0123_0213_0231_1032_1203
Counting sequence:
1, 1, 2, 6, 19, 57, 166, 480, 1389, 4025, 11670, 33838, 98111, 284457, 824730, 2391152, 6932713, 20100161, 58276826, 168963238, 489878683, 1420315593, 4117950942, 11939262000, 34615754101, 100362185881, 290982202062, 843650834926, 2446014657319, 7091781879577, 20561352760946, 59613963675040, 172840022072849, 501118720992705, 1452904075790450, 4212435427011718, 12213202868943331, 35409996640300057, 102664950015240790, 297658654664661312, 863008014747225533, 2502137337001038521, 7254499548359815366, 21033123529594219758, 60981778613686920143, 176805756770091470793, 512616659233166457482, 1486240290609268006800, 4309087817658276059001, 12493429183432456594561, 36222462703548424613322, 105020550006389692199846, 304488295395888259594219, 882809336148523575398697, 2559547725726869072230926, 7420950699111548198717200, 21515718861231651916336709, 62380977436082676590338585, 180862483423354064165672830, 524378412370660402332727598, 1520341389522622357765232951, 4407957852890638626651705657, 12780085161636739944377018850, 37053570426400310388764446272, 107430198154358181388618353953, 311474639087995602514821374081, 903064989748987045226717991906, 2618275369379082444013587625478, 7591220994850730981565739636403, 22009387120472563843176865401593, 63812280230994483379698396303622, 185012289801168222553231190506400, 536410033516490815362050647250701, 1555224922443751901837851484986041, 4509096415542375274389364128257782, 13073318329228643343483383058094766, 37903747533149066895413829336570719, 109895134569210196658338364855405417, 318621281218244181013054272319786106, 923785399991662714553174671588758960, 2678350491765243594372572156123272969, 7765398064100022077962036847633033665, 22514382370540683544477262381536346810, 65276423609284795769359221119448557478, 189257311574941736649824480803038182971, 548717714667809152688777600345064229257, 1590908841960579939239008182788537646910, 4612555555930251415269930512499418594032, 13373279596789245946151069000336690429973, 38773431562024947154691347917271226866841, 112416627814763887603939984875525520582446, 325931899760464505104566616346189974316398, 944981230503641610258040290717536959099463, 2739804010164261142976466015153710207163993, 7943571545978171028686776115312598461473362, 23030964504023387268461887598416816909752800, 66774161083013028855311418253441831166545201, 193599733418079169547748140865878964157280321, 561307789894289508335357441761894764243919442, 1627411512574890800129521927689363679589399686, 4718388518641576535127465643695075420863026819
Generating function in Maple syntax:
-(2*x-1)*(x-1)/(x^4+2*x^3-4*x^2+4*x-1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x -1\right)}{x^{4}+2 x^{3}-4 x^{2}+4 x -1}
Generating function in sympy syntax:
(1 - 2*x)*(x - 1)/(x**4 + 2*x**3 - 4*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^4+2*x^3-4*x^2+4*x-1)*F(x)+(2*x-1)*(x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}+2 x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right) = 0
Explicit closed form in Maple syntax:
7/247920652800000000*((((I*3^(1/2)*163^(1/2)-49/3*I)*(98+6*163^(1/2)*3^(1/2))^(2/3)-200/3*I*(98+6*163^(1/2)*3^(1/2))^(1/3)-12700/21*I)*((40*163^(1/2)*3^(1/2)-2120)*(98+6*163^(1/2)*3^(1/2))^(1/3)+22*(98+6*163^(1/2)*3^(1/2))^(2/3)*163^(1/2)*3^(1/2)-226*(98+6*163^(1/2)*3^(1/2))^(2/3)+18800)^(1/2)+(-210*I*3^(1/2)+270/7*I*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(2/3)-6000/7*I*3^(1/2)*((98+6*163^(1/2)*3^(1/2))^(1/3)+44))*((-6*163^(1/2)*3^(1/2)+98)*(98+6*163^(1/2)*3^(1/2))^(2/3)+40*3^(1/2)*((40*163^(1/2)*3^(1/2)-2120)*(98+6*163^(1/2)*3^(1/2))^(1/3)+22*(98+6*163^(1/2)*3^(1/2))^(2/3)*163^(1/2)*3^(1/2)-226*(98+6*163^(1/2)*3^(1/2))^(2/3)+18800)^(1/2)+400*(98+6*163^(1/2)*3^(1/2))^(1/3)-8800)^(1/2)+176040000/7+((-6520/21*3^(1/2)+440/7*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(2/3)+(-32600/21*3^(1/2)+200/7*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(1/3)+326000/21*3^(1/2))*((40*163^(1/2)*3^(1/2)-2120)*(98+6*163^(1/2)*3^(1/2))^(1/3)+22*(98+6*163^(1/2)*3^(1/2))^(2/3)*163^(1/2)*3^(1/2)-226*(98+6*163^(1/2)*3^(1/2))^(2/3)+18800)^(1/2))*((((((-652/3-141/10*I*163^(1/2))*3^(1/2)+7009/30*I+47/3*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(2/3)+((-1793/6+I*163^(1/2))*3^(1/2)+2771/3*I+713/6*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(1/3)-55/3*163^(1/2)+18745/3*I)*((40*163^(1/2)*3^(1/2)-2120)*(98+6*163^(1/2)*3^(1/2))^(1/3)+22*(98+6*163^(1/2)*3^(1/2))^(2/3)*163^(1/2)*3^(1/2)-226*(98+6*163^(1/2)*3^(1/2))^(2/3)+18800)^(1/2)+((17115/2*I-1491/2*163^(1/2))*3^(1/2)-3015/2*I*163^(1/2)-60147/2)*(98+6*163^(1/2)*3^(1/2))^(2/3)+((31785*I+12675*163^(1/2))*3^(1/2)+315*I*163^(1/2)-256725)*(98+6*163^(1/2)*3^(1/2))^(1/3)+48900*3^(1/2)*(I+47/163*163^(1/2)))*((-6*163^(1/2)*3^(1/2)+98)*(98+6*163^(1/2)*3^(1/2))^(2/3)+40*3^(1/2)*((40*163^(1/2)*3^(1/2)-2120)*(98+6*163^(1/2)*3^(1/2))^(1/3)+22*(98+6*163^(1/2)*3^(1/2))^(2/3)*163^(1/2)*3^(1/2)-226*(98+6*163^(1/2)*3^(1/2))^(2/3)+18800)^(1/2)+400*(98+6*163^(1/2)*3^(1/2))^(1/3)-8800)^(1/2)+(((331/3*I*163^(1/2)+15974/3)*3^(1/2)-1141*I-1228*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(2/3)+((-595/3*I*163^(1/2)+101875/3)*3^(1/2)+10595*I-1225*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(1/3)-1400/3*3^(1/2)*(I*163^(1/2)+7661/7))*((40*163^(1/2)*3^(1/2)-2120)*(98+6*163^(1/2)*3^(1/2))^(1/3)+22*(98+6*163^(1/2)*3^(1/2))^(2/3)*163^(1/2)*3^(1/2)-226*(98+6*163^(1/2)*3^(1/2))^(2/3)+18800)^(1/2)+((102690*I+20430*163^(1/2))*3^(1/2)-61290*I*163^(1/2)-102690)*(98+6*163^(1/2)*3^(1/2))^(2/3)+363816000+((-2493900*I+69300*163^(1/2))*3^(1/2)+207900*I*163^(1/2)-2493900)*(98+6*163^(1/2)*3^(1/2))^(1/3))*(-1/120*(120*((120*2^(1/3)*163^(1/2)*3^(1/2)-6360*2^(1/3))*(3*163^(1/2)*3^(1/2)+49)^(1/3)+56400+(66*2^(2/3)*3^(1/2)*163^(1/2)-678*2^(2/3))*(3*163^(1/2)*3^(1/2)+49)^(2/3))^(1/2)+(18*163^(1/2)*3^(1/2)-294)*2^(2/3)*(3*163^(1/2)*3^(1/2)+49)^(2/3)-1200*2^(1/3)*(3*163^(1/2)*3^(1/2)+49)^(1/3)+26400)^(1/2)+1/648000*(-49*2^(2/3)*(3*163^(1/2)*3^(1/2)+49)^(2/3)-200*(98+6*163^(1/2)*3^(1/2))^(1/3)-2200)*((120*2^(1/3)*163^(1/2)*3^(1/2)-6360*2^(1/3))*(3*163^(1/2)*3^(1/2)+49)^(1/3)+56400+(66*2^(2/3)*3^(1/2)*163^(1/2)-678*2^(2/3))*(3*163^(1/2)*3^(1/2)+49)^(2/3))^(1/2)+1/72000*(6520*2^(1/3)*(163^(1/2)*3^(1/2)-53)*(3*163^(1/2)*3^(1/2)+49)^(1/3)+3064400+(3586*2^(2/3)*3^(1/2)*163^(1/2)-36838*2^(2/3))*(3*163^(1/2)*3^(1/2)+49)^(2/3))^(1/2)*2^(2/3)*(3*163^(1/2)*3^(1/2)+49)^(2/3)-1/2)^(-n)+(((((652/3-141/10*I*163^(1/2))*3^(1/2)+7009/30*I-47/3*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(2/3)+((1793/6+I*163^(1/2))*3^(1/2)+2771/3*I-713/6*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(1/3)+55/3*163^(1/2)+18745/3*I)*((40*163^(1/2)*3^(1/2)-2120)*(98+6*163^(1/2)*3^(1/2))^(1/3)+22*(98+6*163^(1/2)*3^(1/2))^(2/3)*163^(1/2)*3^(1/2)-226*(98+6*163^(1/2)*3^(1/2))^(2/3)+18800)^(1/2)+((17115/2*I+1491/2*163^(1/2))*3^(1/2)-3015/2*I*163^(1/2)+60147/2)*(98+6*163^(1/2)*3^(1/2))^(2/3)+((31785*I-12675*163^(1/2))*3^(1/2)+315*I*163^(1/2)+256725)*(98+6*163^(1/2)*3^(1/2))^(1/3)+48900*3^(1/2)*(I-47/163*163^(1/2)))*((-6*163^(1/2)*3^(1/2)+98)*(98+6*163^(1/2)*3^(1/2))^(2/3)+40*3^(1/2)*((40*163^(1/2)*3^(1/2)-2120)*(98+6*163^(1/2)*3^(1/2))^(1/3)+22*(98+6*163^(1/2)*3^(1/2))^(2/3)*163^(1/2)*3^(1/2)-226*(98+6*163^(1/2)*3^(1/2))^(2/3)+18800)^(1/2)+400*(98+6*163^(1/2)*3^(1/2))^(1/3)-8800)^(1/2)+(((-331/3*I*163^(1/2)+15974/3)*3^(1/2)+1141*I-1228*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(2/3)+((595/3*I*163^(1/2)+101875/3)*3^(1/2)-10595*I-1225*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(1/3)+1400/3*3^(1/2)*(I*163^(1/2)-7661/7))*((40*163^(1/2)*3^(1/2)-2120)*(98+6*163^(1/2)*3^(1/2))^(1/3)+22*(98+6*163^(1/2)*3^(1/2))^(2/3)*163^(1/2)*3^(1/2)-226*(98+6*163^(1/2)*3^(1/2))^(2/3)+18800)^(1/2)+((-102690*I+20430*163^(1/2))*3^(1/2)+61290*I*163^(1/2)-102690)*(98+6*163^(1/2)*3^(1/2))^(2/3)+363816000+((2493900*I+69300*163^(1/2))*3^(1/2)-207900*I*163^(1/2)-2493900)*(98+6*163^(1/2)*3^(1/2))^(1/3))*(1/120*(120*((120*2^(1/3)*163^(1/2)*3^(1/2)-6360*2^(1/3))*(3*163^(1/2)*3^(1/2)+49)^(1/3)+56400+(66*2^(2/3)*3^(1/2)*163^(1/2)-678*2^(2/3))*(3*163^(1/2)*3^(1/2)+49)^(2/3))^(1/2)+(18*163^(1/2)*3^(1/2)-294)*2^(2/3)*(3*163^(1/2)*3^(1/2)+49)^(2/3)-1200*2^(1/3)*(3*163^(1/2)*3^(1/2)+49)^(1/3)+26400)^(1/2)+1/648000*(-49*2^(2/3)*(3*163^(1/2)*3^(1/2)+49)^(2/3)-200*(98+6*163^(1/2)*3^(1/2))^(1/3)-2200)*((120*2^(1/3)*163^(1/2)*3^(1/2)-6360*2^(1/3))*(3*163^(1/2)*3^(1/2)+49)^(1/3)+56400+(66*2^(2/3)*3^(1/2)*163^(1/2)-678*2^(2/3))*(3*163^(1/2)*3^(1/2)+49)^(2/3))^(1/2)+1/72000*(6520*2^(1/3)*(163^(1/2)*3^(1/2)-53)*(3*163^(1/2)*3^(1/2)+49)^(1/3)+3064400+(3586*2^(2/3)*3^(1/2)*163^(1/2)-36838*2^(2/3))*(3*163^(1/2)*3^(1/2)+49)^(2/3))^(1/2)*2^(2/3)*(3*163^(1/2)*3^(1/2)+49)^(2/3)-1/2)^(-n)+((((-577/20*I*163^(1/2)*3^(1/2)+9291/20*I)*(98+6*163^(1/2)*3^(1/2))^(2/3)+14670*I+(-2*I*163^(1/2)*3^(1/2)+1956*I)*(98+6*163^(1/2)*3^(1/2))^(1/3))*((40*163^(1/2)*3^(1/2)-2120)*(98+6*163^(1/2)*3^(1/2))^(1/3)+22*(98+6*163^(1/2)*3^(1/2))^(2/3)*163^(1/2)*3^(1/2)-226*(98+6*163^(1/2)*3^(1/2))^(2/3)+18800)^(1/2)+(6846*I*3^(1/2)-1386*I*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(2/3)+(34230*I*3^(1/2)-630*I*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(1/3)+684600*I*3^(1/2))*((-6*163^(1/2)*3^(1/2)+98)*(98+6*163^(1/2)*3^(1/2))^(2/3)+40*3^(1/2)*((40*163^(1/2)*3^(1/2)-2120)*(98+6*163^(1/2)*3^(1/2))^(1/3)+22*(98+6*163^(1/2)*3^(1/2))^(2/3)*163^(1/2)*3^(1/2)-226*(98+6*163^(1/2)*3^(1/2))^(2/3)+18800)^(1/2)+400*(98+6*163^(1/2)*3^(1/2))^(1/3)-8800)^(1/2)+((-7172*3^(1/2)+1152*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(2/3)+(-21190*3^(1/2)-810*163^(1/2))*(98+6*163^(1/2)*3^(1/2))^(1/3)-130400*3^(1/2))*((40*163^(1/2)*3^(1/2)-2120)*(98+6*163^(1/2)*3^(1/2))^(1/3)+22*(98+6*163^(1/2)*3^(1/2))^(2/3)*163^(1/2)*3^(1/2)-226*(98+6*163^(1/2)*3^(1/2))^(2/3)+18800)^(1/2)+(-40860*163^(1/2)*3^(1/2)+205380)*(98+6*163^(1/2)*3^(1/2))^(2/3)-138600*(98+6*163^(1/2)*3^(1/2))^(1/3)*163^(1/2)*3^(1/2)+4987800*(98+6*163^(1/2)*3^(1/2))^(1/3)+363816000)*(-1/120*I*(120*((120*2^(1/3)*163^(1/2)*3^(1/2)-6360*2^(1/3))*(3*163^(1/2)*3^(1/2)+49)^(1/3)+56400+(66*2^(2/3)*3^(1/2)*163^(1/2)-678*2^(2/3))*(3*163^(1/2)*3^(1/2)+49)^(2/3))^(1/2)+(-18*163^(1/2)*3^(1/2)+294)*2^(2/3)*(3*163^(1/2)*3^(1/2)+49)^(2/3)+1200*2^(1/3)*(3*163^(1/2)*3^(1/2)+49)^(1/3)-26400)^(1/2)+1/648000*(49*2^(2/3)*(3*163^(1/2)*3^(1/2)+49)^(2/3)+200*(98+6*163^(1/2)*3^(1/2))^(1/3)+2200)*((120*2^(1/3)*163^(1/2)*3^(1/2)-6360*2^(1/3))*(3*163^(1/2)*3^(1/2)+49)^(1/3)+56400+(66*2^(2/3)*3^(1/2)*163^(1/2)-678*2^(2/3))*(3*163^(1/2)*3^(1/2)+49)^(2/3))^(1/2)-1/72000*(6520*2^(1/3)*(163^(1/2)*3^(1/2)-53)*(3*163^(1/2)*3^(1/2)+49)^(1/3)+3064400+(3586*2^(2/3)*3^(1/2)*163^(1/2)-36838*2^(2/3))*(3*163^(1/2)*3^(1/2)+49)^(2/3))^(1/2)*2^(2/3)*(3*163^(1/2)*3^(1/2)+49)^(2/3)-1/2)^(-n)+352080000*(1/120*I*(120*((120*2^(1/3)*163^(1/2)*3^(1/2)-6360*2^(1/3))*(3*163^(1/2)*3^(1/2)+49)^(1/3)+56400+(66*2^(2/3)*3^(1/2)*163^(1/2)-678*2^(2/3))*(3*163^(1/2)*3^(1/2)+49)^(2/3))^(1/2)+(-18*163^(1/2)*3^(1/2)+294)*2^(2/3)*(3*163^(1/2)*3^(1/2)+49)^(2/3)+1200*2^(1/3)*(3*163^(1/2)*3^(1/2)+49)^(1/3)-26400)^(1/2)+1/648000*(49*2^(2/3)*(3*163^(1/2)*3^(1/2)+49)^(2/3)+200*(98+6*163^(1/2)*3^(1/2))^(1/3)+2200)*((120*2^(1/3)*163^(1/2)*3^(1/2)-6360*2^(1/3))*(3*163^(1/2)*3^(1/2)+49)^(1/3)+56400+(66*2^(2/3)*3^(1/2)*163^(1/2)-678*2^(2/3))*(3*163^(1/2)*3^(1/2)+49)^(2/3))^(1/2)-1/72000*(6520*2^(1/3)*(163^(1/2)*3^(1/2)-53)*(3*163^(1/2)*3^(1/2)+49)^(1/3)+3064400+(3586*2^(2/3)*3^(1/2)*163^(1/2)-36838*2^(2/3))*(3*163^(1/2)*3^(1/2)+49)^(2/3))^(1/2)*2^(2/3)*(3*163^(1/2)*3^(1/2)+49)^(2/3)-1/2)^(-n))
Explicit closed form in latex syntax:
\frac{7 \left(\left(\left(\left(\mathrm{I} \sqrt{3}\, \sqrt{163}-\frac{49 \,\mathrm{I}}{3}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{200 \,\mathrm{I} \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{12700 \,\mathrm{I}}{21}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(-210 \,\mathrm{I} \sqrt{3}+\frac{270 \,\mathrm{I} \sqrt{163}}{7}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{6000 \,\mathrm{I} \sqrt{3}\, \left(\left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+44\right)}{7}\right) \sqrt{\left(-6 \sqrt{163}\, \sqrt{3}+98\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+40 \sqrt{3}\, \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+400 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-8800}+\frac{176040000}{7}+\left(\left(-\frac{6520 \sqrt{3}}{21}+\frac{440 \sqrt{163}}{7}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(-\frac{32600 \sqrt{3}}{21}+\frac{200 \sqrt{163}}{7}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{326000 \sqrt{3}}{21}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}\right) \left(\left(\left(\left(\left(\left(-\frac{652}{3}-\frac{141 \,\mathrm{I} \sqrt{163}}{10}\right) \sqrt{3}+\frac{7009 \,\mathrm{I}}{30}+\frac{47 \sqrt{163}}{3}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(-\frac{1793}{6}+\mathrm{I} \sqrt{163}\right) \sqrt{3}+\frac{2771 \,\mathrm{I}}{3}+\frac{713 \sqrt{163}}{6}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{55 \sqrt{163}}{3}+\frac{18745 \,\mathrm{I}}{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(\left(\frac{17115 \,\mathrm{I}}{2}-\frac{1491 \sqrt{163}}{2}\right) \sqrt{3}-\frac{3015 \,\mathrm{I} \sqrt{163}}{2}-\frac{60147}{2}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(31785 \,\mathrm{I}+12675 \sqrt{163}\right) \sqrt{3}+315 \,\mathrm{I} \sqrt{163}-256725\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+48900 \sqrt{3}\, \left(\mathrm{I}+\frac{47 \sqrt{163}}{163}\right)\right) \sqrt{\left(-6 \sqrt{163}\, \sqrt{3}+98\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+40 \sqrt{3}\, \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+400 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-8800}+\left(\left(\left(\frac{331 \,\mathrm{I} \sqrt{163}}{3}+\frac{15974}{3}\right) \sqrt{3}-1141 \,\mathrm{I}-1228 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(-\frac{595 \,\mathrm{I} \sqrt{163}}{3}+\frac{101875}{3}\right) \sqrt{3}+10595 \,\mathrm{I}-1225 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{1400 \sqrt{3}\, \left(\mathrm{I} \sqrt{163}+\frac{7661}{7}\right)}{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(\left(102690 \,\mathrm{I}+20430 \sqrt{163}\right) \sqrt{3}-61290 \,\mathrm{I} \sqrt{163}-102690\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+363816000+\left(\left(-2493900 \,\mathrm{I}+69300 \sqrt{163}\right) \sqrt{3}+207900 \,\mathrm{I} \sqrt{163}-2493900\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(-\frac{\sqrt{120 \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}+\left(18 \sqrt{163}\, \sqrt{3}-294\right) 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}-1200 \,2^{\frac{1}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+26400}}{120}+\frac{\left(-49 \,2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}-200 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-2200\right) \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}}{648000}+\frac{\sqrt{6520 \,2^{\frac{1}{3}} \left(\sqrt{163}\, \sqrt{3}-53\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+3064400+\left(3586 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-36838 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}{72000}-\frac{1}{2}\right)^{-n}+\left(\left(\left(\left(\left(\frac{652}{3}-\frac{141 \,\mathrm{I} \sqrt{163}}{10}\right) \sqrt{3}+\frac{7009 \,\mathrm{I}}{30}-\frac{47 \sqrt{163}}{3}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(\frac{1793}{6}+\mathrm{I} \sqrt{163}\right) \sqrt{3}+\frac{2771 \,\mathrm{I}}{3}-\frac{713 \sqrt{163}}{6}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{55 \sqrt{163}}{3}+\frac{18745 \,\mathrm{I}}{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(\left(\frac{17115 \,\mathrm{I}}{2}+\frac{1491 \sqrt{163}}{2}\right) \sqrt{3}-\frac{3015 \,\mathrm{I} \sqrt{163}}{2}+\frac{60147}{2}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(31785 \,\mathrm{I}-12675 \sqrt{163}\right) \sqrt{3}+315 \,\mathrm{I} \sqrt{163}+256725\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+48900 \sqrt{3}\, \left(\mathrm{I}-\frac{47 \sqrt{163}}{163}\right)\right) \sqrt{\left(-6 \sqrt{163}\, \sqrt{3}+98\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+40 \sqrt{3}\, \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+400 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-8800}+\left(\left(\left(-\frac{331 \,\mathrm{I} \sqrt{163}}{3}+\frac{15974}{3}\right) \sqrt{3}+1141 \,\mathrm{I}-1228 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(\left(\frac{595 \,\mathrm{I} \sqrt{163}}{3}+\frac{101875}{3}\right) \sqrt{3}-10595 \,\mathrm{I}-1225 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{1400 \sqrt{3}\, \left(\mathrm{I} \sqrt{163}-\frac{7661}{7}\right)}{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(\left(-102690 \,\mathrm{I}+20430 \sqrt{163}\right) \sqrt{3}+61290 \,\mathrm{I} \sqrt{163}-102690\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+363816000+\left(\left(2493900 \,\mathrm{I}+69300 \sqrt{163}\right) \sqrt{3}-207900 \,\mathrm{I} \sqrt{163}-2493900\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\sqrt{120 \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}+\left(18 \sqrt{163}\, \sqrt{3}-294\right) 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}-1200 \,2^{\frac{1}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+26400}}{120}+\frac{\left(-49 \,2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}-200 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-2200\right) \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}}{648000}+\frac{\sqrt{6520 \,2^{\frac{1}{3}} \left(\sqrt{163}\, \sqrt{3}-53\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+3064400+\left(3586 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-36838 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}{72000}-\frac{1}{2}\right)^{-n}+\left(\left(\left(\left(-\frac{577 \,\mathrm{I} \sqrt{163}\, \sqrt{3}}{20}+\frac{9291 \,\mathrm{I}}{20}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+14670 \,\mathrm{I}+\left(-2 \,\mathrm{I} \sqrt{163}\, \sqrt{3}+1956 \,\mathrm{I}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(6846 \,\mathrm{I} \sqrt{3}-1386 \,\mathrm{I} \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(34230 \,\mathrm{I} \sqrt{3}-630 \,\mathrm{I} \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+684600 \,\mathrm{I} \sqrt{3}\right) \sqrt{\left(-6 \sqrt{163}\, \sqrt{3}+98\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+40 \sqrt{3}\, \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+400 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-8800}+\left(\left(-7172 \sqrt{3}+1152 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(-21190 \sqrt{3}-810 \sqrt{163}\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}-130400 \sqrt{3}\right) \sqrt{\left(40 \sqrt{163}\, \sqrt{3}-2120\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+22 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{163}\, \sqrt{3}-226 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}+18800}+\left(-40860 \sqrt{163}\, \sqrt{3}+205380\right) \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{2}{3}}-138600 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}+4987800 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+363816000\right) \left(-\frac{\mathrm{I} \sqrt{120 \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}+\left(-18 \sqrt{163}\, \sqrt{3}+294\right) 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}+1200 \,2^{\frac{1}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}-26400}}{120}+\frac{\left(49 \,2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}+200 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+2200\right) \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}}{648000}-\frac{\sqrt{6520 \,2^{\frac{1}{3}} \left(\sqrt{163}\, \sqrt{3}-53\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+3064400+\left(3586 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-36838 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}{72000}-\frac{1}{2}\right)^{-n}+352080000 \left(\frac{\mathrm{I} \sqrt{120 \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}+\left(-18 \sqrt{163}\, \sqrt{3}+294\right) 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}+1200 \,2^{\frac{1}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}-26400}}{120}+\frac{\left(49 \,2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}+200 \left(98+6 \sqrt{163}\, \sqrt{3}\right)^{\frac{1}{3}}+2200\right) \sqrt{\left(120 \,2^{\frac{1}{3}} \sqrt{163}\, \sqrt{3}-6360 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+56400+\left(66 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-678 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}}{648000}-\frac{\sqrt{6520 \,2^{\frac{1}{3}} \left(\sqrt{163}\, \sqrt{3}-53\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{1}{3}}+3064400+\left(3586 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{163}-36838 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{163}\, \sqrt{3}+49\right)^{\frac{2}{3}}}{72000}-\frac{1}{2}\right)^{-n}\right)}{247920652800000000}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = a(n)+2*a(n+1)-4*a(n+2)+4*a(n+3), n >= 4
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(n +4\right) = a \! \left(n \right)+2 a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16703/
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[14,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[7,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]+F[17,x]+F[57,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[31,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[1,x]+F[4,x]
F[23,x] = F[16,x]+F[24,x]+F[29,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]+F[4,x]
F[27,x] = F[28,x]
F[28,x] = F[11,x]*F[4,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[7,x]
F[31,x] = F[32,x]+F[37,x]
F[32,x] = F[16,x]+F[17,x]+F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[2,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[2,x]*F[4,x]
F[37,x] = F[16,x]+F[38,x]+F[47,x]+F[55,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[4,x]
F[43,x] = F[44,x]
F[44,x] = 2*F[16,x]+F[38,x]+F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[35,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[35,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[2,x]+F[52,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[15,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[54,x]+F[59,x]
F[59,x] = F[15,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_{14}\! \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_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{57}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{31}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{16}\! \left(x \right)+F_{24}\! \left(x \right)+F_{29}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \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_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{7}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{37}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{47}\! \left(x \right)+F_{55}\! \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_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{35}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{35}\! \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_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{2}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{15}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{15}\! \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_14(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_7(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_57(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_31(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_1(x) + F_4(x))
Eq(F_23(x), F_16(x) + F_24(x) + F_29(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x) + F_4(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_11(x)*F_4(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_7(x))
Eq(F_31(x), F_32(x) + F_37(x))
Eq(F_32(x), F_16(x) + F_17(x) + F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_2(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_2(x)*F_4(x))
Eq(F_37(x), F_16(x) + F_38(x) + F_47(x) + F_55(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_4(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), 2*F_16(x) + F_38(x) + F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_35(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_35(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_2(x) + F_52(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_15(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_54(x) + F_59(x))
Eq(F_59(x), F_15(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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], "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, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}], "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, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"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, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}], "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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}], "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, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}], "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": []}, 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