}+\frac{3944364 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{7727983}+\frac{3944364 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{7727983}-\frac{405479 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{145811}-\frac{405479 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{145811}-\frac{405479 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{145811}-\frac{405479 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{145811}-\frac{405479 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{145811}-\frac{405479 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{145811}-\frac{405479 \mathit{RootOf}\! \left(3 \textit{\_Z}^{7}-16 \textit{\_Z}^{6}+48 \textit{\_Z}^{5}-74 \textit{\_Z}^{4}+65 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{145811}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(6) = 212
a(n+7) = 3*a(n)-16*a(n+1)+48*a(n+2)-74*a(n+3)+65*a(n+4)-33*a(n+5)+9*a(n+6), 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) = 66
a \! \left(6\right) = 212
a \! \left(n +7\right) = 3 a \! \left(n \right)-16 a \! \left(n +1\right)+48 a \! \left(n +2\right)-74 a \! \left(n +3\right)+65 a \! \left(n +4\right)-33 a \! \left(n +5\right)+9 a \! \left(n +6\right), \quad n \geq 7

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21238/
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[14,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[14,x]*F[7,x]
F[7,x] = F[24,x]+F[8,x]
F[8,x] = F[0,x]*F[9,x]
F[9,x] = F[10,x]+F[18,x]
F[10,x] = F[11,x]*F[15,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[9,x]
F[14,x] = x
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[14,x]*F[15,x]
F[18,x] = F[19,x]*F[22,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[14,x]*F[19,x]
F[22,x] = F[23,x]
F[23,x] = F[14,x]*F[15,x]*F[20,x]
F[24,x] = F[19,x]*F[25,x]*F[30,x]
F[25,x] = F[26,x]
F[26,x] = F[14,x]*F[27,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[0,x]*F[11,x]
F[29,x] = F[19,x]*F[25,x]
F[30,x] = F[1,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[14,x]*F[33,x]
F[33,x] = F[15,x]+F[34,x]
F[34,x] = F[20,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[14,x]*F[37,x]
F[37,x] = F[16,x]+F[35,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_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{24}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{0}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{14}\! \left(x \right) F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{20}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right) F_{25}\! \left(x \right) F_{30}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{0}\! \left(x \right) F_{11}\! \left(x \right)
F_{29}\! \left(x \right) = F_{19}\! \left(x \right) F_{25}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{20}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{14}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{16}\! \left(x \right)+F_{35}\! \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_14(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_14(x)*F_7(x))
Eq(F_7(x), F_24(x) + F_8(x))
Eq(F_8(x), F_0(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_18(x))
Eq(F_10(x), F_11(x)*F_15(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_9(x))
Eq(F_14(x), x)
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_14(x)*F_15(x))
Eq(F_18(x), F_19(x)*F_22(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_14(x)*F_19(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_14(x)*F_15(x)*F_20(x))
Eq(F_24(x), F_19(x)*F_25(x)*F_30(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_14(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_0(x)*F_11(x))
Eq(F_29(x), F_19(x)*F_25(x))
Eq(F_30(x), F_1(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_14(x)*F_33(x))
Eq(F_33(x), F_15(x) + F_34(x))
Eq(F_34(x), F_20(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_14(x)*F_37(x))
Eq(F_37(x), F_16(x) + F_35(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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