0123_0321_1032_1302_2031
Counting sequence:
1, 1, 2, 6, 19, 50, 135, 378, 1063, 2967, 8280, 23150, 64753, 181069, 506294, 1415783, 3959173, 11071546, 30960654, 86579159, 242112636, 677051146, 1893325962, 5294553618, 14805850464, 41403529998, 115782087664, 323776544374, 905418561617, 2531939965635, 7080393818488, 19799828317019, 55368841273527, 154835109434463, 432985602767452, 1210814090478780, 3385957298233593, 9468593829294122, 26478263370582010, 74044619904683917, 207060624033356876, 579030618028553975, 1619218806954513301, 4528032651748165090, 12662328035740282026, 35409318707716529276, 99019694309423885787, 276901680658290699799, 774336270033191196528, 2165377464172366194183, 6055327309083126543558, 16933301203512097092213, 47352797794885967635681, 132418801984003556388913, 370299959779197474056237, 1035518054520987692643150, 2895754139099342792263789, 8097774826330681173303425, 22644863475305314853178051, 63324784007062286316767004, 177083349339425652477931099, 495201256584970121916435033, 1384795834493158274780814603, 3872485131508459876870885403, 10829135039420992534451906377, 30282922133863669230139020964, 84684083228006624080937084378, 236813142419588064924453516671, 662231464106979108291444395875, 1851884179959255215155630823575, 5178665167484935710188040577674, 14481776564186554096855895989070, 40497279834153899920311326856426, 113247823338302492431056438985717, 316689652820760172953460400929743, 885600563854836606610887277483239, 2476520314807682089977666788474307, 6925416626835456465455136192437505, 19366445398600942376135120088307606, 54156916123091576744908363506671023, 151446045136164126606660790898105537, 423508320437869006096498756992407922, 1184311530346295866200559728450728554, 3311844734150750833647680902516085876, 9261343204110233223938189263753623828, 25898701427593721917568594999632436067, 72423915284552450973488390031875210575, 202528436408611466944133288975575015356, 566356670900196433744973815627500158838, 1583776996263398628023529487792340969813, 4428922095869752034025425939320111422772, 12385172267031134252008167769270695907753, 34634271898140917063851391598533915993006, 96852329870894497213151688559106317151363, 270840796913766736328243719528594454777515, 757387430644849972362456342400495324067506, 2117981214925489454901698590071436231314054, 5922786998139041366545586586483092570994317, 16562661452386379986394719282415589968067927, 46316329537523210515571633755867052241748174, 129520390668817308196567749861411617808086193
Generating function in Maple syntax:
-(x^4-x^3-2*x+1)/(x^9+2*x^8-2*x^7-3*x^6-5*x^5+2*x^3-x^2+3*x-1)
Generating function in latex syntax:
-\frac{x^{4}-x^{3}-2 x +1}{x^{9}+2 x^{8}-2 x^{7}-3 x^{6}-5 x^{5}+2 x^{3}-x^{2}+3 x -1}
Generating function in sympy syntax:
(-x**4 + x**3 + 2*x - 1)/(x**9 + 2*x**8 - 2*x**7 - 3*x**6 - 5*x**5 + 2*x**3 - x**2 + 3*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^9+2*x^8-2*x^7-3*x^6-5*x^5+2*x^3-x^2+3*x-1)*F(x)+x^4-x^3-2*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{9}+2 x^{8}-2 x^{7}-3 x^{6}-5 x^{5}+2 x^{3}-x^{2}+3 x -1\right) F \! \left(x \right)+x^{4}-x^{3}-2 x +1 = 0
Explicit closed form in Maple syntax:
586857973387/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 1)^(-n+7)+586857973387/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 2)^(-n+7)+586857973387/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 3)^(-n+7)+586857973387/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 4)^(-n+7)+586857973387/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 5)^(-n+7)+586857973387/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 6)^(-n+7)+586857973387/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 7)^(-n+7)+586857973387/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 8)^(-n+7)+586857973387/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 9)^(-n+7)+1740086537342/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 1)^(-n+6)+1740086537342/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 2)^(-n+6)+1740086537342/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 3)^(-n+6)+1740086537342/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 4)^(-n+6)+1740086537342/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 5)^(-n+6)+1740086537342/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 6)^(-n+6)+1740086537342/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 7)^(-n+6)+1740086537342/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 8)^(-n+6)+1740086537342/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 9)^(-n+6)+591137513713/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 1)^(-n+5)+591137513713/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 2)^(-n+5)+591137513713/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 3)^(-n+5)+591137513713/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 4)^(-n+5)+591137513713/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 5)^(-n+5)+591137513713/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 6)^(-n+5)+591137513713/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 7)^(-n+5)+591137513713/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 8)^(-n+5)+591137513713/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 9)^(-n+5)-2806475451896/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 1)^(-n+4)-2806475451896/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 2)^(-n+4)-2806475451896/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 3)^(-n+4)-2806475451896/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 4)^(-n+4)-2806475451896/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 5)^(-n+4)-2806475451896/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 6)^(-n+4)-2806475451896/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 7)^(-n+4)-2806475451896/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 8)^(-n+4)-2806475451896/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 9)^(-n+4)-8332512980487/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 1)^(-n+3)-8332512980487/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 2)^(-n+3)-8332512980487/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 3)^(-n+3)-8332512980487/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 4)^(-n+3)-8332512980487/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 5)^(-n+3)-8332512980487/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 6)^(-n+3)-8332512980487/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 7)^(-n+3)-8332512980487/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 8)^(-n+3)-8332512980487/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 9)^(-n+3)-2997646538106/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 1)^(-n+2)-2997646538106/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 2)^(-n+2)-2997646538106/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 3)^(-n+2)-2997646538106/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 4)^(-n+2)-2997646538106/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 5)^(-n+2)-2997646538106/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 6)^(-n+2)-2997646538106/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 7)^(-n+2)-2997646538106/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 8)^(-n+2)-2997646538106/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 9)^(-n+2)+1503472960610/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 1)^(-n+1)+1503472960610/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 2)^(-n+1)+1503472960610/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 3)^(-n+1)+1503472960610/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 4)^(-n+1)+1503472960610/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 5)^(-n+1)+1503472960610/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 6)^(-n+1)+1503472960610/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 7)^(-n+1)+1503472960610/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 8)^(-n+1)+1503472960610/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 9)^(-n+1)+2036055702173/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 1)^(-n-1)+2036055702173/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 2)^(-n-1)+2036055702173/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 3)^(-n-1)+2036055702173/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 4)^(-n-1)+2036055702173/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 5)^(-n-1)+2036055702173/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 6)^(-n-1)+2036055702173/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 7)^(-n-1)+2036055702173/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 8)^(-n-1)+2036055702173/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 9)^(-n-1)+7925245888861/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 1)^(-n)+7925245888861/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 2)^(-n)+7925245888861/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 3)^(-n)+7925245888861/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 4)^(-n)+7925245888861/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 5)^(-n)+7925245888861/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 6)^(-n)+7925245888861/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 7)^(-n)+7925245888861/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 8)^(-n)+7925245888861/47170387365229*RootOf(_Z^9+2*_Z^8-2*_Z^7-3*_Z^6-5*_Z^5+2*_Z^3-_Z^2+3*_Z-1,index = 9)^(-n)
Explicit closed form in latex syntax:
\frac{586857973387 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +7}}{47170387365229}+\frac{586857973387 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +7}}{47170387365229}+\frac{586857973387 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +7}}{47170387365229}+\frac{586857973387 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +7}}{47170387365229}+\frac{586857973387 \mathit{RootOf}\! 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\left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{47170387365229}+\frac{2036055702173 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{47170387365229}+\frac{2036055702173 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{47170387365229}+\frac{2036055702173 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{47170387365229}+\frac{2036055702173 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{47170387365229}+\frac{2036055702173 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{47170387365229}+\frac{2036055702173 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =8\right)^{-n -1}}{47170387365229}+\frac{2036055702173 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =9\right)^{-n -1}}{47170387365229}+\frac{7925245888861 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{47170387365229}+\frac{7925245888861 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{47170387365229}+\frac{7925245888861 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{47170387365229}+\frac{7925245888861 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{47170387365229}+\frac{7925245888861 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{47170387365229}+\frac{7925245888861 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{47170387365229}+\frac{7925245888861 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{47170387365229}+\frac{7925245888861 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =8\right)^{-n}}{47170387365229}+\frac{7925245888861 \mathit{RootOf}\! \left(\textit{\_Z}^{9}+2 \textit{\_Z}^{8}-2 \textit{\_Z}^{7}-3 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+2 \textit{\_Z}^{3}-\textit{\_Z}^{2}+3 \textit{\_Z} -1, \mathit{index} =9\right)^{-n}}{47170387365229}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 50
a(6) = 135
a(7) = 378
a(8) = 1063
a(n+4) = 1/5*a(n)+2/5*a(n+1)-2/5*a(n+2)-3/5*a(n+3)+2/5*a(n+6)-1/5*a(n+7)+3/5*a(n+8)-1/5*a(n+9), n >= 9
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) = 50
a \! \left(6\right) = 135
a \! \left(7\right) = 378
a \! \left(8\right) = 1063
a \! \left(n +4\right) = \frac{a \! \left(n \right)}{5}+\frac{2 a \! \left(n +1\right)}{5}-\frac{2 a \! \left(n +2\right)}{5}-\frac{3 a \! \left(n +3\right)}{5}+\frac{2 a \! \left(n +6\right)}{5}-\frac{a \! \left(n +7\right)}{5}+\frac{3 a \! \left(n +8\right)}{5}-\frac{a \! \left(n +9\right)}{5}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17719/
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[23,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[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[4,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[15,x]
F[23,x] = F[2,x]+F[24,x]
F[24,x] = F[18,x]+F[25,x]+F[64,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[33,x]
F[27,x] = F[15,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[4,x]
F[32,x] = F[19,x]
F[33,x] = F[34,x]+F[58,x]
F[34,x] = F[18,x]+F[25,x]+F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[2,x]+F[37,x]
F[37,x] = F[18,x]+F[38,x]+F[57,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[42,x]
F[40,x] = F[4,x]+F[41,x]
F[41,x] = F[29,x]
F[42,x] = F[37,x]+F[43,x]
F[43,x] = 2*F[18,x]+F[44,x]+F[51,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[19,x]
F[47,x] = F[48,x]
F[48,x] = 2*F[18,x]+F[44,x]+F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[37,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[37,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[37,x]
F[57,x] = F[2,x]*F[4,x]
F[58,x] = 2*F[18,x]+F[51,x]+F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[19,x]
F[62,x] = F[63,x]
F[63,x] = 2*F[18,x]+F[49,x]+F[59,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[71,x]
F[66,x] = F[2,x]+F[67,x]
F[67,x] = F[18,x]+F[38,x]+F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[2,x]+F[70,x]
F[70,x] = F[57,x]
F[71,x] = F[72,x]+F[74,x]
F[72,x] = F[73,x]
F[73,x] = F[4,x]*F[69,x]
F[74,x] = 2*F[18,x]+F[55,x]+F[75,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[72,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_{23}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)
F_{23}\! \left(x \right) = F_{2}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \left(x \right)+F_{64}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{33}\! \left(x \right)
F_{27}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \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_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{19}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{58}\! \left(x \right)
F_{34}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \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_{2}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)+F_{57}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{29}\! \left(x \right)
F_{42}\! \left(x \right) = F_{37}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{51}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{19}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{37}\! \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_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{37}\! \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_{37}\! \left(x \right)
F_{57}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{51}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{19}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{59}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{71}\! \left(x \right)
F_{66}\! \left(x \right) = F_{2}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{2}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{57}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{74}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{74}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{72}\! \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_23(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_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_4(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_15(x))
Eq(F_23(x), F_2(x) + F_24(x))
Eq(F_24(x), F_18(x) + F_25(x) + F_64(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_33(x))
Eq(F_27(x), F_15(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_4(x))
Eq(F_32(x), F_19(x))
Eq(F_33(x), F_34(x) + F_58(x))
Eq(F_34(x), F_18(x) + F_25(x) + F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_2(x) + F_37(x))
Eq(F_37(x), F_18(x) + F_38(x) + F_57(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_42(x))
Eq(F_40(x), F_4(x) + F_41(x))
Eq(F_41(x), F_29(x))
Eq(F_42(x), F_37(x) + F_43(x))
Eq(F_43(x), 2*F_18(x) + F_44(x) + F_51(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_19(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), 2*F_18(x) + F_44(x) + F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_37(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_37(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_37(x))
Eq(F_57(x), F_2(x)*F_4(x))
Eq(F_58(x), 2*F_18(x) + F_51(x) + F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_19(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), 2*F_18(x) + F_49(x) + F_59(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_71(x))
Eq(F_66(x), F_2(x) + F_67(x))
Eq(F_67(x), F_18(x) + F_38(x) + F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_2(x) + F_70(x))
Eq(F_70(x), F_57(x))
Eq(F_71(x), F_72(x) + F_74(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_4(x)*F_69(x))
Eq(F_74(x), 2*F_18(x) + F_55(x) + F_75(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_72(x))
Pack JSON:
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Specification JSON:
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