0132_0231_1302_2103
Counting sequence:
1, 1, 2, 6, 20, 62, 179, 505, 1431, 4093, 11759, 33785, 96952, 278014, 797123, 2285825, 6555574, 18801498, 53922336, 154645652, 443510461, 1271950763, 3647856113, 10461780619, 30003613185, 86048120529, 246779532516, 707745094944, 2029759635117, 5821197948169, 16694757981690, 47879310393750, 137314260984840, 393806971756894, 1129408772961107, 3239059408756497, 9289378750941367, 26641239536622477, 76405071106935251, 219123996935261217, 628431141240404172, 1802293244038046822, 5168841459887892143, 14823848519604473321, 42513682541322637598, 121926043755259537778, 349674722516753283156, 1002840802516327724868, 2876071990429699795421, 8248358137582093344963, 23655670717634893359249, 67842684297556332500451, 194567715607685569037461, 558005573463971131784849, 1600318012905509371302104, 4589591689078711733233584, 13162603746624290572423417, 37749357487054767902022865, 108262317860242481978264610, 310488184401324404895713462, 890456758715242478408027516, 2553763006056192347617120702, 7324000213677668533679057971, 21004681719776714558392424809, 60239847252486614446512391847, 172763350828888191391989967437, 495472295348382602825999703847, 1420977274867404442634706396121, 4075255941948977805188357631952, 11687527510909817038854734627582, 33518949794585449825750692640699, 96129826798968485563541179874993, 275693112613344282713831552256598, 790667110025935436011525200226906, 2267573799544042532690957613182856, 6503231095839994473414796497435892, 18650777626026644298744531708049597, 53489027366413400119447267694282555, 153402507175484691306736515977142113, 439946852024850074047230669366771851, 1261734479901037353856381616459068089, 3618559583831775572437842327515854657, 10377756707392036311468050346583709212, 29762625647793450686191941258687690288, 85356971687313760780787781640626764709, 244797374460443158003807711185776654809, 702060456903872739264930740851171889594, 2013456583161681481726508995121078269926, 5774442033319353394981831937316512852592, 16560665412415200562372616084869749549278, 47494742750809689803940716622505553648787, 136211349773089332152510487864761331543809, 390643905670814121217214747580785999883495, 1120337338202465615081738986299618592883389, 3213043216981027922305219837866664407479531, 9214766269195001655272969280986752800614305, 26427256548287924779677521145131143867839748, 75791383988086649429165812993733430837323974, 217363988438737158392512954255918281098292135, 623383569264575615312759681841478484571874137, 1787817187291669084027095614705899436960381598
Generating function in Maple syntax:
-(x-1)^4/(x^7-x^6-2*x^4+9*x^3-9*x^2+5*x-1)
Generating function in latex syntax:
-\frac{\left(x -1\right)^{4}}{x^{7}-x^{6}-2 x^{4}+9 x^{3}-9 x^{2}+5 x -1}
Generating function in sympy syntax:
-(x - 1)**4/(x**7 - x**6 - 2*x**4 + 9*x**3 - 9*x**2 + 5*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^7-x^6-2*x^4+9*x^3-9*x^2+5*x-1)*F(x)+(x-1)^4 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}-x^{6}-2 x^{4}+9 x^{3}-9 x^{2}+5 x -1\right) F \! \left(x \right)+\left(x -1\right)^{4} = 0
Explicit closed form in Maple syntax:
86481/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n+5)+86481/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n+5)+86481/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n+5)+86481/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n+5)+86481/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n+5)+86481/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 6)^(-n+5)+86481/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 7)^(-n+5)+50221/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n+4)+50221/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n+4)+50221/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n+4)+50221/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n+4)+50221/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n+4)+50221/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 6)^(-n+4)+50221/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 7)^(-n+4)+67847/7593574*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n+3)+67847/7593574*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n+3)+67847/7593574*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n+3)+67847/7593574*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n+3)+67847/7593574*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n+3)+67847/7593574*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 6)^(-n+3)+67847/7593574*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 7)^(-n+3)-240599/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n+2)-240599/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n+2)-240599/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n+2)-240599/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n+2)-240599/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n+2)-240599/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 6)^(-n+2)-240599/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 7)^(-n+2)+673223/7593574*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n+1)+673223/7593574*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n+1)+673223/7593574*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n+1)+673223/7593574*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n+1)+673223/7593574*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n+1)+673223/7593574*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 6)^(-n+1)+673223/7593574*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 7)^(-n+1)+253101/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n-1)+253101/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n-1)+253101/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n-1)+253101/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n-1)+253101/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n-1)+253101/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 6)^(-n-1)+253101/3796787*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 7)^(-n-1)+11993/122477*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 1)^(-n)+11993/122477*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 2)^(-n)+11993/122477*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 3)^(-n)+11993/122477*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 4)^(-n)+11993/122477*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 5)^(-n)+11993/122477*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 6)^(-n)+11993/122477*RootOf(_Z^7-_Z^6-2*_Z^4+9*_Z^3-9*_Z^2+5*_Z-1,index = 7)^(-n)
Explicit closed form in latex syntax:
\frac{86481 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{3796787}+\frac{86481 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{3796787}+\frac{86481 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{3796787}+\frac{86481 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{3796787}+\frac{86481 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{3796787}+\frac{86481 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{3796787}+\frac{86481 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{3796787}+\frac{50221 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{3796787}+\frac{50221 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{3796787}+\frac{50221 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{3796787}+\frac{50221 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{3796787}+\frac{50221 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{3796787}+\frac{50221 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{3796787}+\frac{50221 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{3796787}+\frac{67847 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{7593574}+\frac{67847 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{7593574}+\frac{67847 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{7593574}+\frac{67847 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{7593574}+\frac{67847 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{7593574}+\frac{67847 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{7593574}+\frac{67847 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{7593574}-\frac{240599 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{3796787}-\frac{240599 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{3796787}-\frac{240599 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{3796787}-\frac{240599 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{3796787}-\frac{240599 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{3796787}-\frac{240599 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{3796787}-\frac{240599 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{3796787}+\frac{673223 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{7593574}+\frac{673223 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{7593574}+\frac{673223 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{7593574}+\frac{673223 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{7593574}+\frac{673223 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{7593574}+\frac{673223 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{7593574}+\frac{673223 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{7593574}+\frac{253101 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{3796787}+\frac{253101 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{3796787}+\frac{253101 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{3796787}+\frac{253101 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{3796787}+\frac{253101 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{3796787}+\frac{253101 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{3796787}+\frac{253101 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{3796787}+\frac{11993 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{122477}+\frac{11993 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{122477}+\frac{11993 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{122477}+\frac{11993 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{122477}+\frac{11993 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{122477}+\frac{11993 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{122477}+\frac{11993 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-\textit{\_Z}^{6}-2 \textit{\_Z}^{4}+9 \textit{\_Z}^{3}-9 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{122477}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 62
a(6) = 179
a(n+1) = a(n)-2*a(n+3)+9*a(n+4)-9*a(n+5)+5*a(n+6)-a(n+7), n >= 7
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 20
a \! \left(5\right) = 62
a \! \left(6\right) = 179
a \! \left(n +1\right) = a \! \left(n \right)-2 a \! \left(n +3\right)+9 a \! \left(n +4\right)-9 a \! \left(n +5\right)+5 a \! \left(n +6\right)-a \! \left(n +7\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15358/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[20,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[61,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[34,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[15,x]+F[22,x]+F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[13,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[7,x]
F[33,x] = F[30,x]
F[34,x] = F[35,x]+F[48,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[2,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[38,x]*F[4,x]
F[41,x] = F[35,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[39,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[45,x]
F[48,x] = 2*F[22,x]+F[49,x]+F[54,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[39,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[51,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[41,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[35,x]
F[60,x] = F[57,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[74,x]
F[63,x] = F[2,x]+F[64,x]
F[64,x] = F[22,x]+F[65,x]+F[73,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[70,x]
F[67,x] = F[11,x]+F[68,x]
F[68,x] = F[15,x]+F[22,x]+F[69,x]
F[69,x] = F[13,x]*F[4,x]
F[70,x] = F[39,x]+F[71,x]
F[71,x] = 2*F[22,x]+F[49,x]+F[72,x]
F[72,x] = F[4,x]*F[41,x]
F[73,x] = F[4,x]*F[63,x]
F[74,x] = F[35,x]+F[75,x]
F[75,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_{20}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{61}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{7}\! \left(x \right)
F_{33}\! \left(x \right) = F_{30}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{48}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{35}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{39}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{49}\! \left(x \right)+F_{54}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{39}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{41}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{35}\! \left(x \right)
F_{60}\! \left(x \right) = F_{57}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{74}\! \left(x \right)
F_{63}\! \left(x \right) = F_{2}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{22}\! \left(x \right)+F_{65}\! \left(x \right)+F_{73}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{70}\! \left(x \right)
F_{67}\! \left(x \right) = F_{11}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{70}\! \left(x \right) = F_{39}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{49}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{74}\! \left(x \right) = F_{35}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \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_20(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_61(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_34(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_15(x) + F_22(x) + F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_13(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_7(x))
Eq(F_33(x), F_30(x))
Eq(F_34(x), F_35(x) + F_48(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_2(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_38(x)*F_4(x))
Eq(F_41(x), F_35(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_39(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_45(x))
Eq(F_48(x), 2*F_22(x) + F_49(x) + F_54(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_39(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_51(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_41(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_35(x))
Eq(F_60(x), F_57(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_74(x))
Eq(F_63(x), F_2(x) + F_64(x))
Eq(F_64(x), F_22(x) + F_65(x) + F_73(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_70(x))
Eq(F_67(x), F_11(x) + F_68(x))
Eq(F_68(x), F_15(x) + F_22(x) + F_69(x))
Eq(F_69(x), F_13(x)*F_4(x))
Eq(F_70(x), F_39(x) + F_71(x))
Eq(F_71(x), 2*F_22(x) + F_49(x) + F_72(x))
Eq(F_72(x), F_4(x)*F_41(x))
Eq(F_73(x), F_4(x)*F_63(x))
Eq(F_74(x), F_35(x) + F_75(x))
Eq(F_75(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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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