0132_0231_0321_2031_2130
Counting sequence:
1, 1, 2, 6, 19, 57, 168, 495, 1460, 4307, 12705, 37477, 110549, 326096, 961914, 2837442, 8369851, 24689282, 72828136, 214827527, 633695559, 1869267254, 5513941225, 16264955034, 47978161439, 141525382041, 417469806285, 1231447226259, 3632507664580, 10715125790098, 31607344374564, 93234950105345, 275023292628838, 811260277427566, 2393045445136383, 7058975598616437, 20822478153574076, 61421886277812139, 181181513847665180, 534446969141517955, 1576505001855273441, 4650354786120205345, 13717558530636616301, 40463883014489946192, 119359857291918662158, 352086217915504507422, 1038579533007176603331, 3063588949227954423918, 9036936461337706142456, 26657042429545032176723, 78632611187505990805491, 231949495466628998518522, 684201727943789421239837, 2018249721041614631125010, 5953407847603681844290767, 17561288194613008117864017, 51802068823218359566428193, 152805096336190836696003995, 450742566788125251027604492, 1329594800082779727907822550, 3922022153363972058042456024, 11569132017152954236925343377, 34126481288616142648227412570, 100665868745861890091427822294, 296942923727077880425535213003, 875918531773559435044186457169, 2583773557134895746100661030976, 7621583003880716995411076111943, 22482050458576885421948684482372, 66317271958415801509408230117475, 195621861453862919589970749845265, 577042926353638411592685056246333, 1702154025015772487117133231474325, 5020992707051709259358296720330288, 14810861645750799404152738374317586, 43688894903490039628936481254125834, 128872957127096316190092728448749531, 380147841124625795197724907181117258, 1121355359054837235513185420317530824, 3307760048198644670203183756637777023, 9757189322822010808163075338375587039, 28781635334533350463181397680722715278, 84899708832386455395169655218984313201, 250436101911679807482239072411535375786, 738733288997951448929658786473996809135, 2179106239507712581512006424226075957721, 6427900398941697497380938113023938408501, 18960940402818114202782464042201904854739, 55930745445030853117212464202916096848980, 164983815126168335568350808557702507244746, 486667199534062574915519831432727417738780, 1435564833563851402537531020066646027668337, 4234611236052629464075884716005865864603486, 12491203393431131635331623469497359093712190, 36846395930671500317720028242733183581701047, 108689039023555511719243742220894249518097549, 320609571315776811048013841605130475162119092, 945730113567468566632908858795647094227477123, 2789702890146762057179591267182016041234023724, 8229030781240948883611527416139972963994446131, 24273892333763376194578457330613638498562001905
Generating function in Maple syntax:
(x-1)*(x^3+2*x-1)/(x^4-3*x^3+4*x^2-4*x+1)
Generating function in latex syntax:
\frac{\left(x -1\right) \left(x^{3}+2 x -1\right)}{x^{4}-3 x^{3}+4 x^{2}-4 x +1}
Generating function in sympy syntax:
(x - 1)*(x**3 + 2*x - 1)/(x**4 - 3*x**3 + 4*x**2 - 4*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^4-3*x^3+4*x^2-4*x+1)*F(x)-(x-1)*(x^3+2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}-3 x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{3}+2 x -1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/6442647552*(((355*(331^(1/2)*3^(1/2)-18867/355)*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+1696*(331^(1/2)*3^(1/2)-993/53)*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)+3968*331^(1/2)*3^(1/2))*((32*331^(1/2)*2^(2/3)*3^(1/2)-672*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+3264+(-5*2^(1/3)*3^(1/2)*331^(1/2)+309*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)+436920*(3^(1/2)-309/1655*331^(1/2))*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)-2796288*(3^(1/2)-21/331*331^(1/2))*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)+709632*331^(1/2))*(48*3^(1/2)*((32*331^(1/2)*2^(2/3)*3^(1/2)-672*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+3264+(-5*2^(1/3)*3^(1/2)*331^(1/2)+309*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)+(-9*331^(1/2)*3^(1/2)+249)*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+384*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)+1920)^(1/2)+15888*((32*331^(1/2)*2^(2/3)*3^(1/2)-672*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+3264+(-5*2^(1/3)*3^(1/2)*331^(1/2)+309*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)*(2^(1/3)*(3^(1/2)+399/331*331^(1/2))*(3*331^(1/2)*3^(1/2)+83)^(2/3)-80*(3^(1/2)-141/1655*331^(1/2))*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)-1024*3^(1/2)))*(1/96*(48*((96*331^(1/2)*2^(2/3)*3^(1/2)-2016*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+9792+(-15*2^(1/3)*3^(1/2)*331^(1/2)+927*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)+(9*331^(1/2)*3^(1/2)-249)*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)-384*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)-1920)^(1/2)+1/202752*(83*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+128*(332+12*331^(1/2)*3^(1/2))^(1/3)-320)*((96*331^(1/2)*2^(2/3)*3^(1/2)-2016*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+9792+(-15*2^(1/3)*3^(1/2)*331^(1/2)+927*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)-1/22528*(10592*2^(2/3)*(331^(1/2)*3^(1/2)-21)*(3*331^(1/2)*3^(1/2)+83)^(1/3)+1080384+(-1655*2^(1/3)*3^(1/2)*331^(1/2)+102279*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+3/4)^(-n)+1/6442647552*(((25728*I*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)*3^(1/2)+16683*I*(3^(1/2)-9/83*331^(1/2))*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+34944*I*3^(1/2))*((32*331^(1/2)*2^(2/3)*3^(1/2)-672*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+3264+(-5*2^(1/3)*3^(1/2)*331^(1/2)+309*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)+37224*I*(331^(1/2)*3^(1/2)-83/3)*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)-1588224*I*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)-4494336*I)*(48*3^(1/2)*((32*331^(1/2)*2^(2/3)*3^(1/2)-672*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+3264+(-5*2^(1/3)*3^(1/2)*331^(1/2)+309*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)+(-9*331^(1/2)*3^(1/2)+249)*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+384*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)+1920)^(1/2)-15888*((32*331^(1/2)*2^(2/3)*3^(1/2)-672*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+3264+(-5*2^(1/3)*3^(1/2)*331^(1/2)+309*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)*(2^(1/3)*(3^(1/2)+399/331*331^(1/2))*(3*331^(1/2)*3^(1/2)+83)^(2/3)-80*(3^(1/2)-141/1655*331^(1/2))*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)-1024*3^(1/2)))*(1/96*I*(48*((96*331^(1/2)*2^(2/3)*3^(1/2)-2016*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+9792+(-15*2^(1/3)*3^(1/2)*331^(1/2)+927*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)+(-9*331^(1/2)*3^(1/2)+249)*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+384*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)+1920)^(1/2)+1/202752*(-83*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)-128*(332+12*331^(1/2)*3^(1/2))^(1/3)+320)*((96*331^(1/2)*2^(2/3)*3^(1/2)-2016*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+9792+(-15*2^(1/3)*3^(1/2)*331^(1/2)+927*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)+1/22528*(10592*2^(2/3)*(331^(1/2)*3^(1/2)-21)*(3*331^(1/2)*3^(1/2)+83)^(1/3)+1080384+(-1655*2^(1/3)*3^(1/2)*331^(1/2)+102279*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+3/4)^(-n)+1/6442647552*((((-355*2^(1/3)*3^(1/2)*331^(1/2)+18867*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3)+(-1696*331^(1/2)*2^(2/3)*3^(1/2)+31776*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)-3968*331^(1/2)*3^(1/2))*((32*331^(1/2)*2^(2/3)*3^(1/2)-672*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+3264+(-5*2^(1/3)*3^(1/2)*331^(1/2)+309*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)-436920*(3^(1/2)-309/1655*331^(1/2))*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+2796288*(3^(1/2)-21/331*331^(1/2))*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)-709632*331^(1/2))*(48*3^(1/2)*((32*331^(1/2)*2^(2/3)*3^(1/2)-672*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+3264+(-5*2^(1/3)*3^(1/2)*331^(1/2)+309*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)+(-9*331^(1/2)*3^(1/2)+249)*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+384*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)+1920)^(1/2)+15888*((32*331^(1/2)*2^(2/3)*3^(1/2)-672*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+3264+(-5*2^(1/3)*3^(1/2)*331^(1/2)+309*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)*(2^(1/3)*(3^(1/2)+399/331*331^(1/2))*(3*331^(1/2)*3^(1/2)+83)^(2/3)-80*(3^(1/2)-141/1655*331^(1/2))*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)-1024*3^(1/2)))*(-1/96*(48*((96*331^(1/2)*2^(2/3)*3^(1/2)-2016*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+9792+(-15*2^(1/3)*3^(1/2)*331^(1/2)+927*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)+(9*331^(1/2)*3^(1/2)-249)*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)-384*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)-1920)^(1/2)+1/202752*(83*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+128*(332+12*331^(1/2)*3^(1/2))^(1/3)-320)*((96*331^(1/2)*2^(2/3)*3^(1/2)-2016*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+9792+(-15*2^(1/3)*3^(1/2)*331^(1/2)+927*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)-1/22528*(10592*2^(2/3)*(331^(1/2)*3^(1/2)-21)*(3*331^(1/2)*3^(1/2)+83)^(1/3)+1080384+(-1655*2^(1/3)*3^(1/2)*331^(1/2)+102279*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+3/4)^(-n)-47/8134656*(((1072/1551*I*3^(1/2)*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)+5561/12408*I*(3^(1/2)-9/83*331^(1/2))*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+1456/1551*I*3^(1/2))*((32*331^(1/2)*2^(2/3)*3^(1/2)-672*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+3264+(-5*2^(1/3)*3^(1/2)*331^(1/2)+309*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)+I*2^(1/3)*(331^(1/2)*3^(1/2)-83/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)-128/3*I*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)-17024/141*I)*(48*3^(1/2)*((32*331^(1/2)*2^(2/3)*3^(1/2)-672*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+3264+(-5*2^(1/3)*3^(1/2)*331^(1/2)+309*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)+(-9*331^(1/2)*3^(1/2)+249)*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+384*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)+1920)^(1/2)+662/1551*((32*331^(1/2)*2^(2/3)*3^(1/2)-672*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+3264+(-5*2^(1/3)*3^(1/2)*331^(1/2)+309*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)*(2^(1/3)*(3^(1/2)+399/331*331^(1/2))*(3*331^(1/2)*3^(1/2)+83)^(2/3)-80*(3^(1/2)-141/1655*331^(1/2))*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)-1024*3^(1/2)))*(-1/96*I*(48*((96*331^(1/2)*2^(2/3)*3^(1/2)-2016*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+9792+(-15*2^(1/3)*3^(1/2)*331^(1/2)+927*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)+(-9*331^(1/2)*3^(1/2)+249)*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+384*2^(2/3)*(3*331^(1/2)*3^(1/2)+83)^(1/3)+1920)^(1/2)+1/202752*(-83*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)-128*(332+12*331^(1/2)*3^(1/2))^(1/3)+320)*((96*331^(1/2)*2^(2/3)*3^(1/2)-2016*2^(2/3))*(3*331^(1/2)*3^(1/2)+83)^(1/3)+9792+(-15*2^(1/3)*3^(1/2)*331^(1/2)+927*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)+1/22528*(10592*2^(2/3)*(331^(1/2)*3^(1/2)-21)*(3*331^(1/2)*3^(1/2)+83)^(1/3)+1080384+(-1655*2^(1/3)*3^(1/2)*331^(1/2)+102279*2^(1/3))*(3*331^(1/2)*3^(1/2)+83)^(2/3))^(1/2)*2^(1/3)*(3*331^(1/2)*3^(1/2)+83)^(2/3)+3/4)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(355 \left(\sqrt{331}\, \sqrt{3}-\frac{18867}{355}\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+1696 \left(\sqrt{331}\, \sqrt{3}-\frac{993}{53}\right) 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3968 \sqrt{331}\, \sqrt{3}\right) \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+436920 \left(\sqrt{3}-\frac{309 \sqrt{331}}{1655}\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-2796288 \left(\sqrt{3}-\frac{21 \sqrt{331}}{331}\right) 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+709632 \sqrt{331}\right) \sqrt{48 \sqrt{3}\, \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(-9 \sqrt{331}\, \sqrt{3}+249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1920}+15888 \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, \left(2^{\frac{1}{3}} \left(\sqrt{3}+\frac{399 \sqrt{331}}{331}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-80 \left(\sqrt{3}-\frac{141 \sqrt{331}}{1655}\right) 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-1024 \sqrt{3}\right)\right) \left(\frac{\sqrt{48 \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(9 \sqrt{331}\, \sqrt{3}-249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-1920}}{96}+\frac{\left(83 \,2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+128 \left(332+12 \sqrt{331}\, \sqrt{3}\right)^{\frac{1}{3}}-320\right) \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}}{202752}-\frac{\sqrt{10592 \,2^{\frac{2}{3}} \left(\sqrt{331}\, \sqrt{3}-21\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1080384+\left(-1655 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+102279 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}{22528}+\frac{3}{4}\right)^{-n}}{6442647552}\\+\\\frac{\left(\left(\left(25728 \,\mathrm{I} \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}} \sqrt{3}+16683 \,\mathrm{I} \left(\sqrt{3}-\frac{9 \sqrt{331}}{83}\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+34944 \,\mathrm{I} \sqrt{3}\right) \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+37224 \,\mathrm{I} \left(\sqrt{331}\, \sqrt{3}-\frac{83}{3}\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-1588224 \,\mathrm{I} \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-4494336 \,\mathrm{I}\right) \sqrt{48 \sqrt{3}\, \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(-9 \sqrt{331}\, \sqrt{3}+249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1920}-15888 \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, \left(2^{\frac{1}{3}} \left(\sqrt{3}+\frac{399 \sqrt{331}}{331}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-80 \left(\sqrt{3}-\frac{141 \sqrt{331}}{1655}\right) 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-1024 \sqrt{3}\right)\right) \left(\frac{\mathrm{I} \sqrt{48 \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(-9 \sqrt{331}\, \sqrt{3}+249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1920}}{96}+\frac{\left(-83 \,2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-128 \left(332+12 \sqrt{331}\, \sqrt{3}\right)^{\frac{1}{3}}+320\right) \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}}{202752}+\frac{\sqrt{10592 \,2^{\frac{2}{3}} \left(\sqrt{331}\, \sqrt{3}-21\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1080384+\left(-1655 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+102279 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}{22528}+\frac{3}{4}\right)^{-n}}{6442647552}\\+\\\frac{\left(\left(\left(\left(-355 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+18867 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+\left(-1696 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}+31776 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-3968 \sqrt{331}\, \sqrt{3}\right) \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}-436920 \left(\sqrt{3}-\frac{309 \sqrt{331}}{1655}\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+2796288 \left(\sqrt{3}-\frac{21 \sqrt{331}}{331}\right) 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-709632 \sqrt{331}\right) \sqrt{48 \sqrt{3}\, \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(-9 \sqrt{331}\, \sqrt{3}+249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1920}+15888 \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, \left(2^{\frac{1}{3}} \left(\sqrt{3}+\frac{399 \sqrt{331}}{331}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-80 \left(\sqrt{3}-\frac{141 \sqrt{331}}{1655}\right) 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-1024 \sqrt{3}\right)\right) \left(-\frac{\sqrt{48 \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(9 \sqrt{331}\, \sqrt{3}-249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-1920}}{96}+\frac{\left(83 \,2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+128 \left(332+12 \sqrt{331}\, \sqrt{3}\right)^{\frac{1}{3}}-320\right) \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}}{202752}-\frac{\sqrt{10592 \,2^{\frac{2}{3}} \left(\sqrt{331}\, \sqrt{3}-21\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1080384+\left(-1655 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+102279 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}{22528}+\frac{3}{4}\right)^{-n}}{6442647552}\\-\\\frac{47 \left(\left(\left(\frac{1072 \,\mathrm{I} \sqrt{3}\, 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}}{1551}+\frac{5561 \,\mathrm{I} \left(\sqrt{3}-\frac{9 \sqrt{331}}{83}\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}{12408}+\frac{1456 \,\mathrm{I} \sqrt{3}}{1551}\right) \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\mathrm{I} \,2^{\frac{1}{3}} \left(\sqrt{331}\, \sqrt{3}-\frac{83}{3}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-\frac{128 \,\mathrm{I} \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}}{3}-\frac{17024 \,\mathrm{I}}{141}\right) \sqrt{48 \sqrt{3}\, \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(-9 \sqrt{331}\, \sqrt{3}+249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1920}+\frac{662 \sqrt{\left(32 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-672 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+3264+\left(-5 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+309 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, \left(2^{\frac{1}{3}} \left(\sqrt{3}+\frac{399 \sqrt{331}}{331}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-80 \left(\sqrt{3}-\frac{141 \sqrt{331}}{1655}\right) 2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}-1024 \sqrt{3}\right)}{1551}\right) \left(-\frac{\mathrm{I} \sqrt{48 \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}+\left(-9 \sqrt{331}\, \sqrt{3}+249\right) 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}+384 \,2^{\frac{2}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1920}}{96}+\frac{\left(-83 \,2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}-128 \left(332+12 \sqrt{331}\, \sqrt{3}\right)^{\frac{1}{3}}+320\right) \sqrt{\left(96 \sqrt{331}\, 2^{\frac{2}{3}} \sqrt{3}-2016 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+9792+\left(-15 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+927 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}}{202752}+\frac{\sqrt{10592 \,2^{\frac{2}{3}} \left(\sqrt{331}\, \sqrt{3}-21\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{1}{3}}+1080384+\left(-1655 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{331}+102279 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}\, 2^{\frac{1}{3}} \left(3 \sqrt{331}\, \sqrt{3}+83\right)^{\frac{2}{3}}}{22528}+\frac{3}{4}\right)^{-n}}{8134656} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+4) = -a(n)+3*a(n+1)-4*a(n+2)+4*a(n+3), n >= 5
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 19
a \! \left(n +4\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18711/
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[12,x]*F[4,x]
F[4,x] = F[18,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[12,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[17,x]
F[15,x] = 0
F[16,x] = F[10,x]*F[12,x]
F[17,x] = F[12,x]*F[13,x]
F[18,x] = F[19,x]+F[41,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]+F[29,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[9,x]
F[26,x] = F[12,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[26,x]
F[29,x] = F[19,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[19,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[12,x]*F[33,x]
F[36,x] = F[37,x]+F[39,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[19,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[36,x]
F[41,x] = F[15,x]+F[42,x]+F[81,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[56,x]
F[44,x] = F[45,x]+F[6,x]
F[45,x] = F[15,x]+F[46,x]+F[54,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[10,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[48,x]
F[51,x] = F[14,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[12,x]*F[51,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[13,x]
F[56,x] = F[41,x]+F[57,x]
F[57,x] = 2*F[15,x]+F[58,x]+F[79,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[76,x]
F[60,x] = F[61,x]+F[74,x]
F[61,x] = F[15,x]+F[62,x]+F[72,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[64,x]+F[66,x]
F[64,x] = F[10,x]+F[65,x]
F[65,x] = F[15,x]+F[17,x]+F[46,x]
F[66,x] = F[61,x]+F[67,x]
F[67,x] = 2*F[15,x]+F[58,x]+F[68,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[37,x]+F[70,x]
F[70,x] = 2*F[15,x]+F[68,x]+F[71,x]
F[71,x] = F[12,x]*F[61,x]
F[72,x] = F[12,x]*F[73,x]
F[73,x] = F[19,x]+F[61,x]
F[74,x] = F[75,x]
F[75,x] = F[12,x]*F[60,x]
F[76,x] = F[70,x]+F[77,x]
F[77,x] = F[78,x]
F[78,x] = F[12,x]*F[76,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[69,x]
F[81,x] = F[12,x]*F[82,x]
F[82,x] = F[69,x]+F[73,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{18}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{41}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{29}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{9}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{19}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{19}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{81}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{56}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{6}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{54}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{14}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13}\! \left(x \right)
F_{56}\! \left(x \right) = F_{41}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{79}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{76}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{74}\! \left(x \right)
F_{61}\! \left(x \right) = F_{15}\! \left(x \right)+F_{62}\! \left(x \right)+F_{72}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = F_{10}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)+F_{46}\! \left(x \right)
F_{66}\! \left(x \right) = F_{61}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{37}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{19}\! \left(x \right)+F_{61}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{76}\! \left(x \right) = F_{70}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{69}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{69}\! \left(x \right)+F_{73}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_18(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_12(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_17(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_10(x)*F_12(x))
Eq(F_17(x), F_12(x)*F_13(x))
Eq(F_18(x), F_19(x) + F_41(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_29(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_9(x))
Eq(F_26(x), F_12(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_26(x))
Eq(F_29(x), F_19(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_19(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_12(x)*F_33(x))
Eq(F_36(x), F_37(x) + F_39(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_19(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_36(x))
Eq(F_41(x), F_15(x) + F_42(x) + F_81(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_56(x))
Eq(F_44(x), F_45(x) + F_6(x))
Eq(F_45(x), F_15(x) + F_46(x) + F_54(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_10(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_48(x))
Eq(F_51(x), F_14(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_12(x)*F_51(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_13(x))
Eq(F_56(x), F_41(x) + F_57(x))
Eq(F_57(x), 2*F_15(x) + F_58(x) + F_79(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_76(x))
Eq(F_60(x), F_61(x) + F_74(x))
Eq(F_61(x), F_15(x) + F_62(x) + F_72(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_66(x))
Eq(F_64(x), F_10(x) + F_65(x))
Eq(F_65(x), F_15(x) + F_17(x) + F_46(x))
Eq(F_66(x), F_61(x) + F_67(x))
Eq(F_67(x), 2*F_15(x) + F_58(x) + F_68(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_37(x) + F_70(x))
Eq(F_70(x), 2*F_15(x) + F_68(x) + F_71(x))
Eq(F_71(x), F_12(x)*F_61(x))
Eq(F_72(x), F_12(x)*F_73(x))
Eq(F_73(x), F_19(x) + F_61(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_12(x)*F_60(x))
Eq(F_76(x), F_70(x) + F_77(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_12(x)*F_76(x))
Eq(F_79(x), F_12(x)*F_80(x))
Eq(F_80(x), F_69(x))
Eq(F_81(x), F_12(x)*F_82(x))
Eq(F_82(x), F_69(x) + F_73(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"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, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "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": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"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, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 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