0123_0132_0213_1320_2103
Counting sequence:
1, 1, 2, 6, 19, 53, 130, 302, 696, 1588, 3577, 7991, 17778, 39425, 87190, 192449, 424228, 934252, 2055929, 4521872, 9941692, 21851358, 48018039, 105502484, 231777551, 509147401, 1118378281, 2456484683, 5395414197, 11850174898, 26026561701, 57161418021, 125540798486, 275717041192, 605536039807, 1329886760213, 2920707466447, 6414466265378, 14087447220470, 30938807583836, 67947654146870, 149226202931129, 327729478321615, 719756798164520, 1580723589168053, 3471570682357332, 7624230717986885, 16744261739173256, 36773582887457728, 80761776486436805, 177368201976065278, 389534251248022440, 855491163961112262, 1878820969945736143, 4126247404617833160, 9062022282443936403, 19901920380122918289, 43708393179877168895, 95991923953001840019, 210816476423665459459, 462992977082139944542, 1016820412533060096615, 2233130525625815852633, 4904378274873052847936, 10770945080685375181698, 23655038698267579755309, 51950952442336069068229, 114094146853222074150233, 250572390534251682469804, 550304503973865911118672, 1208573085091298632111184, 2654255764513363204865190, 5829249178237524889212407, 12802137018996181887741627, 28115921491344196216191524, 61747897254889664464735447, 135610096098731011886638250, 297825496596256837229590375, 654081288737796496033137674, 1436486590852250791990920712, 3154797058438366455403529587, 6928532812823156801176293574, 15216372416049838996826466788, 33418040407487890449891676704, 73392356216069173907283996277, 161183536953725972778848730388, 353989624047622775646671243697, 777428367075041174424776646747, 1707380174090731049908175669649, 3749730756346225177815598827333, 8235120073696512928344950740951, 18085885903517786866896136416618, 39720036379287704889072583282023, 87232734873382484972062822471383, 191579634037229808747878353590114, 420745219453580418993269155708826, 924036318279187222375636717550157, 2029359046807036383792322761273625, 4456857440979061443648844480667165, 9788104416743422391321174435079944, 21496534125629282830354540491159910
Generating function in Maple syntax:
-(x^10+4*x^9+5*x^8-2*x^7-13*x^6-13*x^5-5*x^4-x^3+x^2+x-1)/(x^2+x-1)/(x^5+3*x^4+2*x^3+x^2+x-1)
Generating function in latex syntax:
-\frac{x^{10}+4 x^{9}+5 x^{8}-2 x^{7}-13 x^{6}-13 x^{5}-5 x^{4}-x^{3}+x^{2}+x -1}{\left(x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(-x**10 - 4*x**9 - 5*x**8 + 2*x**7 + 13*x**6 + 13*x**5 + 5*x**4 + x**3 - x**2 - x + 1)/((x**2 + x - 1)*(x**5 + 3*x**4 + 2*x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+x-1)*(x^5+3*x^4+2*x^3+x^2+x-1)*F(x)+x^10+4*x^9+5*x^8-2*x^7-13*x^6-13*x^5-5*x^4-x^3+x^2+x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+x^{10}+4 x^{9}+5 x^{8}-2 x^{7}-13 x^{6}-13 x^{5}-5 x^{4}-x^{3}+x^{2}+x -1 = 0
Explicit closed form in Maple syntax:
8915/7367*(RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)+1)*(RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)^3+2*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)^2+1)*(RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)+1)*(RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)+1)*(RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)^3+2*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)^2+1)*(RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^3+2*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^2+1)*(RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+1)*(RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)+1)*(RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^3+2*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^2+1)*(RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)^3+2*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)^2+1)*(1747/1783*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^(1-n)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+2937/1783*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^(3-n)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+5191/8915*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^(-n+2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^(-n+4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+1747/1783*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^(1-n)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+2937/1783*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^(3-n)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+5191/8915*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^(-n+2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^(-n+4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+1747/1783*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)^(1-n)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+2937/1783*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)^(3-n)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+5191/8915*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)^(-n+2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)^(-n+4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+1747/1783*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)^(1-n)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+2937/1783*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)^(3-n)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+5191/8915*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)^(-n+2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)^(-n+4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+1747/1783*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)^(1-n)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)+2937/1783*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)^(3-n)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)+5191/8915*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)^(-n+2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)-1914/8915*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)^(-n)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 1)*(-1914/8915*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)^(-n)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+(-1914/8915*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3)^(-n)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+(-1914/8915*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)^(-n)*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 4)*(-1914/8915*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)^(-n)+(-14734/8915*(1/2*5^(1/2)-1/2)^(-n)-14734/8915*(-1/2*5^(1/2)-1/2)^(-n)-7367/8915*piecewise(n = 0,-6,Or(n = 1,n = 3),1,0))*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)+RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 5)^(-n+4)))*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 3))*RootOf(_Z^5+3*_Z^4+2*_Z^3+_Z^2+_Z-1,index = 2)))
Explicit closed form in latex syntax:
\frac{8915 \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)+1\right) \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{3}+2 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{2}+1\right) \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)+1\right) \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)+1\right) \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{3}+2 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{2}+1\right) \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{3}+2 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{2}+1\right) \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)+1\right) \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)+1\right) \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{3}+2 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{2}+1\right) \left(\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{3}+2 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{2}+1\right) \left(\frac{1747 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{1-n} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{1783}+\frac{2937 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{3-n} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{1783}+\frac{5191 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +2} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{8915}+\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +4} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)+\frac{1747 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{1-n} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{1783}+\frac{2937 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{3-n} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{1783}+\frac{5191 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +2} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{8915}+\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +4} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)+\frac{1747 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{1-n} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{1783}+\frac{2937 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{3-n} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{1783}+\frac{5191 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +2} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{8915}+\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +4} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)+\frac{1747 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{1-n} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{1783}+\frac{2937 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{3-n} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{1783}+\frac{5191 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +2} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{8915}+\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +4} \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)+\frac{1747 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{1-n} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)}{1783}+\frac{2937 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{3-n} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)}{1783}+\frac{5191 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +2} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)}{8915}-\frac{1914 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{8915}+\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right) \left(-\frac{1914 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{8915}+\left(-\frac{1914 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{8915}+\left(-\frac{1914 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n} \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)}{8915}+\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right) \left(-\frac{1914 \mathit{RootOf}\left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{8915}+\left(-\frac{14734 \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{8915}-\frac{14734 \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{8915}-\frac{7367 \left(\left\{\begin{array}{cc}-6 & n =0 \\ 1 & n =1\text{ or } n =3 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)}{8915}\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)+\mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}\right)\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)\right) \mathit{RootOf}\! \left(\textit{\_Z}^{5}+3 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)\right)\right)}{7367}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(6) = 130
a(7) = 302
a(8) = 696
a(9) = 1588
a(10) = 3577
a(n+2) = -1/4*a(n)-a(n+1)+1/4*a(n+5)+1/2*a(n+6)-1/4*a(n+7), n >= 11
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) = 53
a \! \left(6\right) = 130
a \! \left(7\right) = 302
a \! \left(8\right) = 696
a \! \left(9\right) = 1588
a \! \left(10\right) = 3577
a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{4}-a \! \left(n +1\right)+\frac{a \! \left(n +5\right)}{4}+\frac{a \! \left(n +6\right)}{2}-\frac{a \! \left(n +7\right)}{4}, \quad n \geq 11
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16019/
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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[4,x]*F[7,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]+F[17,x]+F[54,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[33,x]
F[19,x] = F[20,x]+F[7,x]
F[20,x] = F[16,x]+F[21,x]+F[25,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[24,x]
F[23,x] = F[4,x]
F[24,x] = F[12,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[11,x]+F[27,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[23,x]
F[31,x] = F[32,x]
F[32,x] = F[28,x]*F[4,x]
F[33,x] = F[34,x]+F[40,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,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] = 2*F[16,x]+F[41,x]+F[51,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[38,x]
F[44,x] = F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[28,x]+F[7,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[45,x]+F[47,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[64,x]
F[56,x] = F[2,x]+F[57,x]
F[57,x] = F[16,x]+F[39,x]+F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[62,x]
F[60,x] = F[4,x]+F[61,x]
F[61,x] = F[13,x]+F[16,x]+F[21,x]
F[62,x] = F[38,x]+F[63,x]
F[63,x] = 2*F[16,x]+F[41,x]+F[46,x]
F[64,x] = F[47,x]+F[65,x]
F[65,x] = 2*F[16,x]+F[46,x]+F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[12,x]+F[69,x]
F[69,x] = F[32,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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{4}\! \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_{54}\! \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_{33}\! \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_{25}\! \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_{24}\! \left(x \right)
F_{23}\! \left(x \right) = F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \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_{23}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{40}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{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) = 2 F_{16}\! \left(x \right)+F_{41}\! \left(x \right)+F_{51}\! \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_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{38}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{64}\! \left(x \right)
F_{56}\! \left(x \right) = F_{2}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{16}\! \left(x \right)+F_{39}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{62}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{13}\! \left(x \right)+F_{16}\! \left(x \right)+F_{21}\! \left(x \right)
F_{62}\! \left(x \right) = F_{38}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{41}\! \left(x \right)+F_{46}\! \left(x \right)
F_{64}\! \left(x \right) = F_{47}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{46}\! \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_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{32}\! \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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_4(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_54(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_33(x))
Eq(F_19(x), F_20(x) + F_7(x))
Eq(F_20(x), F_16(x) + F_21(x) + F_25(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_24(x))
Eq(F_23(x), F_4(x))
Eq(F_24(x), F_12(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_11(x) + F_27(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_23(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_28(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_40(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(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), 2*F_16(x) + F_41(x) + F_51(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_38(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_28(x) + F_7(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_45(x) + F_47(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_64(x))
Eq(F_56(x), F_2(x) + F_57(x))
Eq(F_57(x), F_16(x) + F_39(x) + F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_62(x))
Eq(F_60(x), F_4(x) + F_61(x))
Eq(F_61(x), F_13(x) + F_16(x) + F_21(x))
Eq(F_62(x), F_38(x) + F_63(x))
Eq(F_63(x), 2*F_16(x) + F_41(x) + F_46(x))
Eq(F_64(x), F_47(x) + F_65(x))
Eq(F_65(x), 2*F_16(x) + F_46(x) + F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_12(x) + F_69(x))
Eq(F_69(x), F_32(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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": 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