0123_0231_0321_2103_3012
Counting sequence:
1, 1, 2, 6, 19, 47, 96, 210, 480, 1084, 2427, 5448, 12250, 27531, 61853, 138977, 312287, 701708, 1576715, 3542841, 7960700, 17887536, 40192920, 90312666, 202930727, 455981210, 1024580470, 2302211413, 5173022097, 11623675147, 26118160967, 58686974974, 131868435779, 296305685575, 665792831944, 1496022913674, 3361532973728, 7553296029196, 16972102118435, 38135967292328, 85690740673906, 192545346521715, 432645466424997, 972145538697793, 2184391197298879, 4908282466870564, 11028810592342203, 24781512454256081, 55683553033983812, 125119807929881512, 281141656439490744, 631719567774879178, 1419460984059367599, 3189499879454123642, 7166741175192735694, 16103521245780227421, 36184283787299066905, 81305347645185625547, 182691457831890090567, 410503979521666739766, 922394069230037944563, 2072600660149852538335, 4657091410008076281456, 10464389410935967156674, 23513269571730158056480, 52833837144388206988188, 118716554449570604876171, 266753676471799258684040, 599390070248780915256074, 1346817262519790484320027, 3026270918816562625212077, 6799969029904134819488097, 15279391716105041779868255, 34332481543297655754012540, 77144385772796218468405227, 173341861372785050910954729, 389495626975068664536302156, 875188729550126161515153824, 1966531224702535936655655064, 4418755551980129370290161786, 9928853599112668515720824823, 22309931525502930465076156378, 50129961098138400075582975782, 112641000122667062100521283109, 253102029817969593811549385633, 568715098660467849648037078603, 1277891227016238231007102259719, 2871395522874974717850692212398, 6451967174105719817060449605923, 14497438644070176120964489164535, 32575448939371097341138735722584, 73196369348706650986181511003770, 164470748992714223192537268565600, 369562418394764000030117312412396, 830399216433535572266590382386611, 1865890102269120921370760808620008, 4192617002577013414977994687214242, 9420724890989612179060159800661891, 21168176682287116851727553479918005, 47564461252986832467537272073283649, 106876374297271169607741937825823423
Generating function in Maple syntax:
-(5*x^7+5*x^6-7*x^5-5*x^4-x^3+x-1)/(x^2+1)/(x^3-x^2-2*x+1)
Generating function in latex syntax:
-\frac{5 x^{7}+5 x^{6}-7 x^{5}-5 x^{4}-x^{3}+x -1}{\left(x^{2}+1\right) \left(x^{3}-x^{2}-2 x +1\right)}
Generating function in sympy syntax:
(-5*x**7 - 5*x**6 + 7*x**5 + 5*x**4 + x**3 - x + 1)/((x**2 + 1)*(x**3 - x**2 - 2*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2+1)*(x^3-x^2-2*x+1)*F(x)+5*x^7+5*x^6-7*x^5-5*x^4-x^3+x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+1\right) \left(x^{3}-x^{2}-2 x +1\right) F \! \left(x \right)+5 x^{7}+5 x^{6}-7 x^{5}-5 x^{4}-x^{3}+x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,1/15288*((688*I*3^(1/2)+1102)*(-28+84*I*3^(1/2))^(2/3)-2618*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)-7294*(-28+84*I*3^(1/2))^(1/3)+49000)*(1/168*(I*3^(1/2)+5)*(-28+84*I*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(-28+84*I*3^(1/2))^(1/3)-1/12*(-28+84*I*3^(1/2))^(1/3)+1/3)^(-n)+1/15288*((207*I*3^(1/2)-1583)*(-28+84*I*3^(1/2))^(2/3)+4956*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)-280*(-28+84*I*3^(1/2))^(1/3)+49000)*(1/84*(I*3^(1/2)-2)*(-28+84*I*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(-28+84*I*3^(1/2))^(1/3)-1/12*(-28+84*I*3^(1/2))^(1/3)+1/3)^(-n)+1/15288*((-895*I*3^(1/2)+481)*(-28+84*I*3^(1/2))^(2/3)-2338*I*(-28+84*I*3^(1/2))^(1/3)*3^(1/2)+7574*(-28+84*I*3^(1/2))^(1/3)+49000)*(1/6*(-28+84*I*3^(1/2))^(1/3)+1/3-1/168*(-28+84*I*3^(1/2))^(2/3)-1/56*I*3^(1/2)*(-28+84*I*3^(1/2))^(2/3))^(-n)-8/13*cos(1/2*n*Pi)+53/13*sin(1/2*n*Pi))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(\left(688 \,\mathrm{I} \sqrt{3}+1102\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}-2618 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-7294 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+49000\right) \left(\frac{\left(\mathrm{I} \sqrt{3}+5\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{168}-\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{3}\right)^{-n}}{15288}\\+\\\frac{\left(\left(207 \,\mathrm{I} \sqrt{3}-1583\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}+4956 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}-280 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+49000\right) \left(\frac{\left(\mathrm{I} \sqrt{3}-2\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{84}+\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{3}\right)^{-n}}{15288}\\+\\\frac{\left(\left(-895 \,\mathrm{I} \sqrt{3}+481\right) \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}-2338 \,\mathrm{I} \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}+7574 \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}+49000\right) \left(\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}-\frac{\left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{168}-\frac{\mathrm{I} \sqrt{3}\, \left(-28+84 \,\mathrm{I} \sqrt{3}\right)^{\frac{2}{3}}}{56}\right)^{-n}}{15288}\\-\frac{8 \cos \left(\frac{n \pi}{2}\right)}{13}+\frac{53 \sin \left(\frac{n \pi}{2}\right)}{13} & \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) = 47
a(6) = 96
a(7) = 210
a(n+2) = a(n)-a(n+1)-2*a(n+4)+a(n+5), n >= 8
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) = 47
a \! \left(6\right) = 96
a \! \left(7\right) = 210
a \! \left(n +2\right) = a \! \left(n \right)-a \! \left(n +1\right)-2 a \! \left(n +4\right)+a \! \left(n +5\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17164/
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[20,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[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[11,x]
F[20,x] = F[21,x]+F[53,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[10,x]+F[24,x]
F[24,x] = F[21,x]+F[25,x]
F[25,x] = F[26,x]+F[27,x]+F[51,x]
F[26,x] = 0
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[33,x]
F[29,x] = F[11,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[4,x]
F[33,x] = F[34,x]+F[39,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[21,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[21,x]*F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[26,x]+F[38,x]+F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[4,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = x^2
F[48,x] = F[37,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[42,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[21,x]+F[42,x]
F[53,x] = F[26,x]+F[54,x]+F[87,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[66,x]
F[56,x] = F[57,x]+F[7,x]
F[57,x] = F[26,x]+F[58,x]+F[59,x]
F[58,x] = F[19,x]*F[4,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[64,x]
F[61,x] = F[4,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[11,x]*F[4,x]
F[64,x] = F[65,x]
F[65,x] = x^2
F[66,x] = F[67,x]+F[77,x]
F[67,x] = F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[21,x]+F[34,x]
F[71,x] = F[72,x]+F[74,x]
F[72,x] = F[73,x]
F[73,x] = F[36,x]*F[4,x]
F[74,x] = F[75,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[34,x]
F[77,x] = 2*F[26,x]+F[78,x]+F[80,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[34,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[82,x]+F[85,x]
F[82,x] = F[42,x]+F[83,x]
F[83,x] = F[84,x]
F[84,x] = F[34,x]*F[4,x]
F[85,x] = F[86,x]
F[86,x] = F[37,x]*F[4,x]
F[87,x] = F[4,x]*F[88,x]
F[88,x] = F[103,x]+F[89,x]
F[89,x] = F[21,x]+F[90,x]
F[90,x] = F[26,x]+F[51,x]+F[91,x]
F[91,x] = F[4,x]*F[92,x]
F[92,x] = F[93,x]+F[98,x]
F[93,x] = F[11,x]+F[94,x]
F[94,x] = F[26,x]+F[58,x]+F[95,x]
F[95,x] = F[4,x]*F[96,x]
F[96,x] = F[4,x]+F[97,x]
F[97,x] = F[65,x]
F[98,x] = F[34,x]+F[99,x]
F[99,x] = 2*F[26,x]+F[100,x]+F[78,x]
F[100,x] = F[101,x]*F[4,x]
F[101,x] = F[102,x]+F[42,x]
F[102,x] = F[86,x]
F[103,x] = F[104,x]+F[106,x]
F[104,x] = F[105,x]+F[26,x]+F[43,x]
F[105,x] = F[4,x]*F[52,x]
F[106,x] = 2*F[26,x]+F[107,x]+F[111,x]
F[107,x] = F[108,x]*F[4,x]
F[108,x] = F[109,x]+F[110,x]
F[109,x] = F[62,x]
F[110,x] = F[84,x]
F[111,x] = F[4,x]*F[79,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_{20}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{53}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{51}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{33}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{39}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{21}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \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_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{26}\! \left(x \right)+F_{38}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = x^{2}
F_{48}\! \left(x \right) = F_{37}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{21}\! \left(x \right)+F_{42}\! \left(x \right)
F_{53}\! \left(x \right) = F_{26}\! \left(x \right)+F_{54}\! \left(x \right)+F_{87}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{66}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{7}\! \left(x \right)
F_{57}\! \left(x \right) = F_{26}\! \left(x \right)+F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{64}\! \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_{63}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = x^{2}
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{77}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{21}\! \left(x \right)+F_{34}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{74}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{34}\! \left(x \right)
F_{77}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{78}\! \left(x \right)+F_{80}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{34}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{85}\! \left(x \right)
F_{82}\! \left(x \right) = F_{42}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{87}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{103}\! \left(x \right)+F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{21}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{26}\! \left(x \right)+F_{51}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{4}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)+F_{98}\! \left(x \right)
F_{93}\! \left(x \right) = F_{11}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{26}\! \left(x \right)+F_{58}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{4}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{4}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{65}\! \left(x \right)
F_{98}\! \left(x \right) = F_{34}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{100}\! \left(x \right)+F_{78}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{4}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{42}\! \left(x \right)
F_{102}\! \left(x \right) = F_{86}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{106}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{26}\! \left(x \right)+F_{43}\! \left(x \right)
F_{105}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{106}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{107}\! \left(x \right)+F_{111}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{4}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{110}\! \left(x \right)
F_{109}\! \left(x \right) = F_{62}\! \left(x \right)
F_{110}\! \left(x \right) = F_{84}\! \left(x \right)
F_{111}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \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_20(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_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_11(x))
Eq(F_20(x), F_21(x) + F_53(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_10(x) + F_24(x))
Eq(F_24(x), F_21(x) + F_25(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_51(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_33(x))
Eq(F_29(x), F_11(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_4(x))
Eq(F_33(x), F_34(x) + F_39(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_21(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_21(x)*F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_26(x) + F_38(x) + F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_4(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), x**2)
Eq(F_48(x), F_37(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_42(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_21(x) + F_42(x))
Eq(F_53(x), F_26(x) + F_54(x) + F_87(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_66(x))
Eq(F_56(x), F_57(x) + F_7(x))
Eq(F_57(x), F_26(x) + F_58(x) + F_59(x))
Eq(F_58(x), F_19(x)*F_4(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_64(x))
Eq(F_61(x), F_4(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_11(x)*F_4(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), x**2)
Eq(F_66(x), F_67(x) + F_77(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_21(x) + F_34(x))
Eq(F_71(x), F_72(x) + F_74(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_36(x)*F_4(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_34(x))
Eq(F_77(x), 2*F_26(x) + F_78(x) + F_80(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_34(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_82(x) + F_85(x))
Eq(F_82(x), F_42(x) + F_83(x))
Eq(F_83(x), F_84(x))
Eq(F_84(x), F_34(x)*F_4(x))
Eq(F_85(x), F_86(x))
Eq(F_86(x), F_37(x)*F_4(x))
Eq(F_87(x), F_4(x)*F_88(x))
Eq(F_88(x), F_103(x) + F_89(x))
Eq(F_89(x), F_21(x) + F_90(x))
Eq(F_90(x), F_26(x) + F_51(x) + F_91(x))
Eq(F_91(x), F_4(x)*F_92(x))
Eq(F_92(x), F_93(x) + F_98(x))
Eq(F_93(x), F_11(x) + F_94(x))
Eq(F_94(x), F_26(x) + F_58(x) + F_95(x))
Eq(F_95(x), F_4(x)*F_96(x))
Eq(F_96(x), F_4(x) + F_97(x))
Eq(F_97(x), F_65(x))
Eq(F_98(x), F_34(x) + F_99(x))
Eq(F_99(x), F_100(x) + 2*F_26(x) + F_78(x))
Eq(F_100(x), F_101(x)*F_4(x))
Eq(F_101(x), F_102(x) + F_42(x))
Eq(F_102(x), F_86(x))
Eq(F_103(x), F_104(x) + F_106(x))
Eq(F_104(x), F_105(x) + F_26(x) + F_43(x))
Eq(F_105(x), F_4(x)*F_52(x))
Eq(F_106(x), F_107(x) + F_111(x) + 2*F_26(x))
Eq(F_107(x), F_108(x)*F_4(x))
Eq(F_108(x), F_109(x) + F_110(x))
Eq(F_109(x), F_62(x))
Eq(F_110(x), F_84(x))
Eq(F_111(x), F_4(x)*F_79(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, 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, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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, 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, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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, 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, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}], "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, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"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, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}, {"patt": [3, 0, 1, 2], "pos": [[1, 2], [1, 0], [1, 0], [1, 2]]}, {"patt": [3, 0, 1, 2], "pos": [[1, 2], [1, 0], [1, 2], [1, 2]]}], "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", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}], "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], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], 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