0132_0231_0312_1302_2103
Counting sequence:
1, 1, 2, 6, 19, 54, 144, 385, 1046, 2860, 7808, 21268, 57899, 157676, 429522, 1170090, 3187329, 8682003, 23649043, 64418588, 175473020, 477979328, 1301989036, 3546544452, 9660588289, 26314905215, 71680339479, 195253254777, 531858987666, 1448754251753, 3946325889928, 10749571944768, 29281235295590, 79760454112244, 217263034667718, 591812405831012, 1612064032257153, 4391172639361470, 11961309701393225, 32581941436028558, 88751393805796936, 241753853678727865, 658524032833389473, 1793782788651275857, 4886164410759410921, 13309639717817089329, 36254717305055502433, 98755830716440434361, 269005382621872666578, 732755679888099020406, 1995985660863165353297, 5436953772886167141661, 14809959263794376198463, 40341504187337731486274, 109888010568371474953521, 299328820526843423590696, 815355035863937222910889, 2220981706133651859861946, 6049830468948626177468105, 16479401249429585164069336, 44888971175897505789284028, 122275057372021001311385038, 333070446117928760250772433, 907265344719277650093088882, 2471340268470255124494173943, 6731793248922560239885771537, 18337029879859618269969460636, 49949048103739896771434511032, 136058425100238877650644752329, 370615572138826178400740488963, 1009536176907785737866664668430, 2749920319332472983211722919486, 7490629791831970261695687383208, 20404058359007626207667989784328, 55579518556872767155976483085057, 151395513023028092398542954976879, 412392945434601790626112739235368, 1123335414956176264853057635627883, 3059903105677342149283967507696007, 8335005637206866478463958989292459, 22704091133922296016971513769074036, 61844679734396611329287965630141691, 168461463129677421541727356980940687, 458879643029467587178455643798926491, 1249962590107386607827130330211586739, 3404828478232656106087825031165655993, 9274563141276204818586431371602339624, 25263393445936004414210245520754201945, 68816076690846578777154633287748654197, 187451160164007124057358541803970122518, 510606520111397863084524784183966463511, 1390863722327244021151200255284041995316, 3788635314848606086243470617487171582778, 10320031587926360432826350179110657612631, 28111191266783020970579915871468193752442, 76573319345473384551890908081048057218691, 208581457112146448548950033018933057346036, 568164272136885619486030258991604086803189, 1547647833140190465793352327908610524326911, 4215706500542605808740959911220999612499608, 11483349711838046766429924684681421445811725
Generating function in Maple syntax:
(x-1)^3/(x^7-2*x^6-x^5+x^4+4*x^3-5*x^2+4*x-1)
Generating function in latex syntax:
\frac{\left(x -1\right)^{3}}{x^{7}-2 x^{6}-x^{5}+x^{4}+4 x^{3}-5 x^{2}+4 x -1}
Generating function in sympy syntax:
(x - 1)**3/(x**7 - 2*x**6 - x**5 + x**4 + 4*x**3 - 5*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^7-2*x^6-x^5+x^4+4*x^3-5*x^2+4*x-1)*F(x)-(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}-2 x^{6}-x^{5}+x^{4}+4 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
2477095/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+5)+2477095/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+5)+2477095/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+5)+2477095/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+5)+2477095/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+5)+2477095/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+5)+2477095/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+5)-883379/17821591*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+4)-883379/17821591*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+4)-883379/17821591*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+4)-883379/17821591*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+4)-883379/17821591*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+4)-883379/17821591*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+4)-883379/17821591*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+4)-5439562/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+3)-5439562/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+3)-5439562/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+3)-5439562/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+3)-5439562/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+3)-5439562/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+3)-5439562/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+3)-3116413/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+2)-3116413/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+2)-3116413/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+2)-3116413/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+2)-3116413/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+2)-3116413/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+2)-3116413/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+2)+8006252/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n+1)+8006252/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n+1)+8006252/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n+1)+8006252/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n+1)+8006252/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n+1)+8006252/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n+1)+8006252/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n+1)+5034812/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n-1)+5034812/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n-1)+5034812/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n-1)+5034812/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n-1)+5034812/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n-1)+5034812/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n-1)+5034812/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n-1)+2615723/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 1)^(-n)+2615723/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 2)^(-n)+2615723/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 3)^(-n)+2615723/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 4)^(-n)+2615723/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 5)^(-n)+2615723/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 6)^(-n)+2615723/53464773*RootOf(_Z^7-2*_Z^6-_Z^5+_Z^4+4*_Z^3-5*_Z^2+4*_Z-1,index = 7)^(-n)
Explicit closed form in latex syntax:
\frac{2477095 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{53464773}+\frac{2477095 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{53464773}+\frac{2477095 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{53464773}+\frac{2477095 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{53464773}+\frac{2477095 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{53464773}+\frac{2477095 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{53464773}+\frac{2477095 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{53464773}-\frac{883379 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{17821591}-\frac{883379 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{17821591}-\frac{883379 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{17821591}-\frac{883379 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{17821591}-\frac{883379 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{17821591}-\frac{883379 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{17821591}-\frac{883379 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{17821591}-\frac{5439562 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{53464773}-\frac{5439562 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{53464773}-\frac{5439562 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{53464773}-\frac{5439562 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{53464773}-\frac{5439562 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{53464773}-\frac{5439562 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{53464773}-\frac{5439562 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{53464773}-\frac{3116413 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{53464773}-\frac{3116413 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{53464773}-\frac{3116413 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{53464773}-\frac{3116413 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{53464773}-\frac{3116413 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{53464773}-\frac{3116413 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{53464773}-\frac{3116413 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{53464773}+\frac{8006252 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{53464773}+\frac{8006252 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{53464773}+\frac{8006252 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{53464773}+\frac{8006252 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{53464773}+\frac{8006252 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{53464773}+\frac{8006252 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{53464773}+\frac{8006252 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{53464773}+\frac{5034812 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{53464773}+\frac{5034812 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{53464773}+\frac{5034812 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{53464773}+\frac{5034812 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{53464773}+\frac{5034812 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{53464773}+\frac{5034812 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{53464773}+\frac{5034812 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{53464773}+\frac{2615723 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{53464773}+\frac{2615723 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{53464773}+\frac{2615723 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{53464773}+\frac{2615723 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{53464773}+\frac{2615723 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{53464773}+\frac{2615723 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{53464773}+\frac{2615723 \mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}-\textit{\_Z}^{5}+\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{53464773}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(6) = 144
a(n+7) = a(n)-2*a(n+1)-a(n+2)+a(n+3)+4*a(n+4)-5*a(n+5)+4*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) = 54
a \! \left(6\right) = 144
a \! \left(n +7\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+4 a \! \left(n +4\right)-5 a \! \left(n +5\right)+4 a \! \left(n +6\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18613/
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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[11,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[64,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[38,x]
F[25,x] = F[26,x]+F[7,x]
F[26,x] = F[22,x]+F[27,x]+F[32,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]
F[29,x] = F[11,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[29,x]*F[4,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[13,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[14,x]+F[35,x]
F[38,x] = F[39,x]+F[52,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[2,x]+F[43,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[42,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[2,x]+F[46,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[43,x]
F[52,x] = 2*F[22,x]+F[53,x]+F[58,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[43,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[55,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[45,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[46,x]+F[61,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[77,x]
F[66,x] = F[2,x]+F[67,x]
F[67,x] = F[22,x]+F[68,x]+F[76,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[73,x]
F[70,x] = F[11,x]+F[71,x]
F[71,x] = F[22,x]+F[27,x]+F[72,x]
F[72,x] = F[13,x]*F[4,x]
F[73,x] = F[43,x]+F[74,x]
F[74,x] = 2*F[22,x]+F[53,x]+F[75,x]
F[75,x] = F[4,x]*F[45,x]
F[76,x] = F[4,x]*F[66,x]
F[77,x] = F[46,x]+F[78,x]
F[78,x] = F[50,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_{17}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \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_{64}\! \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_{38}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \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_{14}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{52}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{2}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{43}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{53}\! \left(x \right)+F_{58}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{43}\! \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_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{45}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{46}\! \left(x \right)+F_{61}\! \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_{77}\! \left(x \right)
F_{66}\! \left(x \right) = F_{2}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{22}\! \left(x \right)+F_{68}\! \left(x \right)+F_{76}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{73}\! \left(x \right)
F_{70}\! \left(x \right) = F_{11}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{73}\! \left(x \right) = F_{43}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{53}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{77}\! \left(x \right) = F_{46}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{50}\! \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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_11(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_64(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_38(x))
Eq(F_25(x), F_26(x) + F_7(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_32(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_11(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_29(x)*F_4(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_13(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_14(x) + F_35(x))
Eq(F_38(x), F_39(x) + F_52(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_2(x) + F_43(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_42(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_2(x) + F_46(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_43(x))
Eq(F_52(x), 2*F_22(x) + F_53(x) + F_58(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_43(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_55(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_45(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_46(x) + F_61(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_77(x))
Eq(F_66(x), F_2(x) + F_67(x))
Eq(F_67(x), F_22(x) + F_68(x) + F_76(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_73(x))
Eq(F_70(x), F_11(x) + F_71(x))
Eq(F_71(x), F_22(x) + F_27(x) + F_72(x))
Eq(F_72(x), F_13(x)*F_4(x))
Eq(F_73(x), F_43(x) + F_74(x))
Eq(F_74(x), 2*F_22(x) + F_53(x) + F_75(x))
Eq(F_75(x), F_4(x)*F_45(x))
Eq(F_76(x), F_4(x)*F_66(x))
Eq(F_77(x), F_46(x) + F_78(x))
Eq(F_78(x), F_50(x))
Pack JSON:
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Specification JSON:
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