0123_0132_0321_1302_2103
Counting sequence:
1, 1, 2, 6, 19, 49, 117, 297, 784, 2046, 5248, 13456, 34718, 89728, 231475, 596499, 1537748, 3966204, 10229575, 26379123, 68021387, 175410317, 452352952, 1166524932, 3008185240, 7757390624, 20004563504, 51587311796, 133031953489, 343058910801, 884670665082, 2281364704118, 5883121055619, 15171229698765, 39123146374645, 100889690031525, 260171553224808, 670923224712250, 1730158285998656, 4461684423933424, 11505668671754534, 29670500880162276, 76513468814984395, 197310821638650283, 508819701362197548, 1312130203610310340, 3383685156992662751, 8725753899021177863, 22501733339083939099, 58026848928595799225, 149638036585237315344, 385882439012594731528, 995102984075076360064, 2566144112315087374480, 6617501615990245317488, 17065030536465755713048, 44006829783982603725713, 113483598139338768960689, 292648370943179452642098, 754673542431745044891654, 1946131303621725505313923, 5018629696135864120957241, 12941903755448026846031413, 33374224231815498817013793, 86064528381817608549516032, 221940830568383260602521590, 572334888711130594152363984, 1475921415618271985509958272, 3806065413880624216387267518, 9814976448912055675600751160, 25310590391161276749326615219, 65270251974996420811254598931, 168317124454245614404945532004, 434051555299631043554560816396, 1119320171782298084211336874407, 2886471967814691007558428253771, 7443554249284529785207791649963, 19195232269651762918691771988037, 49500135223881285328647133817480, 127649582602679547337720825143628, 329179220722150391244301135851064, 848878289657388948891578932994320, 2189063905889383233898734721200064, 5645097586371015932890121652661548, 14557421861425622806357137952164801, 37540277738182682966536357850427361, 96807832188623740435228754364519466, 249645365929954346758647042305556918, 643778579907345582991628442904286995, 1660158434760633305750173333612977013, 4281170754242157812129612328267849653, 11040164988603039082915173799887639261, 28470072784366706328957484224957309784, 73417837974693136598407630073487488114, 189327894371879237992366056917207274928, 488233548904628896474006148458315475072, 1259043201567523008088804942369472310502, 3246785860926259127843450625744588659068, 8372721772840070666004981364486214663531, 21591343836082554759151552208770211554315, 55679161603121965164697624388688649292540
Generating function in Maple syntax:
-1/(x^7+5*x^6+10*x^5+8*x^4+3*x^3+x^2+x-1)
Generating function in latex syntax:
-\frac{1}{x^{7}+5 x^{6}+10 x^{5}+8 x^{4}+3 x^{3}+x^{2}+x -1}
Generating function in sympy syntax:
-1/(x**7 + 5*x**6 + 10*x**5 + 8*x**4 + 3*x**3 + x**2 + x - 1)
Implicit equation for the generating function in Maple syntax:
(x^7+5*x^6+10*x^5+8*x^4+3*x^3+x^2+x-1)*F(x)+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}+5 x^{6}+10 x^{5}+8 x^{4}+3 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+1 = 0
Explicit closed form in Maple syntax:
-12981/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 1)^(-n+5)-12981/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 2)^(-n+5)-12981/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 3)^(-n+5)-12981/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 4)^(-n+5)-12981/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 5)^(-n+5)-12981/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 6)^(-n+5)-12981/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 7)^(-n+5)-12512/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 1)^(-n+4)-12512/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 2)^(-n+4)-12512/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 3)^(-n+4)-12512/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 4)^(-n+4)-12512/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 5)^(-n+4)-12512/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 6)^(-n+4)-12512/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 7)^(-n+4)+70451/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 1)^(-n+3)+70451/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 2)^(-n+3)+70451/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 3)^(-n+3)+70451/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 4)^(-n+3)+70451/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 5)^(-n+3)+70451/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 6)^(-n+3)+70451/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 7)^(-n+3)+422623/3449950*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 1)^(-n+2)+422623/3449950*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 2)^(-n+2)+422623/3449950*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 3)^(-n+2)+422623/3449950*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 4)^(-n+2)+422623/3449950*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 5)^(-n+2)+422623/3449950*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 6)^(-n+2)+422623/3449950*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 7)^(-n+2)+42237/344995*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 1)^(-n+1)+42237/344995*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 2)^(-n+1)+42237/344995*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 3)^(-n+1)+42237/344995*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 4)^(-n+1)+42237/344995*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 5)^(-n+1)+42237/344995*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 6)^(-n+1)+42237/344995*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 7)^(-n+1)+174179/3449950*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 1)^(-n-1)+174179/3449950*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 2)^(-n-1)+174179/3449950*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 3)^(-n-1)+174179/3449950*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 4)^(-n-1)+174179/3449950*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 5)^(-n-1)+174179/3449950*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 6)^(-n-1)+174179/3449950*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 7)^(-n-1)+351739/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 1)^(-n)+351739/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 2)^(-n)+351739/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 3)^(-n)+351739/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 4)^(-n)+351739/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 5)^(-n)+351739/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 6)^(-n)+351739/1724975*RootOf(_Z^7+5*_Z^6+10*_Z^5+8*_Z^4+3*_Z^3+_Z^2+_Z-1,index = 7)^(-n)
Explicit closed form in latex syntax:
-\frac{12981 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{1724975}-\frac{12981 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{1724975}-\frac{12981 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{1724975}-\frac{12981 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{1724975}-\frac{12981 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{1724975}-\frac{12981 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{1724975}-\frac{12981 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{1724975}-\frac{12512 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{1724975}-\frac{12512 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{1724975}-\frac{12512 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{1724975}-\frac{12512 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{1724975}-\frac{12512 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{1724975}-\frac{12512 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{1724975}-\frac{12512 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{1724975}+\frac{70451 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{1724975}+\frac{70451 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{1724975}+\frac{70451 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{1724975}+\frac{70451 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{1724975}+\frac{70451 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{1724975}+\frac{70451 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{1724975}+\frac{70451 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{1724975}+\frac{422623 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{3449950}+\frac{422623 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{3449950}+\frac{422623 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{3449950}+\frac{422623 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{3449950}+\frac{422623 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{3449950}+\frac{422623 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{3449950}+\frac{422623 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{3449950}+\frac{42237 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{344995}+\frac{42237 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{344995}+\frac{42237 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{344995}+\frac{42237 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{344995}+\frac{42237 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{344995}+\frac{42237 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{344995}+\frac{42237 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{344995}+\frac{174179 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{3449950}+\frac{174179 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{3449950}+\frac{174179 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{3449950}+\frac{174179 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{3449950}+\frac{174179 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{3449950}+\frac{174179 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{3449950}+\frac{174179 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{3449950}+\frac{351739 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{1724975}+\frac{351739 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{1724975}+\frac{351739 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{1724975}+\frac{351739 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{1724975}+\frac{351739 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{1724975}+\frac{351739 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{1724975}+\frac{351739 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+5 \textit{\_Z}^{6}+10 \textit{\_Z}^{5}+8 \textit{\_Z}^{4}+3 \textit{\_Z}^{3}+\textit{\_Z}^{2}+\textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{1724975}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 49
a(6) = 117
a(n+7) = a(n)+5*a(n+1)+10*a(n+2)+8*a(n+3)+3*a(n+4)+a(n+5)+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) = 19
a \! \left(5\right) = 49
a \! \left(6\right) = 117
a \! \left(n +7\right) = a \! \left(n \right)+5 a \! \left(n +1\right)+10 a \! \left(n +2\right)+8 a \! \left(n +3\right)+3 a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16261/
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[19,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[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[4,x]
F[18,x] = F[12,x]*F[4,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,x]+F[21,x]+F[51,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[30,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[15,x]+F[25,x]+F[28,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[18,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[11,x]
F[30,x] = F[31,x]+F[44,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[2,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[2,x]*F[4,x]
F[37,x] = F[38,x]+F[40,x]
F[38,x] = F[39,x]
F[39,x] = F[34,x]*F[4,x]
F[40,x] = 2*F[15,x]+F[41,x]+F[43,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[35,x]
F[43,x] = F[38,x]*F[4,x]
F[44,x] = 2*F[15,x]+F[45,x]+F[49,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[35,x]
F[48,x] = F[43,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[61,x]
F[53,x] = F[2,x]+F[54,x]
F[54,x] = F[15,x]+F[36,x]+F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[59,x]
F[57,x] = F[4,x]+F[58,x]
F[58,x] = F[15,x]+F[18,x]+F[25,x]
F[59,x] = F[35,x]+F[60,x]
F[60,x] = 2*F[15,x]+F[43,x]+F[45,x]
F[61,x] = F[38,x]+F[62,x]
F[62,x] = 2*F[15,x]+F[41,x]+F[43,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_{19}\! \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_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{51}\! \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_{30}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{44}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{40}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{35}\! \left(x \right)
F_{43}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{35}\! \left(x \right)
F_{48}\! \left(x \right) = F_{43}\! \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_{61}\! \left(x \right)
F_{53}\! \left(x \right) = F_{2}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{15}\! \left(x \right)+F_{36}\! \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_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{59}\! \left(x \right) = F_{35}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{61}\! \left(x \right) = F_{38}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \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_19(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_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_12(x)*F_4(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(x) + F_21(x) + F_51(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_30(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_15(x) + F_25(x) + F_28(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_18(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_11(x))
Eq(F_30(x), F_31(x) + F_44(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_2(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_2(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_40(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_34(x)*F_4(x))
Eq(F_40(x), 2*F_15(x) + F_41(x) + F_43(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_35(x))
Eq(F_43(x), F_38(x)*F_4(x))
Eq(F_44(x), 2*F_15(x) + F_45(x) + F_49(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_35(x))
Eq(F_48(x), F_43(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_61(x))
Eq(F_53(x), F_2(x) + F_54(x))
Eq(F_54(x), F_15(x) + F_36(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_59(x))
Eq(F_57(x), F_4(x) + F_58(x))
Eq(F_58(x), F_15(x) + F_18(x) + F_25(x))
Eq(F_59(x), F_35(x) + F_60(x))
Eq(F_60(x), 2*F_15(x) + F_43(x) + F_45(x))
Eq(F_61(x), F_38(x) + F_62(x))
Eq(F_62(x), 2*F_15(x) + F_41(x) + F_43(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, 3, 2, 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, 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, 3, 2, 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, 1, 0, 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