0321_1302_2103
Counting sequence:
1, 1, 2, 6, 21, 74, 253, 843, 2772, 9080, 29759, 97686, 321033, 1055596, 3471365, 11415280, 37535830, 123420905, 405812530, 1334326344, 4387329985, 14425801573, 47432972821, 155962779313, 512816020062, 1686173040104, 5544248055746, 18229852104876, 59940951879918, 197089791079973, 648044198117637, 2130811964537770, 7006249960248961, 23037010931226713, 75747207923721743, 249061804234649852, 818931601847766840, 2692700996638202561, 8853778070194005799, 29111804918562767972, 95721530279043324510, 314738690529768277066, 1034881525899036121537, 3402758557723318193901, 11188494056934041624291, 36788504720052547838070, 120963024393657250105395, 397734384199787589981230, 1307776828231235709948425, 4300056269712742811104275, 14138867980788922748691581, 46489528331575593131861543, 152860628405964938524732559, 502615805219862313996689673, 1652633842286052610600797400, 5433968825302729421225998261, 17867247080887839530336944567, 58748684159355685348307633976, 193169539483652037863979752429, 635154157378415158543495252474, 2088428665893393284381745931146, 6866890882880142977277775439640, 22578789099895203224026703339447, 74240544361718917846505500889982, 244107795273659574134639229311046, 802642494417011828981119024957434, 2639141339676336719625928329388072, 8677670394024662883096794038290215, 28532751291208659509781806709634904, 93817563848306683560457680821065673, 308478323614836230435136368908588539, 1014297027516955224994572038390963698, 3335075372473435855153725979003929306, 10965947289925284536171234607524079863, 36056756305401163559801849986510919778, 118556987453470114825216304307994207057, 389823176410817949447731703301260025614, 1281764256422760703703333404708232174611, 4214525221844660268075446712862572522534, 13857636267012831092530216999452166948886, 45564819950176428065435282449529677375479, 149820126397323448572424334627964742259230, 492618434534669168515250542478021454653798, 1619761829594368231205387383003403470923871, 5325883484423787303259512593831144331430215, 17511855367502657302458314738524333259232023, 57580132819129825980275389791426878926287531, 189327265780259053485896498339775077567761788, 622520508600149351966722653291369987212936788, 2046888397350933156807956896060575084311199324, 6730303746347717416562513063116560367514292306, 22129681606835588092475451494695771884436660142, 72763849370939829555024727255125459777847594319, 239252324969799397637553098272513023522246461449, 786677388542825983776812259011295494979441773866, 2586647020975360204082826970835922890155704710485, 8505065619737801488029657809577457232404350337607, 27965215435065351540139772345007510326318462348095, 91951468606508754633824220745400480215773459873212, 302342479661072795451583462924640627601963055562584, 994121968826669952919202561178809833374748097011737
Generating function in Maple syntax:
(x^2-3*x+1)*(-1+x)^5/(4*x^7-23*x^6+55*x^5-78*x^4+66*x^3-33*x^2+9*x-1)
Generating function in latex syntax:
\frac{\left(x^{2}-3 x +1\right) \left(-1+x \right)^{5}}{4 x^{7}-23 x^{6}+55 x^{5}-78 x^{4}+66 x^{3}-33 x^{2}+9 x -1}
Generating function in sympy syntax:
(x - 1)**5*(x**2 - 3*x + 1)/(4*x**7 - 23*x**6 + 55*x**5 - 78*x**4 + 66*x**3 - 33*x**2 + 9*x - 1)
Implicit equation for the generating function in Maple syntax:
(4*x^7-23*x^6+55*x^5-78*x^4+66*x^3-33*x^2+9*x-1)*F(x)-(x^2-3*x+1)*(-1+x)^5 = 0
Implicit equation for the generating function in latex syntax:
\left(4 x^{7}-23 x^{6}+55 x^{5}-78 x^{4}+66 x^{3}-33 x^{2}+9 x -1\right) F \! \left(x \right)-\left(x^{2}-3 x +1\right) \left(-1+x \right)^{5} = 0
Explicit closed form in Maple syntax:
33479224/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+5)+33479224/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+5)+33479224/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+5)+33479224/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+5)+33479224/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+5)+33479224/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+5)+33479224/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+5)-172552762/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+4)-172552762/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+4)-172552762/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+4)-172552762/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+4)-172552762/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+4)-172552762/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+4)-172552762/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+4)+357051072/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+3)+357051072/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+3)+357051072/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+3)+357051072/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+3)+357051072/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+3)+357051072/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+3)+357051072/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+3)-438342235/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+2)-438342235/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+2)-438342235/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+2)-438342235/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+2)-438342235/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+2)-438342235/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+2)-438342235/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+2)+286639494/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n+1)+286639494/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n+1)+286639494/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n+1)+286639494/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n+1)+286639494/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n+1)+286639494/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n+1)+286639494/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n+1)+15176275/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n-1)+15176275/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n-1)+15176275/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n-1)+15176275/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n-1)+15176275/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n-1)+15176275/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n-1)+15176275/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n-1)-96386032/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 1)^(-n)-96386032/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 2)^(-n)-96386032/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 3)^(-n)-96386032/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 4)^(-n)-96386032/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 5)^(-n)-96386032/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 6)^(-n)-96386032/43764829*RootOf(4*_Z^7-23*_Z^6+55*_Z^5-78*_Z^4+66*_Z^3-33*_Z^2+9*_Z-1,index = 7)^(-n)+piecewise(n = 0,1/4,0)
Explicit closed form in latex syntax:
\frac{33479224 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +5}}{43764829}+\frac{33479224 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +5}}{43764829}+\frac{33479224 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +5}}{43764829}+\frac{33479224 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +5}}{43764829}+\frac{33479224 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +5}}{43764829}+\frac{33479224 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +5}}{43764829}+\frac{33479224 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +5}}{43764829}-\frac{172552762 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +4}}{43764829}-\frac{172552762 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +4}}{43764829}-\frac{172552762 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +4}}{43764829}-\frac{172552762 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +4}}{43764829}-\frac{172552762 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +4}}{43764829}-\frac{172552762 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +4}}{43764829}-\frac{172552762 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +4}}{43764829}+\frac{357051072 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{43764829}+\frac{357051072 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{43764829}+\frac{357051072 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{43764829}+\frac{357051072 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{43764829}+\frac{357051072 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{43764829}+\frac{357051072 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +3}}{43764829}+\frac{357051072 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +3}}{43764829}-\frac{438342235 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{43764829}-\frac{438342235 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{43764829}-\frac{438342235 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{43764829}-\frac{438342235 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{43764829}-\frac{438342235 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{43764829}-\frac{438342235 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +2}}{43764829}-\frac{438342235 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +2}}{43764829}+\frac{286639494 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{43764829}+\frac{286639494 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{43764829}+\frac{286639494 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{43764829}+\frac{286639494 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{43764829}+\frac{286639494 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{43764829}+\frac{286639494 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n +1}}{43764829}+\frac{286639494 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n +1}}{43764829}+\frac{15176275 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{43764829}+\frac{15176275 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{43764829}+\frac{15176275 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{43764829}+\frac{15176275 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{43764829}+\frac{15176275 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{43764829}+\frac{15176275 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n -1}}{43764829}+\frac{15176275 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n -1}}{43764829}-\frac{96386032 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{43764829}-\frac{96386032 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{43764829}-\frac{96386032 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{43764829}-\frac{96386032 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{43764829}-\frac{96386032 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{43764829}-\frac{96386032 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =6\right)^{-n}}{43764829}-\frac{96386032 \mathit{RootOf}\! \left(4 \textit{\_Z}^{7}-23 \textit{\_Z}^{6}+55 \textit{\_Z}^{5}-78 \textit{\_Z}^{4}+66 \textit{\_Z}^{3}-33 \textit{\_Z}^{2}+9 \textit{\_Z} -1, \mathit{index} =7\right)^{-n}}{43764829}+\left(\left\{\begin{array}{cc}\frac{1}{4} & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 74
a(6) = 253
a(7) = 843
a(n+7) = 4*a(n)-23*a(n+1)+55*a(n+2)-78*a(n+3)+66*a(n+4)-33*a(n+5)+9*a(n+6), n >= 8
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) = 21
a \! \left(5\right) = 74
a \! \left(6\right) = 253
a \! \left(7\right) = 843
a \! \left(n +7\right) = 4 a \! \left(n \right)-23 a \! \left(n +1\right)+55 a \! \left(n +2\right)-78 a \! \left(n +3\right)+66 a \! \left(n +4\right)-33 a \! \left(n +5\right)+9 a \! \left(n +6\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14782/
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[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,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[14,x]+F[7,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[10,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[15,x]+F[22,x]+F[61,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[36,x]
F[24,x] = F[25,x]+F[7,x]
F[25,x] = F[15,x]+F[18,x]+F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[24,x]+F[28,x]
F[28,x] = F[29,x]+F[33,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]
F[32,x] = F[11,x]+F[29,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[28,x]
F[36,x] = F[37,x]+F[50,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[41,x]
F[40,x] = F[2,x]+F[37,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[2,x]+F[42,x]
F[45,x] = 2*F[15,x]+F[46,x]+F[48,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[37,x]+F[45,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[41,x]
F[50,x] = 2*F[15,x]+F[48,x]+F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[36,x]+F[53,x]
F[53,x] = F[54,x]+F[58,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[42,x]+F[54,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[53,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[20,x]+F[63,x]
F[63,x] = F[42,x]+F[64,x]
F[64,x] = 2*F[15,x]+F[46,x]+F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[63,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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \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_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{61}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{36}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{33}\! \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_{32}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{28}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{50}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{37}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \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_{2}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{37}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{41}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{36}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{58}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{42}\! \left(x \right)+F_{54}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{53}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{20}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{42}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{63}\! \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_6(x))
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_11(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_14(x) + F_7(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_10(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_15(x) + F_22(x) + F_61(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_36(x))
Eq(F_24(x), F_25(x) + F_7(x))
Eq(F_25(x), F_15(x) + F_18(x) + F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_24(x) + F_28(x))
Eq(F_28(x), F_29(x) + F_33(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))
Eq(F_32(x), F_11(x) + F_29(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_28(x))
Eq(F_36(x), F_37(x) + F_50(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_41(x))
Eq(F_40(x), F_2(x) + F_37(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_2(x) + F_42(x))
Eq(F_45(x), 2*F_15(x) + F_46(x) + F_48(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_37(x) + F_45(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_41(x))
Eq(F_50(x), 2*F_15(x) + F_48(x) + F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_36(x) + F_53(x))
Eq(F_53(x), F_54(x) + F_58(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_42(x) + F_54(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_53(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_20(x) + F_63(x))
Eq(F_63(x), F_42(x) + F_64(x))
Eq(F_64(x), 2*F_15(x) + F_46(x) + F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_63(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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