0231_0312_0321_1203_1302
Counting sequence:
1, 1, 2, 6, 19, 59, 181, 553, 1688, 5152, 15725, 47997, 146501, 447165, 1364882, 4166030, 12715975, 38812975, 118468857, 361602533, 1103719536, 3368883520, 10282844329, 31386329289, 95800503705, 292411910473, 892528975106, 2724266498294, 8315279571563, 25380730701843, 77469613067789, 236460526663921, 721748547023880, 2202993338805600, 6724197327218549, 20524269819121317, 62646235841853501, 191215127248855653, 583645998797080658, 1781462883260876990, 5437559772494119487, 16597065567442702631, 50659228951085532721, 154627181985274899981, 471968600860337121248, 1440589923052966487680, 4397113118581940119697, 13421309887154135915729, 40965868793731060425073, 125040135436514800440785, 381660048483447960607618, 1164941097511401577023846, 3555750113388993486697603, 10853217296458284016159659, 33127278894149603565153445, 101114404784734963789158649, 308631532569880037947871864, 942036133221773544440171168, 2875377214071603667482222525, 8776514861405829000397535245, 26788559335978385037091345269, 81766728892917826691612946509, 249576614777807755322659972754, 761782787293883370216198586734, 2325189864178088599933905773943, 7097177823723622115778640282079, 21662718317138030418635503779241, 66121122584675138697422292857461, 201821525251464552133716565866064, 616019911075556414558378025173952, 1880277786864970032250997906876857, 5739172536817886024026279844843033, 17517678311928098551485842224607625, 53469215548334292548422907389075353, 163204105044417844571485100789355394, 498147946069486386664697960835038550, 1520497147456620111495822383196695643, 4641013967166355578537618557670742275, 14165768531339979824234492181882237565, 43238180169930338947989811015351761985, 131975912233158036430751589898957030888, 402830122390007484457574545107443579872, 1229558521391895808362277902914785464389, 3752981898567491119024377787352162879925, 11455227942327435087441429030563252157421, 34964796195997274477150157305303963569781, 106723059478398688454638620032440883271250, 325750831224170421506128386001203805255582, 994289374403813055748039112938055823365999, 3034869800138738232950921451053915841267031, 9263334136821912453014207085433936778241377, 28274477978095605534710192374575203394419997, 86302198876320497835546187354937059066321856, 263420231371133912971005093747076541923294464, 804037662992396070830396020008928927272453665, 2454164435834278149943278991169061286714351265, 7490846953236244885117474092004395424106365921, 22864314736813279003611064488257474378205704353, 69788755750536777429922231697025640167776173314, 213016243227541550858799012314499198823201931462, 650189552611792116030546828258924270213585571827
Generating function in Maple syntax:
-(x-1)*(x^2+2*x-1)/(x^4-3*x^2+4*x-1)
Generating function in latex syntax:
-\frac{\left(x -1\right) \left(x^{2}+2 x -1\right)}{x^{4}-3 x^{2}+4 x -1}
Generating function in sympy syntax:
(1 - x)*(x**2 + 2*x - 1)/(x**4 - 3*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^4-3*x^2+4*x-1)*F(x)+(x-1)*(x^2+2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}-3 x^{2}+4 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{2}+2 x -1\right) = 0
Explicit closed form in Maple syntax:
-29/559872*((((((13/1769*3^(1/2)-387/1769*I)*61^(1/2)+151/1769+I*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+((-653/5307*3^(1/2)+55/1769*I)*61^(1/2)+2945/1769-3/29*I*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(2/3)+498/1769-30/1769*I*61^(1/2))*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2)+((104/1769*3^(1/2)+2568/1769*I)*61^(1/2)-3336/1769-204/29*I*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+((1288/1769*3^(1/2)-1776/1769*I)*61^(1/2)-17336/1769+124/29*I*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(2/3)-5820/1769+1224/1769*I*61^(1/2))*(-3*(81+6*61^(1/2)*3^(1/2))^(1/3)+36-27*(81+6*61^(1/2)*3^(1/2))^(2/3)+2*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)+6*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2))^(1/2)+(((528/1769*I+156/1769*3^(1/2))*61^(1/2)-41/29*I*3^(1/2)-27/29)*(81+6*61^(1/2)*3^(1/2))^(1/3)+((-250/1769*I-306/1769*3^(1/2))*61^(1/2)+17/29*I*3^(1/2)+69/29)*(81+6*61^(1/2)*3^(1/2))^(2/3)-156/1769*I*61^(1/2)+198/29)*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2)+((-4140/1769*I-1380/1769*3^(1/2))*61^(1/2)+324/29*I*3^(1/2)+324/29)*(81+6*61^(1/2)*3^(1/2))^(1/3)-1584/29+((2268/1769*I-756/1769*3^(1/2))*61^(1/2)-156/29*I*3^(1/2)+156/29)*(81+6*61^(1/2)*3^(1/2))^(2/3))*(1/72*(-(81+6*61^(1/2)*3^(1/2))^(1/3)-6)*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2)+1/36*((122*61^(1/2)*3^(5/6)-2013*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(1/3)+(244*61^(1/2)*3^(5/6)-3233*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(2/3)+2379*3^(1/3))^(1/2)*(2*61^(1/2)*3^(1/2)+27)^(2/3)-1/8*((6*61^(1/2)*3^(5/6)-99*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(1/3)+(12*61^(1/2)*3^(5/6)-159*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(2/3)+117*3^(1/3))^(1/2)*(2*61^(1/2)*3^(1/2)+27)^(2/3)-1/6*(-3*(81+6*61^(1/2)*3^(1/2))^(1/3)+36-27*(81+6*61^(1/2)*3^(1/2))^(2/3)+2*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)+6*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2))^(1/2))^(-n)+(((((-13/1769*3^(1/2)-387/1769*I)*61^(1/2)-151/1769+I*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+((653/5307*3^(1/2)+55/1769*I)*61^(1/2)-2945/1769-3/29*I*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(2/3)-498/1769-30/1769*I*61^(1/2))*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2)+((-104/1769*3^(1/2)+2568/1769*I)*61^(1/2)+3336/1769-204/29*I*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+((-1288/1769*3^(1/2)-1776/1769*I)*61^(1/2)+17336/1769+124/29*I*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(2/3)+5820/1769+1224/1769*I*61^(1/2))*(-3*(81+6*61^(1/2)*3^(1/2))^(1/3)+36-27*(81+6*61^(1/2)*3^(1/2))^(2/3)+2*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)+6*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2))^(1/2)+(((-528/1769*I+156/1769*3^(1/2))*61^(1/2)+41/29*I*3^(1/2)-27/29)*(81+6*61^(1/2)*3^(1/2))^(1/3)+((250/1769*I-306/1769*3^(1/2))*61^(1/2)-17/29*I*3^(1/2)+69/29)*(81+6*61^(1/2)*3^(1/2))^(2/3)+156/1769*I*61^(1/2)+198/29)*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2)+((4140/1769*I-1380/1769*3^(1/2))*61^(1/2)-324/29*I*3^(1/2)+324/29)*(81+6*61^(1/2)*3^(1/2))^(1/3)-1584/29+((-2268/1769*I-756/1769*3^(1/2))*61^(1/2)+156/29*I*3^(1/2)+156/29)*(81+6*61^(1/2)*3^(1/2))^(2/3))*(1/72*(-(81+6*61^(1/2)*3^(1/2))^(1/3)-6)*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2)+1/36*((122*61^(1/2)*3^(5/6)-2013*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(1/3)+(244*61^(1/2)*3^(5/6)-3233*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(2/3)+2379*3^(1/3))^(1/2)*(2*61^(1/2)*3^(1/2)+27)^(2/3)-1/8*((6*61^(1/2)*3^(5/6)-99*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(1/3)+(12*61^(1/2)*3^(5/6)-159*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(2/3)+117*3^(1/3))^(1/2)*(2*61^(1/2)*3^(1/2)+27)^(2/3)+1/6*(-3*(81+6*61^(1/2)*3^(1/2))^(1/3)+36-27*(81+6*61^(1/2)*3^(1/2))^(2/3)+2*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)+6*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2))^(1/2))^(-n)+((((-386/1769*I*61^(1/2)+74/87*I*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+(-1394/5307*I*61^(1/2)+106/87*I*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(2/3)+148/1769*I*61^(1/2))*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2)+(3864/1769*I*61^(1/2)-264/29*I*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+(2568/1769*I*61^(1/2)-200/29*I*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(2/3)+504/1769*I*61^(1/2))*(-3*(81+6*61^(1/2)*3^(1/2))^(1/3)+36-27*(81+6*61^(1/2)*3^(1/2))^(2/3)+2*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)+6*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2))^(1/2)+((-312/1769*61^(1/2)*3^(1/2)+90/29)*(81+6*61^(1/2)*3^(1/2))^(1/3)+108/29-852/1769*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)+186/29*(81+6*61^(1/2)*3^(1/2))^(2/3))*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2)+(-648/29+2760/1769*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)-1584/29-312/29*(81+6*61^(1/2)*3^(1/2))^(2/3)+1512/1769*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2))*(1/72*((81+6*61^(1/2)*3^(1/2))^(1/3)+6)*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2)-1/6*I*(3*(81+6*61^(1/2)*3^(1/2))^(1/3)-36+27*(81+6*61^(1/2)*3^(1/2))^(2/3)-2*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)+6*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2))^(1/2)-1/36*((122*61^(1/2)*3^(5/6)-2013*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(1/3)+(244*61^(1/2)*3^(5/6)-3233*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(2/3)+2379*3^(1/3))^(1/2)*(2*61^(1/2)*3^(1/2)+27)^(2/3)+1/8*((6*61^(1/2)*3^(5/6)-99*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(1/3)+(12*61^(1/2)*3^(5/6)-159*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(2/3)+117*3^(1/3))^(1/2)*(2*61^(1/2)*3^(1/2)+27)^(2/3))^(-n)-432/29*(1/72*((81+6*61^(1/2)*3^(1/2))^(1/3)+6)*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2)+1/6*I*(3*(81+6*61^(1/2)*3^(1/2))^(1/3)-36+27*(81+6*61^(1/2)*3^(1/2))^(2/3)-2*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)+6*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2))^(1/2)-1/36*((122*61^(1/2)*3^(5/6)-2013*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(1/3)+(244*61^(1/2)*3^(5/6)-3233*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(2/3)+2379*3^(1/3))^(1/2)*(2*61^(1/2)*3^(1/2)+27)^(2/3)+1/8*((6*61^(1/2)*3^(5/6)-99*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(1/3)+(12*61^(1/2)*3^(5/6)-159*3^(1/3))*(81+6*61^(1/2)*3^(1/2))^(2/3)+117*3^(1/3))^(1/2)*(2*61^(1/2)*3^(1/2)+27)^(2/3))^(-n))*((((-18/61*I*61^(1/2)+I*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+(3*I*3^(1/2)-40/61*I*61^(1/2))*(81+6*61^(1/2)*3^(1/2))^(2/3)+6/61*I*61^(1/2))*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2)+(-24*I*3^(1/2)+342/61*I*61^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+(-12*I*3^(1/2)+150/61*I*61^(1/2))*(81+6*61^(1/2)*3^(1/2))^(2/3)+18/61*I*61^(1/2))*(-3*(81+6*61^(1/2)*3^(1/2))^(1/3)+36-27*(81+6*61^(1/2)*3^(1/2))^(2/3)+2*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)+6*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2))^(1/2)+324+((39/61*61^(1/2)*3^(1/2)-9)*(81+6*61^(1/2)*3^(1/2))^(1/3)+15/61*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-3*(81+6*61^(1/2)*3^(1/2))^(2/3)+18)*((-33+2*61^(1/2)*3^(1/2))*(81+6*61^(1/2)*3^(1/2))^(1/3)+4*(81+6*61^(1/2)*3^(1/2))^(2/3)*61^(1/2)*3^(1/2)-53*(81+6*61^(1/2)*3^(1/2))^(2/3)+39)^(1/2))
Explicit closed form in latex syntax:
-\frac{29 \left(\left(\left(\left(\left(\left(\frac{13 \sqrt{3}}{1769}-\frac{387 \,\mathrm{I}}{1769}\right) \sqrt{61}+\frac{151}{1769}+\mathrm{I} \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(-\frac{653 \sqrt{3}}{5307}+\frac{55 \,\mathrm{I}}{1769}\right) \sqrt{61}+\frac{2945}{1769}-\frac{3 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{498}{1769}-\frac{30 \,\mathrm{I} \sqrt{61}}{1769}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(\left(\frac{104 \sqrt{3}}{1769}+\frac{2568 \,\mathrm{I}}{1769}\right) \sqrt{61}-\frac{3336}{1769}-\frac{204 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(\frac{1288 \sqrt{3}}{1769}-\frac{1776 \,\mathrm{I}}{1769}\right) \sqrt{61}-\frac{17336}{1769}+\frac{124 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{5820}{1769}+\frac{1224 \,\mathrm{I} \sqrt{61}}{1769}\right) \sqrt{-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+36-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}+\left(\left(\left(\frac{528 \,\mathrm{I}}{1769}+\frac{156 \sqrt{3}}{1769}\right) \sqrt{61}-\frac{41 \,\mathrm{I} \sqrt{3}}{29}-\frac{27}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(-\frac{250 \,\mathrm{I}}{1769}-\frac{306 \sqrt{3}}{1769}\right) \sqrt{61}+\frac{17 \,\mathrm{I} \sqrt{3}}{29}+\frac{69}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{156 \,\mathrm{I} \sqrt{61}}{1769}+\frac{198}{29}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(\left(-\frac{4140 \,\mathrm{I}}{1769}-\frac{1380 \sqrt{3}}{1769}\right) \sqrt{61}+\frac{324 \,\mathrm{I} \sqrt{3}}{29}+\frac{324}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{1584}{29}+\left(\left(\frac{2268 \,\mathrm{I}}{1769}-\frac{756 \sqrt{3}}{1769}\right) \sqrt{61}-\frac{156 \,\mathrm{I} \sqrt{3}}{29}+\frac{156}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(-\left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-6\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}{72}+\frac{\sqrt{\left(122 \sqrt{61}\, 3^{\frac{5}{6}}-2013 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(244 \sqrt{61}\, 3^{\frac{5}{6}}-3233 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2379 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{36}-\frac{\sqrt{\left(6 \sqrt{61}\, 3^{\frac{5}{6}}-99 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(12 \sqrt{61}\, 3^{\frac{5}{6}}-159 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+117 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{8}-\frac{\sqrt{-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+36-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}}{6}\right)^{-n}+\left(\left(\left(\left(\left(-\frac{13 \sqrt{3}}{1769}-\frac{387 \,\mathrm{I}}{1769}\right) \sqrt{61}-\frac{151}{1769}+\mathrm{I} \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(\frac{653 \sqrt{3}}{5307}+\frac{55 \,\mathrm{I}}{1769}\right) \sqrt{61}-\frac{2945}{1769}-\frac{3 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{498}{1769}-\frac{30 \,\mathrm{I} \sqrt{61}}{1769}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(\left(-\frac{104 \sqrt{3}}{1769}+\frac{2568 \,\mathrm{I}}{1769}\right) \sqrt{61}+\frac{3336}{1769}-\frac{204 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(-\frac{1288 \sqrt{3}}{1769}-\frac{1776 \,\mathrm{I}}{1769}\right) \sqrt{61}+\frac{17336}{1769}+\frac{124 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{5820}{1769}+\frac{1224 \,\mathrm{I} \sqrt{61}}{1769}\right) \sqrt{-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+36-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}+\left(\left(\left(-\frac{528 \,\mathrm{I}}{1769}+\frac{156 \sqrt{3}}{1769}\right) \sqrt{61}+\frac{41 \,\mathrm{I} \sqrt{3}}{29}-\frac{27}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(\frac{250 \,\mathrm{I}}{1769}-\frac{306 \sqrt{3}}{1769}\right) \sqrt{61}-\frac{17 \,\mathrm{I} \sqrt{3}}{29}+\frac{69}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{156 \,\mathrm{I} \sqrt{61}}{1769}+\frac{198}{29}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(\left(\frac{4140 \,\mathrm{I}}{1769}-\frac{1380 \sqrt{3}}{1769}\right) \sqrt{61}-\frac{324 \,\mathrm{I} \sqrt{3}}{29}+\frac{324}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{1584}{29}+\left(\left(-\frac{2268 \,\mathrm{I}}{1769}-\frac{756 \sqrt{3}}{1769}\right) \sqrt{61}+\frac{156 \,\mathrm{I} \sqrt{3}}{29}+\frac{156}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(-\left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-6\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}{72}+\frac{\sqrt{\left(122 \sqrt{61}\, 3^{\frac{5}{6}}-2013 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(244 \sqrt{61}\, 3^{\frac{5}{6}}-3233 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2379 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{36}-\frac{\sqrt{\left(6 \sqrt{61}\, 3^{\frac{5}{6}}-99 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(12 \sqrt{61}\, 3^{\frac{5}{6}}-159 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+117 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{8}+\frac{\sqrt{-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+36-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}}{6}\right)^{-n}+\left(\left(\left(\left(-\frac{386 \,\mathrm{I} \sqrt{61}}{1769}+\frac{74 \,\mathrm{I} \sqrt{3}}{87}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-\frac{1394 \,\mathrm{I} \sqrt{61}}{5307}+\frac{106 \,\mathrm{I} \sqrt{3}}{87}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{148 \,\mathrm{I} \sqrt{61}}{1769}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(\frac{3864 \,\mathrm{I} \sqrt{61}}{1769}-\frac{264 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\frac{2568 \,\mathrm{I} \sqrt{61}}{1769}-\frac{200 \,\mathrm{I} \sqrt{3}}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{504 \,\mathrm{I} \sqrt{61}}{1769}\right) \sqrt{-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+36-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}+\left(\left(-\frac{312 \sqrt{61}\, \sqrt{3}}{1769}+\frac{90}{29}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{108}{29}-\frac{852 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}}{1769}+\frac{186 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}}{29}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(-\frac{648}{29}+\frac{2760 \sqrt{61}\, \sqrt{3}}{1769}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{1584}{29}-\frac{312 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}}{29}+\frac{1512 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}}{1769}\right) \left(\frac{\left(\left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+6\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}{72}-\frac{\mathrm{I} \sqrt{3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-36+27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}-2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}}{6}-\frac{\sqrt{\left(122 \sqrt{61}\, 3^{\frac{5}{6}}-2013 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(244 \sqrt{61}\, 3^{\frac{5}{6}}-3233 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2379 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{36}+\frac{\sqrt{\left(6 \sqrt{61}\, 3^{\frac{5}{6}}-99 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(12 \sqrt{61}\, 3^{\frac{5}{6}}-159 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+117 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{8}\right)^{-n}-\frac{432 \left(\frac{\left(\left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+6\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}{72}+\frac{\mathrm{I} \sqrt{3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}-36+27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}-2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}}{6}-\frac{\sqrt{\left(122 \sqrt{61}\, 3^{\frac{5}{6}}-2013 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(244 \sqrt{61}\, 3^{\frac{5}{6}}-3233 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2379 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{36}+\frac{\sqrt{\left(6 \sqrt{61}\, 3^{\frac{5}{6}}-99 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(12 \sqrt{61}\, 3^{\frac{5}{6}}-159 \,3^{\frac{1}{3}}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+117 \,3^{\frac{1}{3}}}\, \left(2 \sqrt{61}\, \sqrt{3}+27\right)^{\frac{2}{3}}}{8}\right)^{-n}}{29}\right) \left(\left(\left(\left(-\frac{18 \,\mathrm{I} \sqrt{61}}{61}+\mathrm{I} \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(3 \,\mathrm{I} \sqrt{3}-\frac{40 \,\mathrm{I} \sqrt{61}}{61}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{6 \,\mathrm{I} \sqrt{61}}{61}\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}+\left(-24 \,\mathrm{I} \sqrt{3}+\frac{342 \,\mathrm{I} \sqrt{61}}{61}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-12 \,\mathrm{I} \sqrt{3}+\frac{150 \,\mathrm{I} \sqrt{61}}{61}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{18 \,\mathrm{I} \sqrt{61}}{61}\right) \sqrt{-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+36-27 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+2 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}+6 \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}}+324+\left(\left(\frac{39 \sqrt{61}\, \sqrt{3}}{61}-9\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{15 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}}{61}-3 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+18\right) \sqrt{\left(-33+2 \sqrt{61}\, \sqrt{3}\right) \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{1}{3}}+4 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{61}\, \sqrt{3}-53 \left(81+6 \sqrt{61}\, \sqrt{3}\right)^{\frac{2}{3}}+39}\right)}{559872}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n) = 3*a(n+2)-4*a(n+3)+a(n+4), 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 \right) = 3 a \! \left(n +2\right)-4 a \! \left(n +3\right)+a \! \left(n +4\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20604/
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[10,x]*F[4,x]
F[4,x] = F[13,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = x
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[6,x]
F[13,x] = F[14,x]+F[2,x]
F[14,x] = F[15,x]+F[16,x]+F[20,x]
F[15,x] = 0
F[16,x] = F[10,x]*F[17,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[6,x]
F[19,x] = F[14,x]
F[20,x] = F[10,x]*F[21,x]
F[21,x] = F[13,x]+F[22,x]
F[22,x] = F[23,x]+F[29,x]
F[23,x] = F[15,x]+F[24,x]+F[28,x]
F[24,x] = F[10,x]*F[25,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[10,x]
F[27,x] = F[23,x]
F[28,x] = F[10,x]*F[2,x]
F[29,x] = F[30,x]
F[30,x] = F[10,x]*F[14,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{13}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{6}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{2}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{6}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{29}\! \left(x \right)
F_{23}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{2}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_13(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_6(x))
Eq(F_13(x), F_14(x) + F_2(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_20(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_10(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_6(x))
Eq(F_19(x), F_14(x))
Eq(F_20(x), F_10(x)*F_21(x))
Eq(F_21(x), F_13(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_29(x))
Eq(F_23(x), F_15(x) + F_24(x) + F_28(x))
Eq(F_24(x), F_10(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_10(x))
Eq(F_27(x), F_23(x))
Eq(F_28(x), F_10(x)*F_2(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_10(x)*F_14(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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