0123_0213_0231_0312_1032
Counting sequence:
1, 1, 2, 6, 19, 59, 183, 568, 1763, 5472, 16984, 52715, 163617, 507835, 1576220, 4892277, 15184666, 47130218, 146282931, 454033459, 1409230595, 4373974716, 13575957607, 42137103416, 130785284964, 405931812499, 1259932540913, 3910582907703, 12137680535608, 37672974147737, 116929505350914, 362926196588350, 1126451563912787, 3496284197089275, 10851778787856175, 33681787927481312, 104541647979533211, 324476722726455808, 1007111956106848736, 3125877516300079419, 9702109271626694065, 30113439770986930931, 93466196829260872212, 290100699759409310701, 900415539049214863034, 2794712813976905461314, 8674239141690125694643, 26923132964119591946963, 83564111706335681302203, 259366574260697169601252, 805022855746211100750719, 2498632678944968983554360, 7755264611095604120264332, 24070816689033023461543715, 74711082746044039368185505, 231888512849227722224820847, 719736355236716190136006256, 2233920148456192609776204273, 6933648958217805551553433666, 21520683229890132844796307254, 66795969838126591144165126035, 207321558472597578984048811771, 643485358647682869796942742567, 1997252045781175200534993353736, 6199077695816123180589028872979, 19240718446096052411564029361504, 59719407384069332435227081438248, 185357299848024120486270273187723, 575312617990168413870374848924673, 1785657261354574574046351628212267, 5542329083911747842625325157824524, 17202299869725411941746350322398245, 53392556870530810399285402595407002, 165719999695504179040481532944045530, 514362298956237949063190949154534835, 1596479453739244657588858250059011507, 4955158360913238151807056283121080051, 15379837381695952404484359798517774988, 47735991598827101871041937645612336471, 148163133157394543764932869219958089464, 459869236853879583699282967458392043380, 1427343702160465852968890840020788466611, 4430194239638792102671605389282323489297, 13750451955770255891714099135305362511271, 42678699569471233528111188245936876000424, 132466292948052492687005170127092951490569, 411149330799927733952729609516584216982978, 1276126691969254435386300016795689526949358, 3960846368855815798845905220514161532338643, 12293688437367375130490445271059068813998907, 38157192004071379826857635829972895968946079, 118432422381069955279418812710432849439176880, 367590955580577240968746883402357617131529547, 1140930058745803102733098286037045747363534720, 3541222598618479263478713670821570091529781040, 10991258751436015031404887895867067891720872667, 34114706313053848196947761973638249422526152721, 105885341537780023854321999591736318359108239203, 328647283364776086594370886671076022969045590276, 1020056556407382107980060421986866318329662923549, 3166055010759926347794503265552335273348097009850
Generating function in Maple syntax:
(x^3+x^2+2*x-1)/(x^3+3*x-1)
Generating function in latex syntax:
\frac{x^{3}+x^{2}+2 x -1}{x^{3}+3 x -1}
Generating function in sympy syntax:
(x**3 + x**2 + 2*x - 1)/(x**3 + 3*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^3+3*x-1)*F(x)-x^3-x^2-2*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+3 x -1\right) F \! \left(x \right)-x^{3}-x^{2}-2 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n < 0,1/120*((1+I*3^(1/2))*(5^(1/2)-1)*(4+4*5^(1/2))^(1/3)+4*I*3^(1/2)-4)^(-n)*((-(5+5^(1/2))*(1+I*3^(1/2))*(4+4*5^(1/2))^(-2/3*n+1/3)-3/2*(5^(1/2)-5/3)*(I*3^(1/2)-1)*(4+4*5^(1/2))^(-2/3*n+2/3))*(1/8*(2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)+1/8*(5^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3)*(1+I*3^(1/2)))^n+2*5^(1/2)*(2*(4+4*5^(1/2))^(1/3)+(-5^(1/2)+1)*2^(1/3)*(5^(1/2)+1)^(2/3))^(-n)*(1/8*(-2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)-1/8*(5^(1/2)-1)*(I*3^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3))^n*((4+4*5^(1/2))^(-2/3*n+1/3)-3/2*(4+4*5^(1/2))^(-2/3*n+2/3))*(1/2*(2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)+1/2*(5^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3)*(1+I*3^(1/2)))^n+((I*3^(1/2)-1)*(5+5^(1/2))*(4+4*5^(1/2))^(-2/3*n+1/3)+3/2*(5^(1/2)-5/3)*(4+4*5^(1/2))^(-2/3*n+2/3)*(1+I*3^(1/2)))*(1/8*(-2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)-1/8*(5^(1/2)-1)*(I*3^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3))^n+10*((4+4*5^(1/2))^(-2/3*n+1/3)+1/2*(4+4*5^(1/2))^(-2/3*n+2/3))*(2*(4+4*5^(1/2))^(1/3)+(-5^(1/2)+1)*2^(1/3)*(5^(1/2)+1)^(2/3))^(-n)*(1/16*(-2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)-1/16*(5^(1/2)-1)*(I*3^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3))^n*((2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)+(5^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3)*(1+I*3^(1/2)))^n)*(-1/256*(I*3^(1/2)-1)*(5^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/64*I*3^(1/2)-1/64)^(-n),n = 0,1,0 < n,-1/120*(3*(1/4*(1+I*3^(1/2))*(5^(1/2)-1)*(4+4*5^(1/2))^(1/3)+I*3^(1/2)-1)^(-n)*(1/8*(-2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)-1/8*(5^(1/2)-1)*(I*3^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3))^n*5^(1/2)*(4+4*5^(1/2))^(-2/3*n+2/3)*(2*(4+4*5^(1/2))^(1/3)+(-5^(1/2)+1)*2^(1/3)*(5^(1/2)+1)^(2/3))^(-n)+((1+I*3^(1/2))*(5^(1/2)-1)*(4+4*5^(1/2))^(1/3)+4*I*3^(1/2)-4)^(-n)*((5+5^(1/2))*(1+I*3^(1/2))*(4+4*5^(1/2))^(-2/3*n+1/3)+3/2*(5^(1/2)-5/3)*(I*3^(1/2)-1)*(4+4*5^(1/2))^(-2/3*n+2/3)))*(-1/256*(I*3^(1/2)-1)*(5^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/64*I*3^(1/2)-1/64)^(-n)*(1/8*(2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)+1/8*(5^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3)*(1+I*3^(1/2)))^n+1/120*((1+I*3^(1/2))*(5^(1/2)-1)*(4+4*5^(1/2))^(1/3)+4*I*3^(1/2)-4)^(-n)*((10*(-1/128*(I*3^(1/2)-1)*(5^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/32*I*3^(1/2)-1/32)^(-n)*(2*(4+4*5^(1/2))^(1/3)+(-5^(1/2)+1)*2^(1/3)*(5^(1/2)+1)^(2/3))^(-n)*((4+4*5^(1/2))^(-2/3*n+1/3)+1/2*(4+4*5^(1/2))^(-2/3*n+2/3))*((2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)+(5^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3)*(1+I*3^(1/2)))^n+((I*3^(1/2)-1)*(5+5^(1/2))*(4+4*5^(1/2))^(-2/3*n+1/3)+3/2*(5^(1/2)-5/3)*(4+4*5^(1/2))^(-2/3*n+2/3)*(1+I*3^(1/2)))*(-1/256*(I*3^(1/2)-1)*(5^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/64*I*3^(1/2)-1/64)^(-n))*(1/8*(-2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)-1/8*(5^(1/2)-1)*(I*3^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3))^n+2*(-1/32*(I*3^(1/2)-1)*(5^(1/2)-1)*(4+4*5^(1/2))^(1/3)-1/8*I*3^(1/2)-1/8)^(-n)*((2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)+(5^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3)*(1+I*3^(1/2)))^n*(1/2*(-2*I*3^(1/2)-2)*(4+4*5^(1/2))^(1/3)-1/2*(5^(1/2)-1)*(I*3^(1/2)-1)*2^(1/3)*(5^(1/2)+1)^(2/3))^n*5^(1/2)*(4+4*5^(1/2))^(-2/3*n+1/3)*(2*(4+4*5^(1/2))^(1/3)+(-5^(1/2)+1)*2^(1/3)*(5^(1/2)+1)^(2/3))^(-n)))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}\frac{\left(\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(\left(-\left(5+\sqrt{5}\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\frac{3 \left(\sqrt{5}-\frac{5}{3}\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}+\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{8}\right)^{n}+2 \sqrt{5}\, \left(2 \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(-\sqrt{5}+1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}\right)^{-n} \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{8}\right)^{n} \left(\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}-\frac{3 \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{2}+\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{2}\right)^{n}+\left(\left(\mathrm{I} \sqrt{3}-1\right) \left(5+\sqrt{5}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \left(\sqrt{5}-\frac{5}{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{2}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{8}\right)^{n}+10 \left(\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(2 \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(-\sqrt{5}+1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}\right)^{-n} \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{16}-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{16}\right)^{n} \left(\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)\right)^{n}\right) \left(-\frac{\left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n}}{120} & n <0 \\ 1 & n =0 \\ -\frac{\left(3 \left(\frac{\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{4}+\mathrm{I} \sqrt{3}-1\right)^{-n} \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{8}\right)^{n} \sqrt{5}\, \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}} \left(2 \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(-\sqrt{5}+1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}\right)^{-n}+\left(\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(\left(5+\sqrt{5}\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \left(\sqrt{5}-\frac{5}{3}\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right)\right) \left(-\frac{\left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n} \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}+\frac{\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{8}\right)^{n}}{120}\\+\\\frac{\left(\left(1+\mathrm{I} \sqrt{3}\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+4 \,\mathrm{I} \sqrt{3}-4\right)^{-n} \left(\left(10 \left(-\frac{\left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{128}-\frac{\mathrm{I} \sqrt{3}}{32}-\frac{1}{32}\right)^{-n} \left(2 \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(-\sqrt{5}+1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}\right)^{-n} \left(\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{\left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{2}\right) \left(\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)\right)^{n}+\left(\left(\mathrm{I} \sqrt{3}-1\right) \left(5+\sqrt{5}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}}+\frac{3 \left(\sqrt{5}-\frac{5}{3}\right) \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)}{2}\right) \left(-\frac{\left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{64}-\frac{1}{64}\right)^{-n}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{8}-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{8}\right)^{n}+2 \left(-\frac{\left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{5}-1\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{32}-\frac{\mathrm{I} \sqrt{3}}{8}-\frac{1}{8}\right)^{-n} \left(\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(\sqrt{5}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}} \left(1+\mathrm{I} \sqrt{3}\right)\right)^{n} \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}}{2}-\frac{\left(\sqrt{5}-1\right) \left(\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}}{2}\right)^{n} \sqrt{5}\, \left(4+4 \sqrt{5}\right)^{-\frac{2 n}{3}+\frac{1}{3}} \left(2 \left(4+4 \sqrt{5}\right)^{\frac{1}{3}}+\left(-\sqrt{5}+1\right) 2^{\frac{1}{3}} \left(\sqrt{5}+1\right)^{\frac{2}{3}}\right)^{-n}\right)}{120} & 0<n \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n) = -3*a(n+2)+a(n+3), n >= 4
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(n \right) = -3 a \! \left(n +2\right)+a \! \left(n +3\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16678/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[14,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[11,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]+F[17,x]+F[68,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[31,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[1,x]+F[4,x]
F[23,x] = F[16,x]+F[24,x]+F[29,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]+F[4,x]
F[27,x] = F[28,x]
F[28,x] = F[11,x]*F[4,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[11,x]
F[31,x] = F[32,x]+F[60,x]
F[32,x] = F[16,x]+F[17,x]+F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[2,x]+F[35,x]
F[35,x] = F[16,x]+F[36,x]+F[59,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[40,x]
F[38,x] = F[39,x]+F[4,x]
F[39,x] = F[16,x]+F[24,x]+F[28,x]
F[40,x] = F[35,x]+F[41,x]
F[41,x] = F[16,x]+F[42,x]+F[51,x]+F[55,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[4,x]
F[47,x] = F[48,x]
F[48,x] = 2*F[16,x]+F[42,x]+F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[35,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[35,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[16,x]+F[36,x]+F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[2,x]+F[56,x]
F[59,x] = F[2,x]*F[4,x]
F[60,x] = F[16,x]+F[51,x]+F[61,x]+F[66,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[64,x]
F[63,x] = F[45,x]
F[64,x] = F[65,x]
F[65,x] = 2*F[16,x]+F[49,x]+F[61,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[56,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[58,x]+F[70,x]
F[70,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_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{68}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{31}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{16}\! \left(x \right)+F_{24}\! \left(x \right)+F_{29}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{11}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{60}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{16}\! \left(x \right)+F_{36}\! \left(x \right)+F_{59}\! \left(x \right)
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_{40}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{16}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{40}\! \left(x \right) = F_{35}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{16}\! \left(x \right)+F_{42}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{42}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{35}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{35}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{16}\! \left(x \right)+F_{36}\! \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_{2}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{60}\! \left(x \right) = F_{16}\! \left(x \right)+F_{51}\! \left(x \right)+F_{61}\! \left(x \right)+F_{66}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{64}\! \left(x \right)
F_{63}\! \left(x \right) = F_{45}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{49}\! \left(x \right)+F_{61}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{56}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{58}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \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_14(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_11(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_68(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_31(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_1(x) + F_4(x))
Eq(F_23(x), F_16(x) + F_24(x) + F_29(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x) + F_4(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_11(x)*F_4(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_11(x))
Eq(F_31(x), F_32(x) + F_60(x))
Eq(F_32(x), F_16(x) + F_17(x) + F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_2(x) + F_35(x))
Eq(F_35(x), F_16(x) + F_36(x) + F_59(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_40(x))
Eq(F_38(x), F_39(x) + F_4(x))
Eq(F_39(x), F_16(x) + F_24(x) + F_28(x))
Eq(F_40(x), F_35(x) + F_41(x))
Eq(F_41(x), F_16(x) + F_42(x) + F_51(x) + F_55(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_4(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), 2*F_16(x) + F_42(x) + F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_35(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_35(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_16(x) + F_36(x) + F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_2(x) + F_56(x))
Eq(F_59(x), F_2(x)*F_4(x))
Eq(F_60(x), F_16(x) + F_51(x) + F_61(x) + F_66(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_64(x))
Eq(F_63(x), F_45(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), 2*F_16(x) + F_49(x) + F_61(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_56(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_58(x) + F_70(x))
Eq(F_70(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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 2], [1, 0], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", 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