0123_0132_0231_1302_3102
Counting sequence:
1, 1, 2, 6, 19, 53, 139, 352, 868, 2098, 4995, 11753, 27395, 63368, 145656, 333042, 758123, 1719245, 3886219, 8759936, 19697868, 44199458, 98993859, 221355601, 494250691, 1102175304, 2455066336, 5463103618, 12145823051, 26981593461, 59896052587, 132877452320, 294615546292, 652885241554, 1446164128195, 3201976188217, 7086902389059, 15680098448456, 34682432169416, 76692284896466, 169545704578539, 374735914000797, 828088258246667, 1829570511769088, 4041573518933404, 8926628458902338, 19713613755151491, 43530373680391841, 96110521121996931, 212180946603767176, 468384848097169840, 1033865379734823906, 2281882030910389579, 5036089559591898181, 11113925798266518635, 24525496226139223584, 54118607209261938820, 119414190611573583218, 263479978238852763907, 581330765266100214409, 1282580124302039507587, 2829649033159462770184, 6242646444218087736984, 13771908238004338945970, 30381535959700388590955, 67021859264683360776557, 147847908569500052214411, 326141916702958476450432, 719434819879116280538732, 1586975802744764547014498, 3500610031026551437924675, 7721687899600346891278577, 17032417637281713799352643, 37569577376262489976191944, 82869106793470348722786432, 182787159506912455003173442, 403176952955467487499034699, 889289125835165498753847573, 1961497637438763802576852075, 4326436680356035792784705824, 9542691391593318484587193172, 21047938230717563572279106322, 46424428761975488538398654275, 102395780155912946763495973337, 225847961023280759503493996227, 498137275770011612353832533192, 1098704181618885381088052812648, 2423324024106949940913383167378, 5344920723675708332648165959659, 11788816428353895723319518915389, 26001498479581928741422689365003, 57349000880373940523234081423360, 126488984584170526185268755229500, 278983800438060479942922346758786, 615325267336769898071003068810499, 1357156850418260317651123480589633, 2993332163595681105892866483416323, 6602058919170332091152130386599688, 14561413338043324462546172955702096, 32116436138251129956166789798646690, 70835485792810191853848864791545419
Generating function in Maple syntax:
(3*x^5-3*x^4+x^3-5*x^2+4*x-1)/(x-1)/(2*x-1)/(x^3+2*x-1)
Generating function in latex syntax:
\frac{3 x^{5}-3 x^{4}+x^{3}-5 x^{2}+4 x -1}{\left(x -1\right) \left(2 x -1\right) \left(x^{3}+2 x -1\right)}
Generating function in sympy syntax:
(3*x**5 - 3*x**4 + x**3 - 5*x**2 + 4*x - 1)/((x - 1)*(2*x - 1)*(x**3 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x-1)*(x^3+2*x-1)*F(x)-3*x^5+3*x^4-x^3+5*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x -1\right) \left(x^{3}+2 x -1\right) F \! \left(x \right)-3 x^{5}+3 x^{4}-x^{3}+5 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,97/11328*(5664/97*(108+12*59^(1/2)*3^(1/2))^(-2/3*n)*((-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((-I*59^(1/2)-3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)+2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+5/3)*(1/9*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-(I-1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)+16/3*I*3^(1/2)-16/3)^(-n)*576^n*(-16/3*(108+12*59^(1/2)*3^(1/2))^(1/3)+16/3*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/3*((2*I*59^(1/2)-6)*18^(1/3)-18*I*2^(1/3)*3^(1/6)+2*2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+9440/97*576^n*(108+12*59^(1/2)*3^(1/2))^(-2/3*n)*(1/9*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-(I-1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)+16/3*I*3^(1/2)-16/3)^(-n)*(-16/3*(108+12*59^(1/2)*3^(1/2))^(1/3)-16/3*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/3*((-2*I*59^(1/2)-6)*18^(1/3)+18*I*2^(1/3)*3^(1/6)+2*2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n-39648/97*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((I*59^(1/2)-3)*18^(1/3)-9*I*2^(1/3)*3^(1/6)+2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n*576^n*(108+12*59^(1/2)*3^(1/2))^(-2/3*n)*(1/128*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-9/128*(I-1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)+3/8*I*3^(1/2)-3/8)^(-n)*(-3/4*(108+12*59^(1/2)*3^(1/2))^(1/3)-3/4*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/32*((-3*I*59^(1/2)-9)*18^(1/3)+27*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(1/576*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/64*(I-1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/12*I*3^(1/2)-1/12)^(-n)*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((-I*59^(1/2)-3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)+2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n*(-118/97*(149/177*59^(1/2)+I)*64^n*3^(2*n+1/2)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n+1/3)-2065/582*(-97/2065*59^(1/2)+I)*64^n*3^(2*n+1/2)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n+2/3)+(108+12*59^(1/2)*3^(1/2))^(-2/3*n)*(3^(2*n+2/3)*(9+59^(1/2)*3^(1/2))^(2/3)*(I*59^(1/2)-2065/291)*2^(6*n+1/3)+298/97*2^(6*n+2/3)*(9+59^(1/2)*3^(1/2))^(1/3)*3^(2*n+1/3)*(I*59^(1/2)+59/149)))-(-9440/97*576^n*(108+12*59^(1/2)*3^(1/2))^(-2/3*n)*(1/9*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-(I-1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)+16/3*I*3^(1/2)-16/3)^(-n)*(8*(108+12*59^(1/2)*3^(1/2))^(1/3)+(-2^(1/3)*3^(1/6)*59^(1/2)+3*18^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^(-n)*(-256*(108+12*59^(1/2)*3^(1/2))^(1/3)-256*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((-32*I*59^(1/2)-96)*18^(1/3)+288*I*2^(1/3)*3^(1/6)+32*2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n-596/291*((3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-9*(I-1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)+48*I*3^(1/2)-48)^(-n)*(59^(1/2)*1024^n*3^(3*n+1/2)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n+1/3)-97/596*59^(1/2)*1024^n*3^(3*n+1/2)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n+2/3)-177/149*(108+12*59^(1/2)*3^(1/2))^(-2/3*n)*(3^(3*n+1/3)*2^(10*n+2/3)*(9+59^(1/2)*3^(1/2))^(1/3)-35/6*3^(3*n+2/3)*2^(10*n+1/3)*(9+59^(1/2)*3^(1/2))^(2/3)))*(8*(108+12*59^(1/2)*3^(1/2))^(1/3)+(-2^(1/3)*3^(1/6)*59^(1/2)+3*18^(1/3))*(9+59^(1/2)*3^(1/2))^(2/3))^(-n)*(-48*(108+12*59^(1/2)*3^(1/2))^(1/3)-48*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+((-6*I*59^(1/2)-18)*18^(1/3)+54*I*2^(1/3)*3^(1/6)+6*2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n+(1/576*(3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)-1/64*(I-1/9*59^(1/2))*3^(5/6)*(36+4*59^(1/2)*3^(1/2))^(1/3)+1/12*I*3^(1/2)-1/12)^(-n)*(-118/97*(I-149/177*59^(1/2))*64^n*3^(2*n+1/2)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n+1/3)-2065/582*(I+97/2065*59^(1/2))*64^n*3^(2*n+1/2)*(108+12*59^(1/2)*3^(1/2))^(-2/3*n+2/3)+(3^(2*n+2/3)*(9+59^(1/2)*3^(1/2))^(2/3)*(I*59^(1/2)+2065/291)*2^(6*n+1/3)+298/97*2^(6*n+2/3)*(9+59^(1/2)*3^(1/2))^(1/3)*(I*59^(1/2)-59/149)*3^(2*n+1/3))*(108+12*59^(1/2)*3^(1/2))^(-2/3*n)))*(-1/12*(108+12*59^(1/2)*3^(1/2))^(1/3)+1/12*I*(36+4*59^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1/96*((I*59^(1/2)-3)*18^(1/3)-9*I*2^(1/3)*3^(1/6)+2^(1/3)*3^(1/6)*59^(1/2))*(9+59^(1/2)*3^(1/2))^(2/3))^n)*((-3*I*59^(1/2)-9)*(108+12*59^(1/2)*3^(1/2))^(1/3)+9*3^(5/6)*(I+1/9*59^(1/2))*(36+4*59^(1/2)*3^(1/2))^(1/3)-48*I*3^(1/2)-48)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{97 \left(\frac{5664 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\frac{5}{3}\right) \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{9}-\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{16 \,\mathrm{I} \sqrt{3}}{3}-\frac{16}{3}\right)^{-n} 576^{n} \left(-\frac{16 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}+\frac{16 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{3}+\frac{\left(\left(2 \,\mathrm{I} \sqrt{59}-6\right) 18^{\frac{1}{3}}-18 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{3}\right)^{n}}{97}+\frac{9440 \,576^{n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{9}-\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{16 \,\mathrm{I} \sqrt{3}}{3}-\frac{16}{3}\right)^{-n} \left(-\frac{16 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{16 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{3}+\frac{\left(\left(-2 \,\mathrm{I} \sqrt{59}-6\right) 18^{\frac{1}{3}}+18 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{3}\right)^{n}}{97}-\frac{39648 \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n} 576^{n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{128}-\frac{9 \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{128}+\frac{3 \,\mathrm{I} \sqrt{3}}{8}-\frac{3}{8}\right)^{-n} \left(-\frac{3 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{4}-\frac{3 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{4}+\frac{\left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) 18^{\frac{1}{3}}+27 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{32}\right)^{n}}{97}+\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}-\frac{\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}+\frac{\mathrm{I} \sqrt{3}}{12}-\frac{1}{12}\right)^{-n} \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n} \left(-\frac{118 \left(\frac{149 \sqrt{59}}{177}+\mathrm{I}\right) 64^{n} 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}}{97}-\frac{2065 \left(-\frac{97 \sqrt{59}}{2065}+\mathrm{I}\right) 64^{n} 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{582}+\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(3^{2 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\mathrm{I} \sqrt{59}-\frac{2065}{291}\right) 2^{6 n +\frac{1}{3}}+\frac{298 \,2^{6 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{2 n +\frac{1}{3}} \left(\mathrm{I} \sqrt{59}+\frac{59}{149}\right)}{97}\right)\right)-\left(-\frac{9440 \,576^{n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{9}-\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{16 \,\mathrm{I} \sqrt{3}}{3}-\frac{16}{3}\right)^{-n} \left(8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}+3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{-n} \left(-256 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-256 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-32 \,\mathrm{I} \sqrt{59}-96\right) 18^{\frac{1}{3}}+288 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+32 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{97}-\frac{596 \left(\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(\sqrt{59}\, 1024^{n} 3^{3 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\frac{97 \sqrt{59}\, 1024^{n} 3^{3 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{596}-\frac{177 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(3^{3 n +\frac{1}{3}} 2^{10 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{35 \,3^{3 n +\frac{2}{3}} 2^{10 n +\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{6}\right)}{149}\right) \left(8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}+3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{-n} \left(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-6 \,\mathrm{I} \sqrt{59}-18\right) 18^{\frac{1}{3}}+54 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{291}+\left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}-\frac{\left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}+\frac{\mathrm{I} \sqrt{3}}{12}-\frac{1}{12}\right)^{-n} \left(-\frac{118 \left(\mathrm{I}-\frac{149 \sqrt{59}}{177}\right) 64^{n} 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}}{97}-\frac{2065 \left(\mathrm{I}+\frac{97 \sqrt{59}}{2065}\right) 64^{n} 3^{2 n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{582}+\left(3^{2 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\mathrm{I} \sqrt{59}+\frac{2065}{291}\right) 2^{6 n +\frac{1}{3}}+\frac{298 \,2^{6 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\mathrm{I} \sqrt{59}-\frac{59}{149}\right) 3^{2 n +\frac{1}{3}}}{97}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}}\right)\right) \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}\right) \left(\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{5}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \sqrt{3}-48\right)^{-n}}{11328} & \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(5) = 53
a(n+4) = -2*a(n)+a(n+1)-4*a(n+2)+4*a(n+3)+1, n >= 6
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(5\right) = 53
a \! \left(n +4\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)+1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16132/
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[21,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[16,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[10,x]+F[15,x]
F[15,x] = F[12,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[20,x]
F[19,x] = F[4,x]
F[20,x] = F[16,x]
F[21,x] = F[22,x]+F[34,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[1,x]+F[12,x]
F[26,x] = F[22,x]+F[27,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[22,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[22,x]*F[4,x]
F[33,x] = F[27,x]
F[34,x] = F[35,x]+F[36,x]+F[79,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[51,x]
F[38,x] = F[39,x]+F[7,x]
F[39,x] = F[17,x]+F[35,x]+F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[11,x]+F[42,x]
F[42,x] = F[43,x]+F[47,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[15,x]+F[46,x]
F[46,x] = F[43,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[20,x]+F[50,x]
F[50,x] = F[47,x]
F[51,x] = F[34,x]+F[52,x]
F[52,x] = 2*F[35,x]+F[53,x]+F[67,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[65,x]
F[55,x] = F[56,x]
F[56,x] = F[32,x]+F[35,x]+F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[62,x]
F[59,x] = F[4,x]+F[60,x]
F[60,x] = F[17,x]+F[35,x]+F[61,x]
F[61,x] = F[12,x]*F[4,x]
F[62,x] = F[56,x]+F[63,x]
F[63,x] = 2*F[35,x]+F[53,x]+F[64,x]
F[64,x] = F[27,x]*F[4,x]
F[65,x] = F[66,x]
F[66,x] = F[53,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]+F[70,x]
F[69,x] = F[27,x]+F[66,x]
F[70,x] = F[71,x]+F[75,x]
F[71,x] = F[72,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[33,x]+F[74,x]
F[74,x] = F[71,x]
F[75,x] = F[76,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[65,x]+F[78,x]
F[78,x] = F[75,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[69,x]+F[81,x]
F[81,x] = F[22,x]+F[56,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_{21}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{34}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \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_{22}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{79}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{51}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{7}\! \left(x \right)
F_{39}\! \left(x \right) = F_{17}\! \left(x \right)+F_{35}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{11}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{43}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{20}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{47}\! \left(x \right)
F_{51}\! \left(x \right) = F_{34}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{53}\! \left(x \right)+F_{67}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{65}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{62}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{17}\! \left(x \right)+F_{35}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{62}\! \left(x \right) = F_{56}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{53}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{53}\! \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_{70}\! \left(x \right)
F_{69}\! \left(x \right) = F_{27}\! \left(x \right)+F_{66}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{75}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{33}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{71}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{65}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{75}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{69}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{22}\! \left(x \right)+F_{56}\! \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_21(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_16(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_10(x) + F_15(x))
Eq(F_15(x), F_12(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_20(x))
Eq(F_19(x), F_4(x))
Eq(F_20(x), F_16(x))
Eq(F_21(x), F_22(x) + F_34(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_1(x) + F_12(x))
Eq(F_26(x), F_22(x) + F_27(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_22(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_22(x)*F_4(x))
Eq(F_33(x), F_27(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_79(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_51(x))
Eq(F_38(x), F_39(x) + F_7(x))
Eq(F_39(x), F_17(x) + F_35(x) + F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_11(x) + F_42(x))
Eq(F_42(x), F_43(x) + F_47(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_15(x) + F_46(x))
Eq(F_46(x), F_43(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_20(x) + F_50(x))
Eq(F_50(x), F_47(x))
Eq(F_51(x), F_34(x) + F_52(x))
Eq(F_52(x), 2*F_35(x) + F_53(x) + F_67(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_65(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_32(x) + F_35(x) + F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_62(x))
Eq(F_59(x), F_4(x) + F_60(x))
Eq(F_60(x), F_17(x) + F_35(x) + F_61(x))
Eq(F_61(x), F_12(x)*F_4(x))
Eq(F_62(x), F_56(x) + F_63(x))
Eq(F_63(x), 2*F_35(x) + F_53(x) + F_64(x))
Eq(F_64(x), F_27(x)*F_4(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_53(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_70(x))
Eq(F_69(x), F_27(x) + F_66(x))
Eq(F_70(x), F_71(x) + F_75(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_33(x) + F_74(x))
Eq(F_74(x), F_71(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_65(x) + F_78(x))
Eq(F_78(x), F_75(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_69(x) + F_81(x))
Eq(F_81(x), F_22(x) + F_56(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": 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