0132_0231_0321_1032_1203
Counting sequence:
1, 1, 2, 6, 19, 58, 173, 512, 1512, 4462, 13163, 38824, 114502, 337690, 995921, 2937209, 8662582, 25548264, 75348780, 222224236, 655400501, 1932956887, 5700823267, 16813300913, 49587063402, 146245930493, 431319594421, 1272080470081, 3751716224129, 11064846092167, 32633283463856, 96244555116657, 283851742956232, 837157093018317, 2469007204606391, 7281783344230470, 21475987827663807, 63338612448609048, 186803040638306761, 550933698145697634, 1624855455860728660, 4792146970364118545, 14133363372562065661, 41683187401428575098, 122935218329961192621, 362569871643673022773, 1069318569645969628958, 3153715443056599446604, 9301176822419324904083, 27431736262819769001397, 80903757530885065654255, 238607498989691148585494, 703719336550975964626229, 2075462450814009908622889, 6121111302484249683394072, 18052845794779487561746921, 53242822289119128469367329, 157027770443288564674104557, 463118212563823077686059262, 1365863364186085081178055417, 4028307846711234836838606128, 11880591084998531893203660948, 35039140477851537046577208961, 103340091132070619043132113092, 304778436044543800889693671676, 898875683775457561188824354005, 2651032354417322760180225253640, 7818625724358861267256194315175, 23059284099549304843734273218501, 68008189921040130028826532242436, 200574912749640066628512286609238, 591550748097170289408386202139366, 1744646340747067208756611140211175, 5145443335289549030400490775111721, 15175331812714917169438191153539755, 44756239767829119051799969988595002, 131998497487683185576146474300073698, 389299982067082132992285646084103775, 1148153039026615119230513260561636158, 3386220040459434488135136685972565774, 9986897018650221668661655301291984642, 29454114283604691587807502326850309030, 86868308205394799840594558468080990123, 256198604303912701351375508747542760388, 755600358787654195910825155848573993985, 2228473897237825025153795868786556903887, 6572384267575445331882268415386724319025, 19383774256550457759638165845880959425846, 57168097471500312782048881141097949343080, 168604489778688165811718728100003555264414, 497261151426344902621755602411673204850086, 1466560309529250776374794048202247135647748, 4325290916688696058765941997784689510820969, 12756476083820134122446556752487892732523343, 37622366960152164173748080865371038772758184, 110958738634696043009694784515918711614926935, 327247929202405315223157949173147642806720304, 965144417510310353070331199098581279070586340, 2846477131029220859286766840303687566544647962, 8395046285791512942329661630889302828408338098, 24759307346024270082814375238655581738559655444
Generating function in Maple syntax:
-(-1+2*x)*(x^2+x-1)*(x-1)^2/(x^7+2*x^6-5*x^5+x^4+9*x^3-12*x^2+6*x-1)
Generating function in latex syntax:
-\frac{\left(-1+2 x \right) \left(x^{2}+x -1\right) \left(x -1\right)^{2}}{x^{7}+2 x^{6}-5 x^{5}+x^{4}+9 x^{3}-12 x^{2}+6 x -1}
Generating function in sympy syntax:
(1 - 2*x)*(x - 1)**2*(x**2 + x - 1)/(x**7 + 2*x**6 - 5*x**5 + x**4 + 9*x**3 - 12*x**2 + 6*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^7+2*x^6-5*x^5+x^4+9*x^3-12*x^2+6*x-1)*F(x)+(-1+2*x)*(x^2+x-1)*(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{7}+2 x^{6}-5 x^{5}+x^{4}+9 x^{3}-12 x^{2}+6 x -1\right) F \! \left(x \right)+\left(-1+2 x \right) \left(x^{2}+x -1\right) \left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
556617/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n+5)+556617/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n+5)+556617/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n+5)+556617/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n+5)+556617/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n+5)+556617/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 6)^(-n+5)+556617/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 7)^(-n+5)+856434/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n+4)+856434/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n+4)+856434/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n+4)+856434/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n+4)+856434/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n+4)+856434/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 6)^(-n+4)+856434/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 7)^(-n+4)-4166476/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n+3)-4166476/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n+3)-4166476/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n+3)-4166476/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n+3)-4166476/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n+3)-4166476/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 6)^(-n+3)-4166476/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 7)^(-n+3)-617401/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n+2)-617401/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n+2)-617401/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n+2)-617401/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n+2)-617401/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n+2)-617401/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 6)^(-n+2)-617401/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 7)^(-n+2)+6639857/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n+1)+6639857/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n+1)+6639857/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n+1)+6639857/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n+1)+6639857/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n+1)+6639857/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 6)^(-n+1)+6639857/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 7)^(-n+1)+2293744/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n-1)+2293744/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n-1)+2293744/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n-1)+2293744/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n-1)+2293744/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n-1)+2293744/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 6)^(-n-1)+2293744/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 7)^(-n-1)-6404178/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n)-6404178/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n)-6404178/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n)-6404178/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n)-6404178/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n)-6404178/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 6)^(-n)-6404178/9050009*RootOf(_Z^7+2*_Z^6-5*_Z^5+_Z^4+9*_Z^3-12*_Z^2+6*_Z-1,index = 7)^(-n)
Explicit closed form in latex syntax:
\frac{556617 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{9050009}+\frac{556617 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{9050009}+\frac{556617 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{9050009}+\frac{556617 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{9050009}+\frac{556617 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{9050009}+\frac{556617 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{9050009}+\frac{556617 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{9050009}+\frac{856434 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{9050009}+\frac{856434 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{9050009}+\frac{856434 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{9050009}+\frac{856434 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{9050009}+\frac{856434 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{9050009}+\frac{856434 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{9050009}+\frac{856434 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{9050009}-\frac{4166476 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{9050009}-\frac{4166476 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{9050009}-\frac{4166476 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{9050009}-\frac{4166476 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{9050009}-\frac{4166476 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{9050009}-\frac{4166476 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{9050009}-\frac{4166476 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{9050009}-\frac{617401 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{9050009}-\frac{617401 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{9050009}-\frac{617401 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{9050009}-\frac{617401 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{9050009}-\frac{617401 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{9050009}-\frac{617401 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{9050009}-\frac{617401 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{9050009}+\frac{6639857 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{9050009}+\frac{6639857 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{9050009}+\frac{6639857 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{9050009}+\frac{6639857 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{9050009}+\frac{6639857 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{9050009}+\frac{6639857 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{9050009}+\frac{6639857 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{9050009}+\frac{2293744 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{9050009}+\frac{2293744 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{9050009}+\frac{2293744 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{9050009}+\frac{2293744 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{9050009}+\frac{2293744 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{9050009}+\frac{2293744 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{9050009}+\frac{2293744 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{9050009}-\frac{6404178 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{9050009}-\frac{6404178 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{9050009}-\frac{6404178 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{9050009}-\frac{6404178 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{9050009}-\frac{6404178 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{9050009}-\frac{6404178 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{9050009}-\frac{6404178 \mathit{RootOf}\! \left(\textit{\_Z}^{7}+2 \textit{\_Z}^{6}-5 \textit{\_Z}^{5}+\textit{\_Z}^{4}+9 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{9050009}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 58
a(6) = 173
a(n+7) = a(n)+2*a(n+1)-5*a(n+2)+a(n+3)+9*a(n+4)-12*a(n+5)+6*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) = 58
a \! \left(6\right) = 173
a \! \left(n +7\right) = a \! \left(n \right)+2 a \! \left(n +1\right)-5 a \! \left(n +2\right)+a \! \left(n +3\right)+9 a \! \left(n +4\right)-12 a \! \left(n +5\right)+6 a \! \left(n +6\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18650/
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[12,x]*F[4,x]
F[4,x] = F[18,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[12,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[17,x]
F[15,x] = 0
F[16,x] = F[10,x]*F[12,x]
F[17,x] = F[12,x]*F[13,x]
F[18,x] = F[19,x]+F[2,x]
F[19,x] = F[15,x]+F[20,x]+F[45,x]
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]+F[27,x]
F[22,x] = F[10,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[12,x]+F[26,x]
F[26,x] = F[24,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[15,x]+F[20,x]+F[29,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[2,x]+F[28,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[44,x]
F[34,x] = F[15,x]+F[35,x]+F[43,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[12,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = x^2
F[40,x] = F[34,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[34,x]
F[43,x] = F[12,x]*F[2,x]
F[44,x] = F[32,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[30,x]+F[47,x]
F[47,x] = F[34,x]+F[48,x]
F[48,x] = F[15,x]+F[49,x]+F[50,x]+F[51,x]
F[49,x] = 0
F[50,x] = F[12,x]*F[28,x]
F[51,x] = F[12,x]*F[47,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{18}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{2}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{45}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{2}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{44}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{43}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = x^{2}
F_{40}\! \left(x \right) = F_{34}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{2}\! \left(x \right)
F_{44}\! \left(x \right) = F_{32}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{30}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{34}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{15}\! \left(x \right)+F_{49}\! \left(x \right)+F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{49}\! \left(x \right) = 0
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_18(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_12(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_17(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_10(x)*F_12(x))
Eq(F_17(x), F_12(x)*F_13(x))
Eq(F_18(x), F_19(x) + F_2(x))
Eq(F_19(x), F_15(x) + F_20(x) + F_45(x))
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_27(x))
Eq(F_22(x), F_10(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_12(x) + F_26(x))
Eq(F_26(x), F_24(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_15(x) + F_20(x) + F_29(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_2(x) + F_28(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_44(x))
Eq(F_34(x), F_15(x) + F_35(x) + F_43(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_12(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), x**2)
Eq(F_40(x), F_34(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_34(x))
Eq(F_43(x), F_12(x)*F_2(x))
Eq(F_44(x), F_32(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_30(x) + F_47(x))
Eq(F_47(x), F_34(x) + F_48(x))
Eq(F_48(x), F_15(x) + F_49(x) + F_50(x) + F_51(x))
Eq(F_49(x), 0)
Eq(F_50(x), F_12(x)*F_28(x))
Eq(F_51(x), F_12(x)*F_47(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:
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