0132_0231_0321_2031_2103
Counting sequence:
1, 1, 2, 6, 19, 55, 151, 411, 1128, 3116, 8619, 23819, 65773, 181589, 501386, 1384510, 3823239, 10557539, 29153387, 80503319, 222299736, 613853436, 1695081727, 4680761991, 12925352809, 35691781625, 98558492146, 272157233686, 751528953323, 2075255395071, 5730564240559, 15824253033859, 43696741466248, 120663213053004, 333196721256771, 920082038660915, 2540694142038069, 7015816472927789, 19373320057421242, 53497056471859214, 147725585634079871, 407926156879636171, 1126438245293003427, 3110521595782011551, 8589325369815653624, 23718372638404206620, 65495388332959607991, 180857513214862024911, 499415927118082539793, 1379076068368262673201, 3808150079075045701474, 10515741196146061623974, 29037934589800151221123, 80184708763138742243719, 221420276967213505066631, 611425043608506975556395, 1688375559240315497088424, 4662242835550719873081964, 12874213997402543414177947, 35550569092426343255212059, 98168553284135850939374781, 271080466499572581011232581, 748555589944718985783082602, 2067044809509982785088878686, 5707891707598762386424406839, 15761645609123584144396796915, 43523858726485174771603750811, 120185818500227810695217524711, 331878454512578467886862939288, 916441806064239101302175933756, 2530642084421460483326862888239, 6988058943915188109039241166039, 19296671032322984892618216897273, 53285399553466080059886289245609, 147141120912331608695937900048466, 406312228955203162163832262720950, 1121981580505341910307802924521179, 3098215060448144345255688273919439, 8555342375999012086835005424004127, 23624532752731897424456624287224979, 65236260953230448554982026049078216, 180141964613706499790240692730123916, 497440027075600358170714143248008627, 1373619861799522625376037837824952771, 3793083431228145158366525432029341829, 10474136488831075439059009804714761885, 28923048273457266824078277553260158106, 79867464236385793359860200755409143086, 220544244964809286233651979906224171183, 609005988259968320870679035865606363611, 1681695633434827292577975324506201328979, 4643797036535090504252974239077079325679, 12823278176970908123838388457145960283448, 35409916047207457465271262967088856184924, 97780157083550478551482981863440141954151, 270007957842583349879182140685977207351455, 745593988318381744744237093752209504658209, 2058866708442019601252240861999801405877089, 5685308880630053559355989523845507071577282, 15699285891426226560018884301077836482907974, 43351659984641098593048790126835225701912051
Generating function in Maple syntax:
(x-1)^3/(x^4+4*x^3-5*x^2+4*x-1)
Generating function in latex syntax:
\frac{\left(x -1\right)^{3}}{x^{4}+4 x^{3}-5 x^{2}+4 x -1}
Generating function in sympy syntax:
(x - 1)**3/(x**4 + 4*x**3 - 5*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^4+4*x^3-5*x^2+4*x-1)*F(x)-(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}+4 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
-89/4373768415988125000*(((I*5^(2/3)*(17^(1/2)*3^(1/2)-11/6)*(6*17^(1/2)*3^(1/2)+11)^(2/3)-245/6*I*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)+2695/8*I)*((350*17^(1/2)*5^(1/3)*3^(1/2)-9625*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+211925+(220*17^(1/2)*3^(1/2)*5^(2/3)+5*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)-46585/72*I*5^(2/3)*(3^(1/2)-18/11*17^(1/2))*(6*17^(1/2)*3^(1/2)+11)^(2/3)-1037575/72*I*3^(1/2)*(5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)-61/11))*(70*3^(1/2)*((350*17^(1/2)*5^(1/3)*3^(1/2)-9625*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+211925+(220*17^(1/2)*3^(1/2)*5^(2/3)+5*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+(-30*17^(1/2)*3^(1/2)*5^(2/3)+55*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3)+1225*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)-53900)^(1/2)-280616875/4+((32725/24*3^(1/2)-10675/24*17^(1/2))*(6*17^(1/2)*3^(1/2)+11)^(2/3)+20825/24*(3^(1/2)+11/17*17^(1/2))*(6*17^(1/2)*3^(1/2)+11)^(1/3)*5^(2/3))*((220*17^(1/2)*3^(1/2)+5)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+(70*17^(1/2)*3^(1/2)*5^(2/3)-1925*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+42385*5^(1/3))^(1/2))*((((2381/3115*((I*17^(1/2)-15351/2381)*3^(1/2)-731/2381*I+5943/2381*17^(1/2))*5^(2/3)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+((1785/89+I*17^(1/2))*3^(1/2)+861/89*17^(1/2)-2941/89*I)*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)-15120/89*17^(1/2)+97580/89*I)*((350*17^(1/2)*5^(1/3)*3^(1/2)-9625*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+211925+(220*17^(1/2)*3^(1/2)*5^(2/3)+5*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)-53856/89*((I-223/816*17^(1/2))*3^(1/2)-128/51*I*17^(1/2)+209/48)*5^(2/3)*(6*17^(1/2)*3^(1/2)+11)^(2/3)-1884960/89*((I-77/816*17^(1/2))*3^(1/2)-2/51*I*17^(1/2)-7/48)*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)+26114550/89*(-45/323*17^(1/2)+I)*3^(1/2))*(70*3^(1/2)*((350*17^(1/2)*5^(1/3)*3^(1/2)-9625*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+211925+(220*17^(1/2)*3^(1/2)*5^(2/3)+5*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+(-30*17^(1/2)*3^(1/2)*5^(2/3)+55*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3)+1225*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)-53900)^(1/2)-78400/89*3^(1/2)*(I*17^(1/2)+119/32)*((350*17^(1/2)*5^(1/3)*3^(1/2)-9625*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+211925+(220*17^(1/2)*3^(1/2)*5^(2/3)+5*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+(((1090/89*I*17^(1/2)+551905/89)*3^(1/2)+9435/89*I-229590/89*17^(1/2))*(6*17^(1/2)*3^(1/2)+11)^(2/3)-1960/89*(6*17^(1/2)*3^(1/2)+11)^(1/3)*5^(2/3)*(I*3^(1/2)*17^(1/2)-51/8*I-2023/8*3^(1/2)-108*17^(1/2)))*((220*17^(1/2)*3^(1/2)+5)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+(70*17^(1/2)*3^(1/2)*5^(2/3)-1925*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+42385*5^(1/3))^(1/2)-6636630/89*((I-237/2873*17^(1/2))*3^(1/2)+711/2873*I*17^(1/2)-1)*5^(2/3)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+15201417000/89-34361250/89*((I-81/425*17^(1/2))*3^(1/2)-243/425*I*17^(1/2)+1)*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3))*(-1/210*(210*((1050*17^(1/2)*5^(1/3)*3^(1/2)-28875*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+635775+(660*17^(1/2)*3^(1/2)*5^(2/3)+15*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+(90*17^(1/2)*3^(1/2)-165)*5^(2/3)*(6*17^(1/2)*3^(1/2)+11)^(2/3)-3675*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)+161700)^(1/2)-1/630*((1050*17^(1/2)*5^(1/3)*3^(1/2)-28875*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+635775+(660*17^(1/2)*3^(1/2)*5^(2/3)+15*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+1/679140*(-11*(6*17^(1/2)*3^(1/2)+11)^(2/3)-49*(275+150*17^(1/2)*3^(1/2))^(1/3))*((660*17^(1/2)*3^(1/2)+15)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+(210*17^(1/2)*3^(1/2)*5^(2/3)-5775*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+127155*5^(1/3))^(1/2)+1/37730*((3740*17^(1/2)*3^(1/2)+85)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+(1190*17^(1/2)*3^(1/2)*5^(2/3)-32725*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+720545*5^(1/3))^(1/2)*(6*17^(1/2)*3^(1/2)+11)^(2/3)-1)^(-n)+(((2381/3115*5^(2/3)*((I*17^(1/2)+15351/2381)*3^(1/2)-731/2381*I-5943/2381*17^(1/2))*(6*17^(1/2)*3^(1/2)+11)^(2/3)+((-1785/89+I*17^(1/2))*3^(1/2)-861/89*17^(1/2)-2941/89*I)*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)+15120/89*17^(1/2)+97580/89*I)*((350*17^(1/2)*5^(1/3)*3^(1/2)-9625*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+211925+(220*17^(1/2)*3^(1/2)*5^(2/3)+5*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)-53856/89*((I+223/816*17^(1/2))*3^(1/2)-128/51*I*17^(1/2)-209/48)*5^(2/3)*(6*17^(1/2)*3^(1/2)+11)^(2/3)-1884960/89*((I+77/816*17^(1/2))*3^(1/2)-2/51*I*17^(1/2)+7/48)*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)+26114550/89*3^(1/2)*(45/323*17^(1/2)+I))*(70*3^(1/2)*((350*17^(1/2)*5^(1/3)*3^(1/2)-9625*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+211925+(220*17^(1/2)*3^(1/2)*5^(2/3)+5*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+(-30*17^(1/2)*3^(1/2)*5^(2/3)+55*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3)+1225*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)-53900)^(1/2)+78400/89*(I*17^(1/2)-119/32)*3^(1/2)*((350*17^(1/2)*5^(1/3)*3^(1/2)-9625*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+211925+(220*17^(1/2)*3^(1/2)*5^(2/3)+5*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+(((-1090/89*I*17^(1/2)+551905/89)*3^(1/2)-9435/89*I-229590/89*17^(1/2))*(6*17^(1/2)*3^(1/2)+11)^(2/3)+1960/89*(I*3^(1/2)*17^(1/2)-51/8*I+2023/8*3^(1/2)+108*17^(1/2))*(6*17^(1/2)*3^(1/2)+11)^(1/3)*5^(2/3))*((220*17^(1/2)*3^(1/2)+5)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+(70*17^(1/2)*3^(1/2)*5^(2/3)-1925*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+42385*5^(1/3))^(1/2)+6636630/89*((I+237/2873*17^(1/2))*3^(1/2)+711/2873*I*17^(1/2)+1)*5^(2/3)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+15201417000/89+34361250/89*((I+81/425*17^(1/2))*3^(1/2)-243/425*I*17^(1/2)-1)*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3))*(1/210*(210*((1050*17^(1/2)*5^(1/3)*3^(1/2)-28875*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+635775+(660*17^(1/2)*3^(1/2)*5^(2/3)+15*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+(90*17^(1/2)*3^(1/2)-165)*5^(2/3)*(6*17^(1/2)*3^(1/2)+11)^(2/3)-3675*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)+161700)^(1/2)-1/630*((1050*17^(1/2)*5^(1/3)*3^(1/2)-28875*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+635775+(660*17^(1/2)*3^(1/2)*5^(2/3)+15*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+1/679140*(-11*(6*17^(1/2)*3^(1/2)+11)^(2/3)-49*(275+150*17^(1/2)*3^(1/2))^(1/3))*((660*17^(1/2)*3^(1/2)+15)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+(210*17^(1/2)*3^(1/2)*5^(2/3)-5775*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+127155*5^(1/3))^(1/2)+1/37730*((3740*17^(1/2)*3^(1/2)+85)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+(1190*17^(1/2)*3^(1/2)*5^(2/3)-32725*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+720545*5^(1/3))^(1/2)*(6*17^(1/2)*3^(1/2)+11)^(2/3)-1)^(-n)+(((11558/3115*I*5^(2/3)*(17^(1/2)*3^(1/2)-14229/5779)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+171360/89*I-2*I*(17^(1/2)*3^(1/2)+6579/89)*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3))*((350*17^(1/2)*5^(1/3)*3^(1/2)-9625*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+211925+(220*17^(1/2)*3^(1/2)*5^(2/3)+5*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)-303688/89*I*(-654/493*17^(1/2)+3^(1/2))*5^(2/3)*(6*17^(1/2)*3^(1/2)+11)^(2/3)-5393080/89*I*(3^(1/2)+48/1751*17^(1/2))*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)+43066100/89*I*3^(1/2))*(70*3^(1/2)*((350*17^(1/2)*5^(1/3)*3^(1/2)-9625*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+211925+(220*17^(1/2)*3^(1/2)*5^(2/3)+5*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+(-30*17^(1/2)*3^(1/2)*5^(2/3)+55*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3)+1225*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)-53900)^(1/2)-583100/89*3^(1/2)*((350*17^(1/2)*5^(1/3)*3^(1/2)-9625*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+211925+(220*17^(1/2)*3^(1/2)*5^(2/3)+5*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+((-85510/89*3^(1/2)-71220/89*17^(1/2))*(6*17^(1/2)*3^(1/2)+11)^(2/3)+234430/89*(6*17^(1/2)*3^(1/2)+11)^(1/3)*5^(2/3)*(3^(1/2)-948/3349*17^(1/2)))*((220*17^(1/2)*3^(1/2)+5)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+(70*17^(1/2)*3^(1/2)*5^(2/3)-1925*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+42385*5^(1/3))^(1/2)-1094940/89*(17^(1/2)*3^(1/2)+2873/237)*5^(2/3)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+15201417000/89-13097700/89*5^(1/3)*(17^(1/2)*3^(1/2)-425/81)*(6*17^(1/2)*3^(1/2)+11)^(1/3))*(-1/210*I*(210*((1050*17^(1/2)*5^(1/3)*3^(1/2)-28875*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+635775+(660*17^(1/2)*3^(1/2)*5^(2/3)+15*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+(-90*17^(1/2)*3^(1/2)+165)*5^(2/3)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+3675*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)-161700)^(1/2)+1/630*((1050*17^(1/2)*5^(1/3)*3^(1/2)-28875*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+635775+(660*17^(1/2)*3^(1/2)*5^(2/3)+15*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+1/679140*(11*(6*17^(1/2)*3^(1/2)+11)^(2/3)+49*(275+150*17^(1/2)*3^(1/2))^(1/3))*((660*17^(1/2)*3^(1/2)+15)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+(210*17^(1/2)*3^(1/2)*5^(2/3)-5775*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+127155*5^(1/3))^(1/2)-1/37730*((3740*17^(1/2)*3^(1/2)+85)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+(1190*17^(1/2)*3^(1/2)*5^(2/3)-32725*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+720545*5^(1/3))^(1/2)*(6*17^(1/2)*3^(1/2)+11)^(2/3)-1)^(-n)+15586263000/89*(1/210*I*(210*((1050*17^(1/2)*5^(1/3)*3^(1/2)-28875*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+635775+(660*17^(1/2)*3^(1/2)*5^(2/3)+15*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+(-90*17^(1/2)*3^(1/2)+165)*5^(2/3)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+3675*5^(1/3)*(6*17^(1/2)*3^(1/2)+11)^(1/3)-161700)^(1/2)+1/630*((1050*17^(1/2)*5^(1/3)*3^(1/2)-28875*5^(1/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+635775+(660*17^(1/2)*3^(1/2)*5^(2/3)+15*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(2/3))^(1/2)+1/679140*(11*(6*17^(1/2)*3^(1/2)+11)^(2/3)+49*(275+150*17^(1/2)*3^(1/2))^(1/3))*((660*17^(1/2)*3^(1/2)+15)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+(210*17^(1/2)*3^(1/2)*5^(2/3)-5775*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+127155*5^(1/3))^(1/2)-1/37730*((3740*17^(1/2)*3^(1/2)+85)*(6*17^(1/2)*3^(1/2)+11)^(2/3)+(1190*17^(1/2)*3^(1/2)*5^(2/3)-32725*5^(2/3))*(6*17^(1/2)*3^(1/2)+11)^(1/3)+720545*5^(1/3))^(1/2)*(6*17^(1/2)*3^(1/2)+11)^(2/3)-1)^(-n))
Explicit closed form in latex syntax:
-\frac{89 \left(\left(\left(\mathrm{I} \,5^{\frac{2}{3}} \left(\sqrt{17}\, \sqrt{3}-\frac{11}{6}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}-\frac{245 \,\mathrm{I} \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{6}+\frac{2695 \,\mathrm{I}}{8}\right) \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}-\frac{46585 \,\mathrm{I} \,5^{\frac{2}{3}} \left(\sqrt{3}-\frac{18 \sqrt{17}}{11}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{72}-\frac{1037575 \,\mathrm{I} \sqrt{3}\, \left(5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-\frac{61}{11}\right)}{72}\right) \sqrt{70 \sqrt{3}\, \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-30 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+55 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+1225 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-53900}-\frac{280616875}{4}+\left(\left(\frac{32725 \sqrt{3}}{24}-\frac{10675 \sqrt{17}}{24}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\frac{20825 \left(\sqrt{3}+\frac{11 \sqrt{17}}{17}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}} 5^{\frac{2}{3}}}{24}\right) \sqrt{\left(220 \sqrt{17}\, \sqrt{3}+5\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(70 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-1925 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+42385 \,5^{\frac{1}{3}}}\right) \left(\left(\left(\left(\frac{2381 \left(\left(\mathrm{I} \sqrt{17}-\frac{15351}{2381}\right) \sqrt{3}-\frac{731 \,\mathrm{I}}{2381}+\frac{5943 \sqrt{17}}{2381}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{3115}+\left(\left(\frac{1785}{89}+\mathrm{I} \sqrt{17}\right) \sqrt{3}+\frac{861 \sqrt{17}}{89}-\frac{2941 \,\mathrm{I}}{89}\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-\frac{15120 \sqrt{17}}{89}+\frac{97580 \,\mathrm{I}}{89}\right) \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}-\frac{53856 \left(\left(\mathrm{I}-\frac{223 \sqrt{17}}{816}\right) \sqrt{3}-\frac{128 \,\mathrm{I} \sqrt{17}}{51}+\frac{209}{48}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}-\frac{1884960 \left(\left(\mathrm{I}-\frac{77 \sqrt{17}}{816}\right) \sqrt{3}-\frac{2 \,\mathrm{I} \sqrt{17}}{51}-\frac{7}{48}\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}+\frac{26114550 \left(-\frac{45 \sqrt{17}}{323}+\mathrm{I}\right) \sqrt{3}}{89}\right) \sqrt{70 \sqrt{3}\, \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-30 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+55 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+1225 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-53900}-\frac{78400 \sqrt{3}\, \left(\mathrm{I} \sqrt{17}+\frac{119}{32}\right) \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{89}+\left(\left(\left(\frac{1090 \,\mathrm{I} \sqrt{17}}{89}+\frac{551905}{89}\right) \sqrt{3}+\frac{9435 \,\mathrm{I}}{89}-\frac{229590 \sqrt{17}}{89}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}-\frac{1960 \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}} 5^{\frac{2}{3}} \left(\mathrm{I} \sqrt{3}\, \sqrt{17}-\frac{51 \,\mathrm{I}}{8}-\frac{2023 \sqrt{3}}{8}-108 \sqrt{17}\right)}{89}\right) \sqrt{\left(220 \sqrt{17}\, \sqrt{3}+5\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(70 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-1925 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+42385 \,5^{\frac{1}{3}}}-\frac{6636630 \left(\left(\mathrm{I}-\frac{237 \sqrt{17}}{2873}\right) \sqrt{3}+\frac{711 \,\mathrm{I} \sqrt{17}}{2873}-1\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}+\frac{15201417000}{89}-\frac{34361250 \left(\left(\mathrm{I}-\frac{81 \sqrt{17}}{425}\right) \sqrt{3}-\frac{243 \,\mathrm{I} \sqrt{17}}{425}+1\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}\right) \left(-\frac{\sqrt{210 \sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(90 \sqrt{17}\, \sqrt{3}-165\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}-3675 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+161700}}{210}-\frac{\sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{630}+\frac{\left(-11 \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}-49 \left(275+150 \sqrt{17}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(660 \sqrt{17}\, \sqrt{3}+15\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(210 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-5775 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+127155 \,5^{\frac{1}{3}}}}{679140}+\frac{\sqrt{\left(3740 \sqrt{17}\, \sqrt{3}+85\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(1190 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-32725 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+720545 \,5^{\frac{1}{3}}}\, \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{37730}-1\right)^{-n}+\left(\left(\left(\frac{2381 \,5^{\frac{2}{3}} \left(\left(\mathrm{I} \sqrt{17}+\frac{15351}{2381}\right) \sqrt{3}-\frac{731 \,\mathrm{I}}{2381}-\frac{5943 \sqrt{17}}{2381}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{3115}+\left(\left(-\frac{1785}{89}+\mathrm{I} \sqrt{17}\right) \sqrt{3}-\frac{861 \sqrt{17}}{89}-\frac{2941 \,\mathrm{I}}{89}\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+\frac{15120 \sqrt{17}}{89}+\frac{97580 \,\mathrm{I}}{89}\right) \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}-\frac{53856 \left(\left(\mathrm{I}+\frac{223 \sqrt{17}}{816}\right) \sqrt{3}-\frac{128 \,\mathrm{I} \sqrt{17}}{51}-\frac{209}{48}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}-\frac{1884960 \left(\left(\mathrm{I}+\frac{77 \sqrt{17}}{816}\right) \sqrt{3}-\frac{2 \,\mathrm{I} \sqrt{17}}{51}+\frac{7}{48}\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}+\frac{26114550 \sqrt{3}\, \left(\frac{45 \sqrt{17}}{323}+\mathrm{I}\right)}{89}\right) \sqrt{70 \sqrt{3}\, \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-30 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+55 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+1225 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-53900}+\frac{78400 \left(\mathrm{I} \sqrt{17}-\frac{119}{32}\right) \sqrt{3}\, \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{89}+\left(\left(\left(-\frac{1090 \,\mathrm{I} \sqrt{17}}{89}+\frac{551905}{89}\right) \sqrt{3}-\frac{9435 \,\mathrm{I}}{89}-\frac{229590 \sqrt{17}}{89}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\frac{1960 \left(\mathrm{I} \sqrt{3}\, \sqrt{17}-\frac{51 \,\mathrm{I}}{8}+\frac{2023 \sqrt{3}}{8}+108 \sqrt{17}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}} 5^{\frac{2}{3}}}{89}\right) \sqrt{\left(220 \sqrt{17}\, \sqrt{3}+5\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(70 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-1925 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+42385 \,5^{\frac{1}{3}}}+\frac{6636630 \left(\left(\mathrm{I}+\frac{237 \sqrt{17}}{2873}\right) \sqrt{3}+\frac{711 \,\mathrm{I} \sqrt{17}}{2873}+1\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}+\frac{15201417000}{89}+\frac{34361250 \left(\left(\mathrm{I}+\frac{81 \sqrt{17}}{425}\right) \sqrt{3}-\frac{243 \,\mathrm{I} \sqrt{17}}{425}-1\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}\right) \left(\frac{\sqrt{210 \sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(90 \sqrt{17}\, \sqrt{3}-165\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}-3675 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+161700}}{210}-\frac{\sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{630}+\frac{\left(-11 \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}-49 \left(275+150 \sqrt{17}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(660 \sqrt{17}\, \sqrt{3}+15\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(210 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-5775 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+127155 \,5^{\frac{1}{3}}}}{679140}+\frac{\sqrt{\left(3740 \sqrt{17}\, \sqrt{3}+85\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(1190 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-32725 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+720545 \,5^{\frac{1}{3}}}\, \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{37730}-1\right)^{-n}+\left(\left(\left(\frac{11558 \,\mathrm{I} \,5^{\frac{2}{3}} \left(\sqrt{17}\, \sqrt{3}-\frac{14229}{5779}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{3115}+\frac{171360 \,\mathrm{I}}{89}-2 \,\mathrm{I} \left(\sqrt{17}\, \sqrt{3}+\frac{6579}{89}\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}\right) \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}-\frac{303688 \,\mathrm{I} \left(-\frac{654 \sqrt{17}}{493}+\sqrt{3}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}-\frac{5393080 \,\mathrm{I} \left(\sqrt{3}+\frac{48 \sqrt{17}}{1751}\right) 5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}+\frac{43066100 \,\mathrm{I} \sqrt{3}}{89}\right) \sqrt{70 \sqrt{3}\, \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-30 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+55 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+1225 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-53900}-\frac{583100 \sqrt{3}\, \sqrt{\left(350 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-9625 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+211925+\left(220 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+5 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{89}+\left(\left(-\frac{85510 \sqrt{3}}{89}-\frac{71220 \sqrt{17}}{89}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\frac{234430 \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}} 5^{\frac{2}{3}} \left(\sqrt{3}-\frac{948 \sqrt{17}}{3349}\right)}{89}\right) \sqrt{\left(220 \sqrt{17}\, \sqrt{3}+5\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(70 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-1925 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+42385 \,5^{\frac{1}{3}}}-\frac{1094940 \left(\sqrt{17}\, \sqrt{3}+\frac{2873}{237}\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{89}+\frac{15201417000}{89}-\frac{13097700 \,5^{\frac{1}{3}} \left(\sqrt{17}\, \sqrt{3}-\frac{425}{81}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}}{89}\right) \left(-\frac{\mathrm{I} \sqrt{210 \sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-90 \sqrt{17}\, \sqrt{3}+165\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+3675 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-161700}}{210}+\frac{\sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{630}+\frac{\left(11 \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+49 \left(275+150 \sqrt{17}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(660 \sqrt{17}\, \sqrt{3}+15\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(210 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-5775 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+127155 \,5^{\frac{1}{3}}}}{679140}-\frac{\sqrt{\left(3740 \sqrt{17}\, \sqrt{3}+85\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(1190 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-32725 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+720545 \,5^{\frac{1}{3}}}\, \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{37730}-1\right)^{-n}+\frac{15586263000 \left(\frac{\mathrm{I} \sqrt{210 \sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}+\left(-90 \sqrt{17}\, \sqrt{3}+165\right) 5^{\frac{2}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+3675 \,5^{\frac{1}{3}} \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}-161700}}{210}+\frac{\sqrt{\left(1050 \sqrt{17}\, 5^{\frac{1}{3}} \sqrt{3}-28875 \,5^{\frac{1}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+635775+\left(660 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}+15 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}}{630}+\frac{\left(11 \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+49 \left(275+150 \sqrt{17}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(660 \sqrt{17}\, \sqrt{3}+15\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(210 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-5775 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+127155 \,5^{\frac{1}{3}}}}{679140}-\frac{\sqrt{\left(3740 \sqrt{17}\, \sqrt{3}+85\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}+\left(1190 \sqrt{17}\, \sqrt{3}\, 5^{\frac{2}{3}}-32725 \,5^{\frac{2}{3}}\right) \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{1}{3}}+720545 \,5^{\frac{1}{3}}}\, \left(6 \sqrt{17}\, \sqrt{3}+11\right)^{\frac{2}{3}}}{37730}-1\right)^{-n}}{89}\right)}{4373768415988125000}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = a(n)+4*a(n+1)-5*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)+4 a \! \left(n +1\right)-5 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/18710/
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[17,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[11,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]+F[14,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[43,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[26,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[16,x]
F[26,x] = F[27,x]+F[37,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[2,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[30,x]*F[4,x]
F[33,x] = F[31,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[31,x]+F[34,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[31,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[54,x]
F[45,x] = F[2,x]+F[46,x]
F[46,x] = F[19,x]+F[47,x]+F[53,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[51,x]
F[49,x] = F[11,x]+F[50,x]
F[50,x] = F[24,x]
F[51,x] = F[31,x]+F[52,x]
F[52,x] = F[38,x]
F[53,x] = F[4,x]*F[45,x]
F[54,x] = F[46,x]+F[55,x]
F[55,x] = 2*F[19,x]+F[42,x]+F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[65,x]
F[58,x] = F[14,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[14,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[62,x]
F[65,x] = F[34,x]+F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]
F[69,x] = F[34,x]+F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[69,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_{17}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{43}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{37}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{2}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \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_{40}\! \left(x \right) = F_{31}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{54}\! \left(x \right)
F_{45}\! \left(x \right) = F_{2}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{19}\! \left(x \right)+F_{47}\! \left(x \right)+F_{53}\! \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_{51}\! \left(x \right)
F_{49}\! \left(x \right) = F_{11}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{24}\! \left(x \right)
F_{51}\! \left(x \right) = F_{31}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{38}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{54}\! \left(x \right) = F_{46}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{65}\! \left(x \right)
F_{58}\! \left(x \right) = F_{14}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{14}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{34}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{34}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \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_17(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_11(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_43(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_26(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_16(x))
Eq(F_26(x), F_27(x) + F_37(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_2(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_30(x)*F_4(x))
Eq(F_33(x), F_31(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_31(x) + F_34(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))
Eq(F_40(x), F_31(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_40(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_54(x))
Eq(F_45(x), F_2(x) + F_46(x))
Eq(F_46(x), F_19(x) + F_47(x) + F_53(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_51(x))
Eq(F_49(x), F_11(x) + F_50(x))
Eq(F_50(x), F_24(x))
Eq(F_51(x), F_31(x) + F_52(x))
Eq(F_52(x), F_38(x))
Eq(F_53(x), F_4(x)*F_45(x))
Eq(F_54(x), F_46(x) + F_55(x))
Eq(F_55(x), 2*F_19(x) + F_42(x) + F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_65(x))
Eq(F_58(x), F_14(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_14(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_62(x))
Eq(F_65(x), F_34(x) + F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_34(x) + F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_69(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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