0132_1023_1032_1302
Counting sequence:
1, 1, 2, 6, 20, 68, 231, 781, 2629, 8821, 29530, 98706, 329592, 1099792, 3668127, 12230505, 40771337, 135895689, 452914658, 1509385902, 5029980252, 16761785436, 55855539047, 186125915029, 620217261197, 2066704787645, 6886704234970, 22947920663130, 76467083518464, 254803406631720, 849054583023231, 2829213845232913, 9427483784173585, 31414179178173585, 104678038025214082, 348807157661010070, 1162291798642440356, 3872977188167224180, 12905495735197767719, 43003562566423115869, 143296035000587474389, 477489590091917534341, 1591085945076125157914, 5301800346973502385762, 17666604956446845398408, 58868480459158631630144, 196160948633301220181919, 653645506687034883997945, 2178070870669384783290649, 7257745471220006108861977, 24184185198741027136075234, 80586294456236605976996286, 268528825764220004153382636, 894789005346698067585771916, 2981606767064193760752715047, 9935279557518825299295105381, 33106236870093586454433186845, 110316263708283342709073155533, 367594725011775299146534771226, 1224895380910673888306914714026, 4081583853309330122758000007184, 13600611947088059707355918832216, 45319819948065977270636637006335, 151014240249841913152985719030561, 503208105953434614918213619441697, 1676784901067364407354615155537697, 5587365487921141970286860977925122, 18618162100417104294694883070909990, 62039249221607229079800935648917812, 206726551376089535005649611940569764, 688852098954212194444380986459955303, 2295385914750389628634298658878795437, 7648661455242966738304779600786580837, 25486791428394663386257565349830782101, 84926825577985659254590409740174110682, 282992299090188798989551624687004370994, 942984043019611888198131132518821666712, 3142201778099331312925374621853751950960, 10470412609182638532274552494942138411935, 34889401747090776818196541656503593257865, 116258107460095988949937092344496361564969, 387394075948299388333829972859857038507945, 1290870575468016909244120527845250850017506, 4301425721418400751107975266194416568027598, 14333166770163336978592029117808922700457340, 47760831632719782609115074183430516884030140, 159148154404890911963957934459426289721700327, 530311851461382592855857006127293843723702325, 1767099724480102230953430959785413309960873901, 5888311618253631322408732361093816064530991325, 19620971716161405084377554658758512372750317914, 65380801160891143613135871174627272987311787514, 217861236552267233893721178944768365144530637344, 725954982951083956246069848196277694420536229256, 2419019765111234567907373229769733094654407834239, 8060632906204745636945289741260067811311895474737, 26859558481368196733688564363703418333864908785713, 89501145903655578348061911840502486574267593074609, 298234802467959250731359613141251274257549644655234, 993774956790467762205347528156036385326769676853430, 3311446741196468182577048120522012097348327078080964
Generating function in Maple syntax:
-(x^2-3*x+1)^2/(x-1)/(x^4-4*x^3+10*x^2-6*x+1)
Generating function in latex syntax:
-\frac{\left(x^{2}-3 x +1\right)^{2}}{\left(x -1\right) \left(x^{4}-4 x^{3}+10 x^{2}-6 x +1\right)}
Generating function in sympy syntax:
-(x**2 - 3*x + 1)**2/((x - 1)*(x**4 - 4*x**3 + 10*x**2 - 6*x + 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^4-4*x^3+10*x^2-6*x+1)*F(x)+(x^2-3*x+1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{4}-4 x^{3}+10 x^{2}-6 x +1\right) F \! \left(x \right)+\left(x^{2}-3 x +1\right)^{2} = 0
Explicit closed form in Maple syntax:
1/2972160000*((((-172*3^(1/2)-92*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(2/3)+(1720*3^(1/2)-680*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3)-2000*43^(1/2))*((485-5*43^(1/2)*3^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3)-(262+6*43^(1/2)*3^(1/2))^(2/3)*43^(1/2)*3^(1/2)+77*(262+6*43^(1/2)*3^(1/2))^(2/3)+800)^(1/2)+(5895*43^(1/2)*3^(1/2)-17415)*(262+6*43^(1/2)*3^(1/2))^(2/3)+36000*3^(1/2)*43^(1/2)*((262+6*43^(1/2)*3^(1/2))^(1/3)+4))*((-6*43^(1/2)*3^(1/2)+262)*(262+6*43^(1/2)*3^(1/2))^(2/3)+320*3^(1/2)*((485-5*43^(1/2)*3^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3)-(262+6*43^(1/2)*3^(1/2))^(2/3)*43^(1/2)*3^(1/2)+77*(262+6*43^(1/2)*3^(1/2))^(2/3)+800)^(1/2)+1600*(262+6*43^(1/2)*3^(1/2))^(1/3)+25600)^(1/2)+371520000+((-15480*3^(1/2)-8280*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(2/3)-154800*(3^(1/2)-17/43*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3))*((485-5*43^(1/2)*3^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3)-(262+6*43^(1/2)*3^(1/2))^(2/3)*43^(1/2)*3^(1/2)+77*(262+6*43^(1/2)*3^(1/2))^(2/3)+800)^(1/2))*(-1/240*(960*((-15*43^(1/2)*2^(1/3)*3^(1/2)+1455*2^(1/3))*(3*43^(1/2)*3^(1/2)+131)^(1/3)+2400+(-3*2^(2/3)*3^(1/2)*43^(1/2)+231*2^(2/3))*(3*43^(1/2)*3^(1/2)+131)^(2/3))^(1/2)+(18*43^(1/2)*3^(1/2)-786)*2^(2/3)*(3*43^(1/2)*3^(1/2)+131)^(2/3)-4800*2^(1/3)*(3*43^(1/2)*3^(1/2)+131)^(1/3)-76800)^(1/2)+1/864000*(-131*2^(2/3)*(3*43^(1/2)*3^(1/2)+131)^(2/3)-800*(262+6*43^(1/2)*3^(1/2))^(1/3)+6400)*((-15*43^(1/2)*2^(1/3)*3^(1/2)+1455*2^(1/3))*(3*43^(1/2)*3^(1/2)+131)^(1/3)+2400+(-3*2^(2/3)*3^(1/2)*43^(1/2)+231*2^(2/3))*(3*43^(1/2)*3^(1/2)+131)^(2/3))^(1/2)+1/96000*(-215*2^(1/3)*(43^(1/2)*3^(1/2)-97)*(3*43^(1/2)*3^(1/2)+131)^(1/3)+34400+(-43*2^(2/3)*3^(1/2)*43^(1/2)+3311*2^(2/3))*(3*43^(1/2)*3^(1/2)+131)^(2/3))^(1/2)*2^(2/3)*(3*43^(1/2)*3^(1/2)+131)^(2/3)+1)^(-n)+1/2972160000*((((172*3^(1/2)+92*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(2/3)+(-1720*3^(1/2)+680*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3)+2000*43^(1/2))*((485-5*43^(1/2)*3^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3)-(262+6*43^(1/2)*3^(1/2))^(2/3)*43^(1/2)*3^(1/2)+77*(262+6*43^(1/2)*3^(1/2))^(2/3)+800)^(1/2)+(-5895*43^(1/2)*3^(1/2)+17415)*(262+6*43^(1/2)*3^(1/2))^(2/3)-36000*3^(1/2)*43^(1/2)*((262+6*43^(1/2)*3^(1/2))^(1/3)+4))*((-6*43^(1/2)*3^(1/2)+262)*(262+6*43^(1/2)*3^(1/2))^(2/3)+320*3^(1/2)*((485-5*43^(1/2)*3^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3)-(262+6*43^(1/2)*3^(1/2))^(2/3)*43^(1/2)*3^(1/2)+77*(262+6*43^(1/2)*3^(1/2))^(2/3)+800)^(1/2)+1600*(262+6*43^(1/2)*3^(1/2))^(1/3)+25600)^(1/2)+371520000+((-15480*3^(1/2)-8280*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(2/3)-154800*(3^(1/2)-17/43*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3))*((485-5*43^(1/2)*3^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3)-(262+6*43^(1/2)*3^(1/2))^(2/3)*43^(1/2)*3^(1/2)+77*(262+6*43^(1/2)*3^(1/2))^(2/3)+800)^(1/2))*(1/240*(960*((-15*43^(1/2)*2^(1/3)*3^(1/2)+1455*2^(1/3))*(3*43^(1/2)*3^(1/2)+131)^(1/3)+2400+(-3*2^(2/3)*3^(1/2)*43^(1/2)+231*2^(2/3))*(3*43^(1/2)*3^(1/2)+131)^(2/3))^(1/2)+(18*43^(1/2)*3^(1/2)-786)*2^(2/3)*(3*43^(1/2)*3^(1/2)+131)^(2/3)-4800*2^(1/3)*(3*43^(1/2)*3^(1/2)+131)^(1/3)-76800)^(1/2)+1/864000*(-131*2^(2/3)*(3*43^(1/2)*3^(1/2)+131)^(2/3)-800*(262+6*43^(1/2)*3^(1/2))^(1/3)+6400)*((-15*43^(1/2)*2^(1/3)*3^(1/2)+1455*2^(1/3))*(3*43^(1/2)*3^(1/2)+131)^(1/3)+2400+(-3*2^(2/3)*3^(1/2)*43^(1/2)+231*2^(2/3))*(3*43^(1/2)*3^(1/2)+131)^(2/3))^(1/2)+1/96000*(-215*2^(1/3)*(43^(1/2)*3^(1/2)-97)*(3*43^(1/2)*3^(1/2)+131)^(1/3)+34400+(-43*2^(2/3)*3^(1/2)*43^(1/2)+3311*2^(2/3))*(3*43^(1/2)*3^(1/2)+131)^(2/3))^(1/2)*2^(2/3)*(3*43^(1/2)*3^(1/2)+131)^(2/3)+1)^(-n)+1/2972160000*((((-1834*I+42*I*3^(1/2)*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(2/3)-11200*I*(262+6*43^(1/2)*3^(1/2))^(1/3)-94000*I)*((485-5*43^(1/2)*3^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3)-(262+6*43^(1/2)*3^(1/2))^(2/3)*43^(1/2)*3^(1/2)+77*(262+6*43^(1/2)*3^(1/2))^(2/3)+800)^(1/2)+(100215*I*3^(1/2)-6885*I*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(2/3)+612000*I*3^(1/2)*((262+6*43^(1/2)*3^(1/2))^(1/3)+92/17))*((-6*43^(1/2)*3^(1/2)+262)*(262+6*43^(1/2)*3^(1/2))^(2/3)+320*3^(1/2)*((485-5*43^(1/2)*3^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3)-(262+6*43^(1/2)*3^(1/2))^(2/3)*43^(1/2)*3^(1/2)+77*(262+6*43^(1/2)*3^(1/2))^(2/3)+800)^(1/2)+1600*(262+6*43^(1/2)*3^(1/2))^(1/3)+25600)^(1/2)+371520000+((15480*3^(1/2)+8280*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(2/3)+154800*(3^(1/2)-17/43*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3))*((485-5*43^(1/2)*3^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3)-(262+6*43^(1/2)*3^(1/2))^(2/3)*43^(1/2)*3^(1/2)+77*(262+6*43^(1/2)*3^(1/2))^(2/3)+800)^(1/2))*(-1/240*I*(960*((-15*43^(1/2)*2^(1/3)*3^(1/2)+1455*2^(1/3))*(3*43^(1/2)*3^(1/2)+131)^(1/3)+2400+(-3*2^(2/3)*3^(1/2)*43^(1/2)+231*2^(2/3))*(3*43^(1/2)*3^(1/2)+131)^(2/3))^(1/2)+(-18*43^(1/2)*3^(1/2)+786)*2^(2/3)*(3*43^(1/2)*3^(1/2)+131)^(2/3)+4800*2^(1/3)*(3*43^(1/2)*3^(1/2)+131)^(1/3)+76800)^(1/2)+1/864000*(131*2^(2/3)*(3*43^(1/2)*3^(1/2)+131)^(2/3)+800*(262+6*43^(1/2)*3^(1/2))^(1/3)-6400)*((-15*43^(1/2)*2^(1/3)*3^(1/2)+1455*2^(1/3))*(3*43^(1/2)*3^(1/2)+131)^(1/3)+2400+(-3*2^(2/3)*3^(1/2)*43^(1/2)+231*2^(2/3))*(3*43^(1/2)*3^(1/2)+131)^(2/3))^(1/2)-1/96000*(-215*2^(1/3)*(43^(1/2)*3^(1/2)-97)*(3*43^(1/2)*3^(1/2)+131)^(1/3)+34400+(-43*2^(2/3)*3^(1/2)*43^(1/2)+3311*2^(2/3))*(3*43^(1/2)*3^(1/2)+131)^(2/3))^(1/2)*2^(2/3)*(3*43^(1/2)*3^(1/2)+131)^(2/3)+1)^(-n)+1/2+1/2972160000*((((-42*I*3^(1/2)*43^(1/2)+1834*I)*(262+6*43^(1/2)*3^(1/2))^(2/3)+11200*I*(262+6*43^(1/2)*3^(1/2))^(1/3)+94000*I)*((485-5*43^(1/2)*3^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3)-(262+6*43^(1/2)*3^(1/2))^(2/3)*43^(1/2)*3^(1/2)+77*(262+6*43^(1/2)*3^(1/2))^(2/3)+800)^(1/2)+(-100215*I*3^(1/2)+6885*I*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(2/3)-612000*I*3^(1/2)*((262+6*43^(1/2)*3^(1/2))^(1/3)+92/17))*((-6*43^(1/2)*3^(1/2)+262)*(262+6*43^(1/2)*3^(1/2))^(2/3)+320*3^(1/2)*((485-5*43^(1/2)*3^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3)-(262+6*43^(1/2)*3^(1/2))^(2/3)*43^(1/2)*3^(1/2)+77*(262+6*43^(1/2)*3^(1/2))^(2/3)+800)^(1/2)+1600*(262+6*43^(1/2)*3^(1/2))^(1/3)+25600)^(1/2)+371520000+((15480*3^(1/2)+8280*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(2/3)+154800*(3^(1/2)-17/43*43^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3))*((485-5*43^(1/2)*3^(1/2))*(262+6*43^(1/2)*3^(1/2))^(1/3)-(262+6*43^(1/2)*3^(1/2))^(2/3)*43^(1/2)*3^(1/2)+77*(262+6*43^(1/2)*3^(1/2))^(2/3)+800)^(1/2))*(1/240*I*(960*((-15*43^(1/2)*2^(1/3)*3^(1/2)+1455*2^(1/3))*(3*43^(1/2)*3^(1/2)+131)^(1/3)+2400+(-3*2^(2/3)*3^(1/2)*43^(1/2)+231*2^(2/3))*(3*43^(1/2)*3^(1/2)+131)^(2/3))^(1/2)+(-18*43^(1/2)*3^(1/2)+786)*2^(2/3)*(3*43^(1/2)*3^(1/2)+131)^(2/3)+4800*2^(1/3)*(3*43^(1/2)*3^(1/2)+131)^(1/3)+76800)^(1/2)+1/864000*(131*2^(2/3)*(3*43^(1/2)*3^(1/2)+131)^(2/3)+800*(262+6*43^(1/2)*3^(1/2))^(1/3)-6400)*((-15*43^(1/2)*2^(1/3)*3^(1/2)+1455*2^(1/3))*(3*43^(1/2)*3^(1/2)+131)^(1/3)+2400+(-3*2^(2/3)*3^(1/2)*43^(1/2)+231*2^(2/3))*(3*43^(1/2)*3^(1/2)+131)^(2/3))^(1/2)-1/96000*(-215*2^(1/3)*(43^(1/2)*3^(1/2)-97)*(3*43^(1/2)*3^(1/2)+131)^(1/3)+34400+(-43*2^(2/3)*3^(1/2)*43^(1/2)+3311*2^(2/3))*(3*43^(1/2)*3^(1/2)+131)^(2/3))^(1/2)*2^(2/3)*(3*43^(1/2)*3^(1/2)+131)^(2/3)+1)^(-n)
Explicit closed form in latex syntax:
\frac{\left(\left(\left(\left(-172 \sqrt{3}-92 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(1720 \sqrt{3}-680 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-2000 \sqrt{43}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+\left(5895 \sqrt{43}\, \sqrt{3}-17415\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+36000 \sqrt{3}\, \sqrt{43}\, \left(\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+4\right)\right) \sqrt{\left(-6 \sqrt{43}\, \sqrt{3}+262\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+320 \sqrt{3}\, \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+1600 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+25600}+371520000+\left(\left(-15480 \sqrt{3}-8280 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-154800 \left(\sqrt{3}-\frac{17 \sqrt{43}}{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}\right) \left(-\frac{\sqrt{960 \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}+\left(18 \sqrt{43}\, \sqrt{3}-786\right) 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}-4800 \,2^{\frac{1}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}-76800}}{240}+\frac{\left(-131 \,2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}-800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+6400\right) \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}}{864000}+\frac{\sqrt{-215 \,2^{\frac{1}{3}} \left(\sqrt{43}\, \sqrt{3}-97\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+34400+\left(-43 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+3311 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}{96000}+1\right)^{-n}}{2972160000}+\frac{\left(\left(\left(\left(172 \sqrt{3}+92 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(-1720 \sqrt{3}+680 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+2000 \sqrt{43}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+\left(-5895 \sqrt{43}\, \sqrt{3}+17415\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-36000 \sqrt{3}\, \sqrt{43}\, \left(\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+4\right)\right) \sqrt{\left(-6 \sqrt{43}\, \sqrt{3}+262\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+320 \sqrt{3}\, \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+1600 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+25600}+371520000+\left(\left(-15480 \sqrt{3}-8280 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-154800 \left(\sqrt{3}-\frac{17 \sqrt{43}}{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}\right) \left(\frac{\sqrt{960 \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}+\left(18 \sqrt{43}\, \sqrt{3}-786\right) 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}-4800 \,2^{\frac{1}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}-76800}}{240}+\frac{\left(-131 \,2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}-800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+6400\right) \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}}{864000}+\frac{\sqrt{-215 \,2^{\frac{1}{3}} \left(\sqrt{43}\, \sqrt{3}-97\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+34400+\left(-43 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+3311 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}{96000}+1\right)^{-n}}{2972160000}+\frac{\left(\left(\left(\left(-1834 \,\mathrm{I}+42 \,\mathrm{I} \sqrt{3}\, \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-11200 \,\mathrm{I} \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-94000 \,\mathrm{I}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+\left(100215 \,\mathrm{I} \sqrt{3}-6885 \,\mathrm{I} \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+612000 \,\mathrm{I} \sqrt{3}\, \left(\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{92}{17}\right)\right) \sqrt{\left(-6 \sqrt{43}\, \sqrt{3}+262\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+320 \sqrt{3}\, \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+1600 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+25600}+371520000+\left(\left(15480 \sqrt{3}+8280 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+154800 \left(\sqrt{3}-\frac{17 \sqrt{43}}{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}\right) \left(-\frac{\mathrm{I} \sqrt{960 \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}+\left(-18 \sqrt{43}\, \sqrt{3}+786\right) 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}+4800 \,2^{\frac{1}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+76800}}{240}+\frac{\left(131 \,2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}+800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-6400\right) \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}}{864000}-\frac{\sqrt{-215 \,2^{\frac{1}{3}} \left(\sqrt{43}\, \sqrt{3}-97\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+34400+\left(-43 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+3311 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}{96000}+1\right)^{-n}}{2972160000}+\frac{1}{2}+\frac{\left(\left(\left(\left(-42 \,\mathrm{I} \sqrt{3}\, \sqrt{43}+1834 \,\mathrm{I}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+11200 \,\mathrm{I} \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+94000 \,\mathrm{I}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+\left(-100215 \,\mathrm{I} \sqrt{3}+6885 \,\mathrm{I} \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-612000 \,\mathrm{I} \sqrt{3}\, \left(\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{92}{17}\right)\right) \sqrt{\left(-6 \sqrt{43}\, \sqrt{3}+262\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+320 \sqrt{3}\, \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+1600 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+25600}+371520000+\left(\left(15480 \sqrt{3}+8280 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+154800 \left(\sqrt{3}-\frac{17 \sqrt{43}}{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}\right) \left(\frac{\mathrm{I} \sqrt{960 \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}+\left(-18 \sqrt{43}\, \sqrt{3}+786\right) 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}+4800 \,2^{\frac{1}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+76800}}{240}+\frac{\left(131 \,2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}+800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-6400\right) \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}}{864000}-\frac{\sqrt{-215 \,2^{\frac{1}{3}} \left(\sqrt{43}\, \sqrt{3}-97\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+34400+\left(-43 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+3311 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}{96000}+1\right)^{-n}}{2972160000}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+4) = -a(n)+4*a(n+1)-10*a(n+2)+6*a(n+3)+1, n >= 5
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(n +4\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+6 a \! \left(n +3\right)+1, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15473/
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[27,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[16,x]+F[26,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]+F[4,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[22,x]+F[24,x]
F[22,x] = F[15,x]+F[16,x]+F[23,x]
F[23,x] = F[4,x]*F[7,x]
F[24,x] = F[25,x]
F[25,x] = F[21,x]*F[4,x]
F[26,x] = F[13,x]*F[4,x]
F[27,x] = F[2,x]+F[28,x]
F[28,x] = F[15,x]+F[29,x]+F[74,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[22,x]+F[4,x]
F[32,x] = F[33,x]+F[35,x]
F[33,x] = F[15,x]+F[29,x]+F[34,x]
F[34,x] = F[2,x]*F[4,x]
F[35,x] = F[15,x]+F[36,x]+F[37,x]+F[52,x]
F[36,x] = 0
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[33,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[39,x]*F[4,x]
F[42,x] = F[43,x]+F[72,x]
F[43,x] = 2*F[15,x]+F[44,x]+F[52,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[33,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[46,x]
F[49,x] = F[43,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[49,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[59,x]
F[56,x] = F[2,x]+F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[4,x]*F[56,x]
F[59,x] = F[53,x]+F[60,x]
F[60,x] = 2*F[15,x]+F[61,x]+F[71,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[67,x]
F[63,x] = F[64,x]+F[65,x]
F[64,x] = F[34,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[63,x]
F[67,x] = F[68,x]+F[69,x]
F[68,x] = 2*F[15,x]+F[52,x]+F[61,x]
F[69,x] = F[70,x]
F[70,x] = F[4,x]*F[67,x]
F[71,x] = F[4,x]*F[59,x]
F[72,x] = F[73,x]
F[73,x] = F[4,x]*F[42,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[76,x]+F[79,x]
F[76,x] = F[2,x]+F[77,x]
F[77,x] = F[15,x]+F[29,x]+F[78,x]
F[78,x] = F[4,x]*F[76,x]
F[79,x] = F[53,x]+F[80,x]
F[80,x] = 2*F[15,x]+F[44,x]+F[81,x]
F[81,x] = F[4,x]*F[79,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_{27}\! \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_{16}\! \left(x \right)+F_{26}\! \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_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{2}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{74}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{22}\! \left(x \right)+F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{52}\! \left(x \right)
F_{36}\! \left(x \right) = 0
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{72}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{52}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{33}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{43}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{59}\! \left(x \right)
F_{56}\! \left(x \right) = F_{2}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{53}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{61}\! \left(x \right)+F_{71}\! \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_{67}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{65}\! \left(x \right)
F_{64}\! \left(x \right) = F_{34}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)+F_{69}\! \left(x \right)
F_{68}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{52}\! \left(x \right)+F_{61}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{79}\! \left(x \right)
F_{76}\! \left(x \right) = F_{2}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{53}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \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_27(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) + F_16(x) + F_26(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x) + F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_24(x))
Eq(F_22(x), F_15(x) + F_16(x) + F_23(x))
Eq(F_23(x), F_4(x)*F_7(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_21(x)*F_4(x))
Eq(F_26(x), F_13(x)*F_4(x))
Eq(F_27(x), F_2(x) + F_28(x))
Eq(F_28(x), F_15(x) + F_29(x) + F_74(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_22(x) + F_4(x))
Eq(F_32(x), F_33(x) + F_35(x))
Eq(F_33(x), F_15(x) + F_29(x) + F_34(x))
Eq(F_34(x), F_2(x)*F_4(x))
Eq(F_35(x), F_15(x) + F_36(x) + F_37(x) + F_52(x))
Eq(F_36(x), 0)
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_33(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_39(x)*F_4(x))
Eq(F_42(x), F_43(x) + F_72(x))
Eq(F_43(x), 2*F_15(x) + F_44(x) + F_52(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_33(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_46(x))
Eq(F_49(x), F_43(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_49(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_59(x))
Eq(F_56(x), F_2(x) + F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_4(x)*F_56(x))
Eq(F_59(x), F_53(x) + F_60(x))
Eq(F_60(x), 2*F_15(x) + F_61(x) + F_71(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_67(x))
Eq(F_63(x), F_64(x) + F_65(x))
Eq(F_64(x), F_34(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_63(x))
Eq(F_67(x), F_68(x) + F_69(x))
Eq(F_68(x), 2*F_15(x) + F_52(x) + F_61(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_4(x)*F_67(x))
Eq(F_71(x), F_4(x)*F_59(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_4(x)*F_42(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_79(x))
Eq(F_76(x), F_2(x) + F_77(x))
Eq(F_77(x), F_15(x) + F_29(x) + F_78(x))
Eq(F_78(x), F_4(x)*F_76(x))
Eq(F_79(x), F_53(x) + F_80(x))
Eq(F_80(x), 2*F_15(x) + F_44(x) + F_81(x))
Eq(F_81(x), F_4(x)*F_79(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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, 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