0123_0321_1302_1320_2130
Counting sequence:
1, 1, 2, 6, 19, 51, 120, 263, 554, 1133, 2269, 4474, 8716, 16820, 32214, 61318, 116127, 219005, 411574, 771179, 1441358, 2688187, 5004403, 9301706, 17265736, 32011046, 59289132, 109715762, 202876435, 374891675, 692354716, 1278004019, 2358002450, 4348994421, 8018386169, 14779401558, 27234185948, 50173395996, 92415809626, 170193640018, 313381920215, 576960692885, 1062104650722, 1954984984455, 3598156446302, 6621889920355, 12185781308231, 22423221470886, 41259036452592, 75913576780830, 139669516006552, 256961348091342, 472737341032339, 869680324135219, 1599884032417504, 2943118835748655, 5414005349623578, 9959147513275533, 18319733151455765, 33698486762648674, 61986429628481772, 114019312491981812, 209727954033609662, 385772084253966254, 709581464029948143, 1305182003667807485, 2400698166552395726, 4415724750201108627, 8122030650972942958, 14939142414229035707, 27478012392457306811, 50540988881216093394, 92961061688513463352, 170984784386354698918, 314494474380863118468, 578452681304677978906, 1063951940345621357267, 1956931457153834713787, 3599388440200531870084, 6620356560219058020411, 12176813541488892503682, 22396780348343020372853, 41194309340400976775025, 75768483927017433508318, 139360513202895980391036, 256324826754233484265980, 471456283755200541491778, 867145603867302743066842, 1594933154402763149069095, 2933545462206265550790261, 5395641080683356940333242, 9924146977680410254762191, 18253377661165082857862334, 33573237140511924779504003, 61750877340935542730651407, 113577679125173749933086862, 208902096150760541846834272, 384231142143570358479233662, 706711709490344498631407920, 1299846229382215771298390102, 2390791154684510849122199059
Generating function in Maple syntax:
(2*x^8+5*x^7+2*x^6-4*x^5-6*x^4-3*x^3+2*x-1)/(x-1)/(x^2+x-1)/(x^3+x^2+x-1)
Generating function in latex syntax:
\frac{2 x^{8}+5 x^{7}+2 x^{6}-4 x^{5}-6 x^{4}-3 x^{3}+2 x -1}{\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(2*x**8 + 5*x**7 + 2*x**6 - 4*x**5 - 6*x**4 - 3*x**3 + 2*x - 1)/((x - 1)*(x**2 + x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(x^2+x-1)*(x^3+x^2+x-1)*F(x)+2*x^8+5*x^7+2*x^6-4*x^5-6*x^4-3*x^3+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+2 x^{8}+5 x^{7}+2 x^{6}-4 x^{5}-6 x^{4}-3 x^{3}+2 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,n = 2,2,1/264*(((-56*11^(1/2)+220*I)*3^(1/2)-168*I*11^(1/2)+220)*(17+3*11^(1/2)*3^(1/2))^(1/3)+836+((73*11^(1/2)+451*I)*3^(1/2)-219*I*11^(1/2)-451)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(1/24*((17*I+3*11^(1/2))*3^(1/2)-9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/264*(((73*11^(1/2)-451*I)*3^(1/2)+219*I*11^(1/2)-451)*(17+3*11^(1/2)*3^(1/2))^(2/3)+836+((-56*11^(1/2)-220*I)*3^(1/2)+168*I*11^(1/2)+220)*(17+3*11^(1/2)*3^(1/2))^(1/3))*(1/24*((-17*I+3*11^(1/2))*3^(1/2)+9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/264*((-146*11^(1/2)*3^(1/2)+902)*(17+3*11^(1/2)*3^(1/2))^(2/3)+112*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-440*(17+3*11^(1/2)*3^(1/2))^(1/3)+836)*(1/3*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3+17/12*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/4*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2))^(-n)+1/264*(660*5^(1/2)-1452)*(-1/2*5^(1/2)-1/2)^(-n)+3/2+1/264*(-660*5^(1/2)-1452)*(1/2*5^(1/2)-1/2)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(\left(\left(-56 \sqrt{11}+220 \,\mathrm{I}\right) \sqrt{3}-168 \,\mathrm{I} \sqrt{11}+220\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+836+\left(\left(73 \sqrt{11}+451 \,\mathrm{I}\right) \sqrt{3}-219 \,\mathrm{I} \sqrt{11}-451\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{264}\\+\\\frac{\left(\left(\left(73 \sqrt{11}-451 \,\mathrm{I}\right) \sqrt{3}+219 \,\mathrm{I} \sqrt{11}-451\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+836+\left(\left(-56 \sqrt{11}-220 \,\mathrm{I}\right) \sqrt{3}+168 \,\mathrm{I} \sqrt{11}+220\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{264}\\+\\\frac{\left(\left(-146 \sqrt{11}\, \sqrt{3}+902\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+112 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-440 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+836\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{264}\\+\frac{\left(660 \sqrt{5}-1452\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{264}+\frac{3}{2}+\\\frac{\left(-660 \sqrt{5}-1452\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{264} & \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) = 51
a(6) = 120
a(7) = 263
a(8) = 554
a(n+5) = -a(n)-2*a(n+1)-a(n+2)+a(n+3)+2*a(n+4)+3, n >= 9
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) = 51
a \! \left(6\right) = 120
a \! \left(7\right) = 263
a \! \left(8\right) = 554
a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)+3, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17814/
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[26,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[19,x]+F[24,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[11,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[4,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[15,x]
F[26,x] = F[2,x]+F[27,x]
F[27,x] = F[18,x]+F[233,x]+F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[45,x]
F[30,x] = F[31,x]+F[7,x]
F[31,x] = F[18,x]+F[32,x]+F[43,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[36,x]
F[34,x] = F[11,x]+F[35,x]
F[35,x] = F[22,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[11,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[35,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[14,x]
F[45,x] = F[106,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[55,x]
F[49,x] = F[50,x]+F[7,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[11,x]
F[54,x] = F[37,x]
F[55,x] = F[46,x]+F[56,x]
F[56,x] = 2*F[18,x]+F[57,x]+F[78,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[63,x]
F[59,x] = F[50,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[54,x]
F[63,x] = F[64,x]+F[69,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[68,x]
F[67,x] = F[50,x]
F[68,x] = F[64,x]
F[69,x] = F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]+F[73,x]
F[72,x] = F[60,x]
F[73,x] = F[74,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[76,x]+F[77,x]
F[76,x] = F[60,x]
F[77,x] = F[74,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[80,x]
F[80,x] = F[81,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[83,x]+F[85,x]
F[83,x] = F[11,x]+F[84,x]
F[84,x] = F[51,x]
F[85,x] = F[80,x]+F[86,x]
F[86,x] = 2*F[18,x]+F[78,x]+F[87,x]
F[87,x] = F[4,x]*F[88,x]
F[88,x] = F[89,x]+F[91,x]
F[89,x] = F[84,x]+F[90,x]
F[90,x] = F[61,x]
F[91,x] = F[92,x]+F[97,x]
F[92,x] = F[93,x]
F[93,x] = F[4,x]*F[94,x]
F[94,x] = F[95,x]+F[96,x]
F[95,x] = F[84,x]
F[96,x] = F[92,x]
F[97,x] = F[98,x]
F[98,x] = F[4,x]*F[99,x]
F[99,x] = F[100,x]+F[101,x]
F[100,x] = F[90,x]
F[101,x] = F[102,x]
F[102,x] = F[103,x]*F[4,x]
F[103,x] = F[104,x]+F[105,x]
F[104,x] = F[90,x]
F[105,x] = F[102,x]
F[106,x] = 2*F[18,x]+F[107,x]+F[204,x]
F[107,x] = F[108,x]*F[4,x]
F[108,x] = F[109,x]+F[119,x]
F[109,x] = F[110,x]+F[31,x]
F[110,x] = 2*F[18,x]+F[111,x]+F[117,x]
F[111,x] = F[112,x]*F[4,x]
F[112,x] = F[113,x]+F[114,x]
F[113,x] = F[35,x]
F[114,x] = F[115,x]
F[115,x] = F[116,x]*F[4,x]
F[116,x] = F[35,x]
F[117,x] = F[118,x]*F[4,x]
F[118,x] = F[36,x]
F[119,x] = F[120,x]+F[180,x]
F[120,x] = F[121,x]
F[121,x] = F[122,x]*F[4,x]
F[122,x] = F[123,x]+F[129,x]
F[123,x] = F[124,x]+F[31,x]
F[124,x] = F[125,x]
F[125,x] = F[126,x]*F[4,x]
F[126,x] = F[127,x]+F[128,x]
F[127,x] = F[35,x]
F[128,x] = F[115,x]
F[129,x] = F[120,x]+F[130,x]
F[130,x] = 3*F[18,x]+F[131,x]+F[152,x]
F[131,x] = F[132,x]*F[4,x]
F[132,x] = F[133,x]+F[137,x]
F[133,x] = F[124,x]+F[134,x]
F[134,x] = F[135,x]
F[135,x] = F[136,x]*F[4,x]
F[136,x] = F[128,x]
F[137,x] = F[138,x]+F[143,x]
F[138,x] = F[139,x]
F[139,x] = F[140,x]*F[4,x]
F[140,x] = F[141,x]+F[142,x]
F[141,x] = F[124,x]
F[142,x] = F[138,x]
F[143,x] = F[144,x]
F[144,x] = F[145,x]*F[4,x]
F[145,x] = F[146,x]+F[147,x]
F[146,x] = F[134,x]
F[147,x] = F[148,x]
F[148,x] = F[149,x]*F[4,x]
F[149,x] = F[150,x]+F[151,x]
F[150,x] = F[134,x]
F[151,x] = F[148,x]
F[152,x] = F[153,x]*F[4,x]
F[153,x] = F[154,x]
F[154,x] = F[155,x]
F[155,x] = F[156,x]*F[4,x]
F[156,x] = F[157,x]+F[159,x]
F[157,x] = F[158,x]+F[35,x]
F[158,x] = F[125,x]
F[159,x] = F[154,x]+F[160,x]
F[160,x] = 3*F[18,x]+F[152,x]+F[161,x]
F[161,x] = F[162,x]*F[4,x]
F[162,x] = F[163,x]+F[165,x]
F[163,x] = F[158,x]+F[164,x]
F[164,x] = F[135,x]
F[165,x] = F[166,x]+F[171,x]
F[166,x] = F[167,x]
F[167,x] = F[168,x]*F[4,x]
F[168,x] = F[169,x]+F[170,x]
F[169,x] = F[158,x]
F[170,x] = F[166,x]
F[171,x] = F[172,x]
F[172,x] = F[173,x]*F[4,x]
F[173,x] = F[174,x]+F[175,x]
F[174,x] = F[164,x]
F[175,x] = F[176,x]
F[176,x] = F[177,x]*F[4,x]
F[177,x] = F[178,x]+F[179,x]
F[178,x] = F[164,x]
F[179,x] = F[176,x]
F[180,x] = 3*F[18,x]+F[181,x]+F[202,x]
F[181,x] = F[182,x]*F[4,x]
F[182,x] = F[183,x]+F[187,x]
F[183,x] = F[110,x]+F[184,x]
F[184,x] = F[185,x]
F[185,x] = F[186,x]*F[4,x]
F[186,x] = F[114,x]
F[187,x] = F[188,x]+F[193,x]
F[188,x] = F[189,x]
F[189,x] = F[190,x]*F[4,x]
F[190,x] = F[191,x]+F[192,x]
F[191,x] = F[110,x]
F[192,x] = F[188,x]
F[193,x] = F[194,x]
F[194,x] = F[195,x]*F[4,x]
F[195,x] = F[196,x]+F[197,x]
F[196,x] = F[184,x]
F[197,x] = F[198,x]
F[198,x] = F[199,x]*F[4,x]
F[199,x] = F[200,x]+F[201,x]
F[200,x] = F[184,x]
F[201,x] = F[198,x]
F[202,x] = F[203,x]*F[4,x]
F[203,x] = F[154,x]
F[204,x] = F[205,x]*F[4,x]
F[205,x] = F[206,x]
F[206,x] = F[207,x]+F[80,x]
F[207,x] = F[208,x]
F[208,x] = F[209,x]*F[4,x]
F[209,x] = F[210,x]+F[212,x]
F[210,x] = F[211,x]+F[35,x]
F[211,x] = F[111,x]
F[212,x] = F[154,x]+F[213,x]
F[213,x] = 3*F[18,x]+F[202,x]+F[214,x]
F[214,x] = F[215,x]*F[4,x]
F[215,x] = F[216,x]+F[218,x]
F[216,x] = F[211,x]+F[217,x]
F[217,x] = F[185,x]
F[218,x] = F[219,x]+F[224,x]
F[219,x] = F[220,x]
F[220,x] = F[221,x]*F[4,x]
F[221,x] = F[222,x]+F[223,x]
F[222,x] = F[211,x]
F[223,x] = F[219,x]
F[224,x] = F[225,x]
F[225,x] = F[226,x]*F[4,x]
F[226,x] = F[227,x]+F[228,x]
F[227,x] = F[217,x]
F[228,x] = F[229,x]
F[229,x] = F[230,x]*F[4,x]
F[230,x] = F[231,x]+F[232,x]
F[231,x] = F[217,x]
F[232,x] = F[229,x]
F[233,x] = F[234,x]*F[4,x]
F[234,x] = F[235,x]
F[235,x] = F[2,x]+F[236,x]
F[236,x] = F[18,x]+F[237,x]+F[294,x]
F[237,x] = F[238,x]*F[4,x]
F[238,x] = F[239,x]+F[241,x]
F[239,x] = F[11,x]+F[240,x]
F[240,x] = F[18,x]+F[24,x]+F[32,x]
F[241,x] = F[242,x]+F[80,x]
F[242,x] = 2*F[18,x]+F[204,x]+F[243,x]
F[243,x] = F[244,x]*F[4,x]
F[244,x] = F[245,x]+F[247,x]
F[245,x] = F[240,x]+F[246,x]
F[246,x] = 2*F[18,x]+F[111,x]+F[117,x]
F[247,x] = F[248,x]+F[274,x]
F[248,x] = F[249,x]
F[249,x] = F[250,x]*F[4,x]
F[250,x] = F[251,x]+F[253,x]
F[251,x] = F[240,x]+F[252,x]
F[252,x] = F[125,x]
F[253,x] = F[248,x]+F[254,x]
F[254,x] = 3*F[18,x]+F[152,x]+F[255,x]
F[255,x] = F[256,x]*F[4,x]
F[256,x] = F[257,x]+F[259,x]
F[257,x] = F[252,x]+F[258,x]
F[258,x] = F[135,x]
F[259,x] = F[260,x]+F[265,x]
F[260,x] = F[261,x]
F[261,x] = F[262,x]*F[4,x]
F[262,x] = F[263,x]+F[264,x]
F[263,x] = F[252,x]
F[264,x] = F[260,x]
F[265,x] = F[266,x]
F[266,x] = F[267,x]*F[4,x]
F[267,x] = F[268,x]+F[269,x]
F[268,x] = F[258,x]
F[269,x] = F[270,x]
F[270,x] = F[271,x]*F[4,x]
F[271,x] = F[272,x]+F[273,x]
F[272,x] = F[258,x]
F[273,x] = F[270,x]
F[274,x] = 3*F[18,x]+F[202,x]+F[275,x]
F[275,x] = F[276,x]*F[4,x]
F[276,x] = F[277,x]+F[279,x]
F[277,x] = F[246,x]+F[278,x]
F[278,x] = F[185,x]
F[279,x] = F[280,x]+F[285,x]
F[280,x] = F[281,x]
F[281,x] = F[282,x]*F[4,x]
F[282,x] = F[283,x]+F[284,x]
F[283,x] = F[246,x]
F[284,x] = F[280,x]
F[285,x] = F[286,x]
F[286,x] = F[287,x]*F[4,x]
F[287,x] = F[288,x]+F[289,x]
F[288,x] = F[278,x]
F[289,x] = F[290,x]
F[290,x] = F[291,x]*F[4,x]
F[291,x] = F[292,x]+F[293,x]
F[292,x] = F[278,x]
F[293,x] = F[290,x]
F[294,x] = F[295,x]*F[4,x]
F[295,x] = F[2,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_{26}\! \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_{19}\! \left(x \right)+F_{24}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right)+F_{21}\! \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_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)
F_{26}\! \left(x \right) = F_{2}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18}\! \left(x \right)+F_{233}\! \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_{45}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{7}\! \left(x \right)
F_{31}\! \left(x \right) = F_{18}\! \left(x \right)+F_{32}\! \left(x \right)+F_{43}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{11}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{22}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{35}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{14}\! \left(x \right)
F_{45}\! \left(x \right) = F_{106}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{55}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{7}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \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_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{11}\! \left(x \right)
F_{54}\! \left(x \right) = F_{37}\! \left(x \right)
F_{55}\! \left(x \right) = F_{46}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{57}\! \left(x \right)+F_{78}\! \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_{63}\! \left(x \right)
F_{59}\! \left(x \right) = F_{50}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{54}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{69}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{68}\! \left(x \right)
F_{67}\! \left(x \right) = F_{50}\! \left(x \right)
F_{68}\! \left(x \right) = F_{64}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{73}\! \left(x \right)
F_{72}\! \left(x \right) = F_{60}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{76}\! \left(x \right) = F_{60}\! \left(x \right)
F_{77}\! \left(x \right) = F_{74}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{85}\! \left(x \right)
F_{83}\! \left(x \right) = F_{11}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{51}\! \left(x \right)
F_{85}\! \left(x \right) = F_{80}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{78}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{91}\! \left(x \right)
F_{89}\! \left(x \right) = F_{84}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{61}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)+F_{97}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{4}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)+F_{96}\! \left(x \right)
F_{95}\! \left(x \right) = F_{84}\! \left(x \right)
F_{96}\! \left(x \right) = F_{92}\! \left(x \right)
F_{97}\! \left(x \right) = F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{4}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{101}\! \left(x \right)
F_{100}\! \left(x \right) = F_{90}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right) F_{4}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{105}\! \left(x \right)
F_{104}\! \left(x \right) = F_{90}\! \left(x \right)
F_{105}\! \left(x \right) = F_{102}\! \left(x \right)
F_{106}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{107}\! \left(x \right)+F_{204}\! \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_{119}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)+F_{31}\! \left(x \right)
F_{110}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{111}\! \left(x \right)+F_{117}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right) F_{4}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)+F_{114}\! \left(x \right)
F_{113}\! \left(x \right) = F_{35}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right) F_{4}\! \left(x \right)
F_{116}\! \left(x \right) = F_{35}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right) F_{4}\! \left(x \right)
F_{118}\! \left(x \right) = F_{36}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)+F_{180}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right) F_{4}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)+F_{129}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)+F_{31}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right) F_{4}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)+F_{128}\! \left(x \right)
F_{127}\! \left(x \right) = F_{35}\! \left(x \right)
F_{128}\! \left(x \right) = F_{115}\! \left(x \right)
F_{129}\! \left(x \right) = F_{120}\! \left(x \right)+F_{130}\! \left(x \right)
F_{130}\! \left(x \right) = 3 F_{18}\! \left(x \right)+F_{131}\! \left(x \right)+F_{152}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right) F_{4}\! \left(x \right)
F_{132}\! \left(x \right) = F_{133}\! \left(x \right)+F_{137}\! \left(x \right)
F_{133}\! \left(x \right) = F_{124}\! \left(x \right)+F_{134}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right) F_{4}\! \left(x \right)
F_{136}\! \left(x \right) = F_{128}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right)+F_{143}\! \left(x \right)
F_{138}\! \left(x \right) = F_{139}\! \left(x \right)
F_{139}\! \left(x \right) = F_{140}\! \left(x \right) F_{4}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right)+F_{142}\! \left(x \right)
F_{141}\! \left(x \right) = F_{124}\! \left(x \right)
F_{142}\! \left(x \right) = F_{138}\! \left(x \right)
F_{143}\! \left(x \right) = F_{144}\! \left(x \right)
F_{144}\! \left(x \right) = F_{145}\! \left(x \right) F_{4}\! \left(x \right)
F_{145}\! \left(x \right) = F_{146}\! \left(x \right)+F_{147}\! \left(x \right)
F_{146}\! \left(x \right) = F_{134}\! \left(x \right)
F_{147}\! \left(x \right) = F_{148}\! \left(x \right)
F_{148}\! \left(x \right) = F_{149}\! \left(x \right) F_{4}\! \left(x \right)
F_{149}\! \left(x \right) = F_{150}\! \left(x \right)+F_{151}\! \left(x \right)
F_{150}\! \left(x \right) = F_{134}\! \left(x \right)
F_{151}\! \left(x \right) = F_{148}\! \left(x \right)
F_{152}\! \left(x \right) = F_{153}\! \left(x \right) F_{4}\! \left(x \right)
F_{153}\! \left(x \right) = F_{154}\! \left(x \right)
F_{154}\! \left(x \right) = F_{155}\! \left(x \right)
F_{155}\! \left(x \right) = F_{156}\! \left(x \right) F_{4}\! \left(x \right)
F_{156}\! \left(x \right) = F_{157}\! \left(x \right)+F_{159}\! \left(x \right)
F_{157}\! \left(x \right) = F_{158}\! \left(x \right)+F_{35}\! \left(x \right)
F_{158}\! \left(x \right) = F_{125}\! \left(x \right)
F_{159}\! \left(x \right) = F_{154}\! \left(x \right)+F_{160}\! \left(x \right)
F_{160}\! \left(x \right) = 3 F_{18}\! \left(x \right)+F_{152}\! \left(x \right)+F_{161}\! \left(x \right)
F_{161}\! \left(x \right) = F_{162}\! \left(x \right) F_{4}\! \left(x \right)
F_{162}\! \left(x \right) = F_{163}\! \left(x \right)+F_{165}\! \left(x \right)
F_{163}\! \left(x \right) = F_{158}\! \left(x \right)+F_{164}\! \left(x \right)
F_{164}\! \left(x \right) = F_{135}\! \left(x \right)
F_{165}\! \left(x \right) = F_{166}\! \left(x \right)+F_{171}\! \left(x \right)
F_{166}\! \left(x \right) = F_{167}\! \left(x \right)
F_{167}\! \left(x \right) = F_{168}\! \left(x \right) F_{4}\! \left(x \right)
F_{168}\! \left(x \right) = F_{169}\! \left(x \right)+F_{170}\! \left(x \right)
F_{169}\! \left(x \right) = F_{158}\! \left(x \right)
F_{170}\! \left(x \right) = F_{166}\! \left(x \right)
F_{171}\! \left(x \right) = F_{172}\! \left(x \right)
F_{172}\! \left(x \right) = F_{173}\! \left(x \right) F_{4}\! \left(x \right)
F_{173}\! \left(x \right) = F_{174}\! \left(x \right)+F_{175}\! \left(x \right)
F_{174}\! \left(x \right) = F_{164}\! \left(x \right)
F_{175}\! \left(x \right) = F_{176}\! \left(x \right)
F_{176}\! \left(x \right) = F_{177}\! \left(x \right) F_{4}\! \left(x \right)
F_{177}\! \left(x \right) = F_{178}\! \left(x \right)+F_{179}\! \left(x \right)
F_{178}\! \left(x \right) = F_{164}\! \left(x \right)
F_{179}\! \left(x \right) = F_{176}\! \left(x \right)
F_{180}\! \left(x \right) = 3 F_{18}\! \left(x \right)+F_{181}\! \left(x \right)+F_{202}\! \left(x \right)
F_{181}\! \left(x \right) = F_{182}\! \left(x \right) F_{4}\! \left(x \right)
F_{182}\! \left(x \right) = F_{183}\! \left(x \right)+F_{187}\! \left(x \right)
F_{183}\! \left(x \right) = F_{110}\! \left(x \right)+F_{184}\! \left(x \right)
F_{184}\! \left(x \right) = F_{185}\! \left(x \right)
F_{185}\! \left(x \right) = F_{186}\! \left(x \right) F_{4}\! \left(x \right)
F_{186}\! \left(x \right) = F_{114}\! \left(x \right)
F_{187}\! \left(x \right) = F_{188}\! \left(x \right)+F_{193}\! \left(x \right)
F_{188}\! \left(x \right) = F_{189}\! \left(x \right)
F_{189}\! \left(x \right) = F_{190}\! \left(x \right) F_{4}\! \left(x \right)
F_{190}\! \left(x \right) = F_{191}\! \left(x \right)+F_{192}\! \left(x \right)
F_{191}\! \left(x \right) = F_{110}\! \left(x \right)
F_{192}\! \left(x \right) = F_{188}\! \left(x \right)
F_{193}\! \left(x \right) = F_{194}\! \left(x \right)
F_{194}\! \left(x \right) = F_{195}\! \left(x \right) F_{4}\! \left(x \right)
F_{195}\! \left(x \right) = F_{196}\! \left(x \right)+F_{197}\! \left(x \right)
F_{196}\! \left(x \right) = F_{184}\! \left(x \right)
F_{197}\! \left(x \right) = F_{198}\! \left(x \right)
F_{198}\! \left(x \right) = F_{199}\! \left(x \right) F_{4}\! \left(x \right)
F_{199}\! \left(x \right) = F_{200}\! \left(x \right)+F_{201}\! \left(x \right)
F_{200}\! \left(x \right) = F_{184}\! \left(x \right)
F_{201}\! \left(x \right) = F_{198}\! \left(x \right)
F_{202}\! \left(x \right) = F_{203}\! \left(x \right) F_{4}\! \left(x \right)
F_{203}\! \left(x \right) = F_{154}\! \left(x \right)
F_{204}\! \left(x \right) = F_{205}\! \left(x \right) F_{4}\! \left(x \right)
F_{205}\! \left(x \right) = F_{206}\! \left(x \right)
F_{206}\! \left(x \right) = F_{207}\! \left(x \right)+F_{80}\! \left(x \right)
F_{207}\! \left(x \right) = F_{208}\! \left(x \right)
F_{208}\! \left(x \right) = F_{209}\! \left(x \right) F_{4}\! \left(x \right)
F_{209}\! \left(x \right) = F_{210}\! \left(x \right)+F_{212}\! \left(x \right)
F_{210}\! \left(x \right) = F_{211}\! \left(x \right)+F_{35}\! \left(x \right)
F_{211}\! \left(x \right) = F_{111}\! \left(x \right)
F_{212}\! \left(x \right) = F_{154}\! \left(x \right)+F_{213}\! \left(x \right)
F_{213}\! \left(x \right) = 3 F_{18}\! \left(x \right)+F_{202}\! \left(x \right)+F_{214}\! \left(x \right)
F_{214}\! \left(x \right) = F_{215}\! \left(x \right) F_{4}\! \left(x \right)
F_{215}\! \left(x \right) = F_{216}\! \left(x \right)+F_{218}\! \left(x \right)
F_{216}\! \left(x \right) = F_{211}\! \left(x \right)+F_{217}\! \left(x \right)
F_{217}\! \left(x \right) = F_{185}\! \left(x \right)
F_{218}\! \left(x \right) = F_{219}\! \left(x \right)+F_{224}\! \left(x \right)
F_{219}\! \left(x \right) = F_{220}\! \left(x \right)
F_{220}\! \left(x \right) = F_{221}\! \left(x \right) F_{4}\! \left(x \right)
F_{221}\! \left(x \right) = F_{222}\! \left(x \right)+F_{223}\! \left(x \right)
F_{222}\! \left(x \right) = F_{211}\! \left(x \right)
F_{223}\! \left(x \right) = F_{219}\! \left(x \right)
F_{224}\! \left(x \right) = F_{225}\! \left(x \right)
F_{225}\! \left(x \right) = F_{226}\! \left(x \right) F_{4}\! \left(x \right)
F_{226}\! \left(x \right) = F_{227}\! \left(x \right)+F_{228}\! \left(x \right)
F_{227}\! \left(x \right) = F_{217}\! \left(x \right)
F_{228}\! \left(x \right) = F_{229}\! \left(x \right)
F_{229}\! \left(x \right) = F_{230}\! \left(x \right) F_{4}\! \left(x \right)
F_{230}\! \left(x \right) = F_{231}\! \left(x \right)+F_{232}\! \left(x \right)
F_{231}\! \left(x \right) = F_{217}\! \left(x \right)
F_{232}\! \left(x \right) = F_{229}\! \left(x \right)
F_{233}\! \left(x \right) = F_{234}\! \left(x \right) F_{4}\! \left(x \right)
F_{234}\! \left(x \right) = F_{235}\! \left(x \right)
F_{235}\! \left(x \right) = F_{2}\! \left(x \right)+F_{236}\! \left(x \right)
F_{236}\! \left(x \right) = F_{18}\! \left(x \right)+F_{237}\! \left(x \right)+F_{294}\! \left(x \right)
F_{237}\! \left(x \right) = F_{238}\! \left(x \right) F_{4}\! \left(x \right)
F_{238}\! \left(x \right) = F_{239}\! \left(x \right)+F_{241}\! \left(x \right)
F_{239}\! \left(x \right) = F_{11}\! \left(x \right)+F_{240}\! \left(x \right)
F_{240}\! \left(x \right) = F_{18}\! \left(x \right)+F_{24}\! \left(x \right)+F_{32}\! \left(x \right)
F_{241}\! \left(x \right) = F_{242}\! \left(x \right)+F_{80}\! \left(x \right)
F_{242}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{204}\! \left(x \right)+F_{243}\! \left(x \right)
F_{243}\! \left(x \right) = F_{244}\! \left(x \right) F_{4}\! \left(x \right)
F_{244}\! \left(x \right) = F_{245}\! \left(x \right)+F_{247}\! \left(x \right)
F_{245}\! \left(x \right) = F_{240}\! \left(x \right)+F_{246}\! \left(x \right)
F_{246}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{111}\! \left(x \right)+F_{117}\! \left(x \right)
F_{247}\! \left(x \right) = F_{248}\! \left(x \right)+F_{274}\! \left(x \right)
F_{248}\! \left(x \right) = F_{249}\! \left(x \right)
F_{249}\! \left(x \right) = F_{250}\! \left(x \right) F_{4}\! \left(x \right)
F_{250}\! \left(x \right) = F_{251}\! \left(x \right)+F_{253}\! \left(x \right)
F_{251}\! \left(x \right) = F_{240}\! \left(x \right)+F_{252}\! \left(x \right)
F_{252}\! \left(x \right) = F_{125}\! \left(x \right)
F_{253}\! \left(x \right) = F_{248}\! \left(x \right)+F_{254}\! \left(x \right)
F_{254}\! \left(x \right) = 3 F_{18}\! \left(x \right)+F_{152}\! \left(x \right)+F_{255}\! \left(x \right)
F_{255}\! \left(x \right) = F_{256}\! \left(x \right) F_{4}\! \left(x \right)
F_{256}\! \left(x \right) = F_{257}\! \left(x \right)+F_{259}\! \left(x \right)
F_{257}\! \left(x \right) = F_{252}\! \left(x \right)+F_{258}\! \left(x \right)
F_{258}\! \left(x \right) = F_{135}\! \left(x \right)
F_{259}\! \left(x \right) = F_{260}\! \left(x \right)+F_{265}\! \left(x \right)
F_{260}\! \left(x \right) = F_{261}\! \left(x \right)
F_{261}\! \left(x \right) = F_{262}\! \left(x \right) F_{4}\! \left(x \right)
F_{262}\! \left(x \right) = F_{263}\! \left(x \right)+F_{264}\! \left(x \right)
F_{263}\! \left(x \right) = F_{252}\! \left(x \right)
F_{264}\! \left(x \right) = F_{260}\! \left(x \right)
F_{265}\! \left(x \right) = F_{266}\! \left(x \right)
F_{266}\! \left(x \right) = F_{267}\! \left(x \right) F_{4}\! \left(x \right)
F_{267}\! \left(x \right) = F_{268}\! \left(x \right)+F_{269}\! \left(x \right)
F_{268}\! \left(x \right) = F_{258}\! \left(x \right)
F_{269}\! \left(x \right) = F_{270}\! \left(x \right)
F_{270}\! \left(x \right) = F_{271}\! \left(x \right) F_{4}\! \left(x \right)
F_{271}\! \left(x \right) = F_{272}\! \left(x \right)+F_{273}\! \left(x \right)
F_{272}\! \left(x \right) = F_{258}\! \left(x \right)
F_{273}\! \left(x \right) = F_{270}\! \left(x \right)
F_{274}\! \left(x \right) = 3 F_{18}\! \left(x \right)+F_{202}\! \left(x \right)+F_{275}\! \left(x \right)
F_{275}\! \left(x \right) = F_{276}\! \left(x \right) F_{4}\! \left(x \right)
F_{276}\! \left(x \right) = F_{277}\! \left(x \right)+F_{279}\! \left(x \right)
F_{277}\! \left(x \right) = F_{246}\! \left(x \right)+F_{278}\! \left(x \right)
F_{278}\! \left(x \right) = F_{185}\! \left(x \right)
F_{279}\! \left(x \right) = F_{280}\! \left(x \right)+F_{285}\! \left(x \right)
F_{280}\! \left(x \right) = F_{281}\! \left(x \right)
F_{281}\! \left(x \right) = F_{282}\! \left(x \right) F_{4}\! \left(x \right)
F_{282}\! \left(x \right) = F_{283}\! \left(x \right)+F_{284}\! \left(x \right)
F_{283}\! \left(x \right) = F_{246}\! \left(x \right)
F_{284}\! \left(x \right) = F_{280}\! \left(x \right)
F_{285}\! \left(x \right) = F_{286}\! \left(x \right)
F_{286}\! \left(x \right) = F_{287}\! \left(x \right) F_{4}\! \left(x \right)
F_{287}\! \left(x \right) = F_{288}\! \left(x \right)+F_{289}\! \left(x \right)
F_{288}\! \left(x \right) = F_{278}\! \left(x \right)
F_{289}\! \left(x \right) = F_{290}\! \left(x \right)
F_{290}\! \left(x \right) = F_{291}\! \left(x \right) F_{4}\! \left(x \right)
F_{291}\! \left(x \right) = F_{292}\! \left(x \right)+F_{293}\! \left(x \right)
F_{292}\! \left(x \right) = F_{278}\! \left(x \right)
F_{293}\! \left(x \right) = F_{290}\! \left(x \right)
F_{294}\! \left(x \right) = F_{295}\! \left(x \right) F_{4}\! \left(x \right)
F_{295}\! \left(x \right) = F_{2}\! \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_26(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) + F_19(x) + F_24(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_11(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_4(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_15(x))
Eq(F_26(x), F_2(x) + F_27(x))
Eq(F_27(x), F_18(x) + F_233(x) + F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_45(x))
Eq(F_30(x), F_31(x) + F_7(x))
Eq(F_31(x), F_18(x) + F_32(x) + F_43(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_36(x))
Eq(F_34(x), F_11(x) + F_35(x))
Eq(F_35(x), F_22(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_11(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_35(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_14(x))
Eq(F_45(x), F_106(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_55(x))
Eq(F_49(x), F_50(x) + F_7(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_11(x))
Eq(F_54(x), F_37(x))
Eq(F_55(x), F_46(x) + F_56(x))
Eq(F_56(x), 2*F_18(x) + F_57(x) + F_78(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_63(x))
Eq(F_59(x), F_50(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_54(x))
Eq(F_63(x), F_64(x) + F_69(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_68(x))
Eq(F_67(x), F_50(x))
Eq(F_68(x), F_64(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_73(x))
Eq(F_72(x), F_60(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_77(x))
Eq(F_76(x), F_60(x))
Eq(F_77(x), F_74(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_85(x))
Eq(F_83(x), F_11(x) + F_84(x))
Eq(F_84(x), F_51(x))
Eq(F_85(x), F_80(x) + F_86(x))
Eq(F_86(x), 2*F_18(x) + F_78(x) + F_87(x))
Eq(F_87(x), F_4(x)*F_88(x))
Eq(F_88(x), F_89(x) + F_91(x))
Eq(F_89(x), F_84(x) + F_90(x))
Eq(F_90(x), F_61(x))
Eq(F_91(x), F_92(x) + F_97(x))
Eq(F_92(x), F_93(x))
Eq(F_93(x), F_4(x)*F_94(x))
Eq(F_94(x), F_95(x) + F_96(x))
Eq(F_95(x), F_84(x))
Eq(F_96(x), F_92(x))
Eq(F_97(x), F_98(x))
Eq(F_98(x), F_4(x)*F_99(x))
Eq(F_99(x), F_100(x) + F_101(x))
Eq(F_100(x), F_90(x))
Eq(F_101(x), F_102(x))
Eq(F_102(x), F_103(x)*F_4(x))
Eq(F_103(x), F_104(x) + F_105(x))
Eq(F_104(x), F_90(x))
Eq(F_105(x), F_102(x))
Eq(F_106(x), F_107(x) + 2*F_18(x) + F_204(x))
Eq(F_107(x), F_108(x)*F_4(x))
Eq(F_108(x), F_109(x) + F_119(x))
Eq(F_109(x), F_110(x) + F_31(x))
Eq(F_110(x), F_111(x) + F_117(x) + 2*F_18(x))
Eq(F_111(x), F_112(x)*F_4(x))
Eq(F_112(x), F_113(x) + F_114(x))
Eq(F_113(x), F_35(x))
Eq(F_114(x), F_115(x))
Eq(F_115(x), F_116(x)*F_4(x))
Eq(F_116(x), F_35(x))
Eq(F_117(x), F_118(x)*F_4(x))
Eq(F_118(x), F_36(x))
Eq(F_119(x), F_120(x) + F_180(x))
Eq(F_120(x), F_121(x))
Eq(F_121(x), F_122(x)*F_4(x))
Eq(F_122(x), F_123(x) + F_129(x))
Eq(F_123(x), F_124(x) + F_31(x))
Eq(F_124(x), F_125(x))
Eq(F_125(x), F_126(x)*F_4(x))
Eq(F_126(x), F_127(x) + F_128(x))
Eq(F_127(x), F_35(x))
Eq(F_128(x), F_115(x))
Eq(F_129(x), F_120(x) + F_130(x))
Eq(F_130(x), F_131(x) + F_152(x) + 3*F_18(x))
Eq(F_131(x), F_132(x)*F_4(x))
Eq(F_132(x), F_133(x) + F_137(x))
Eq(F_133(x), F_124(x) + F_134(x))
Eq(F_134(x), F_135(x))
Eq(F_135(x), F_136(x)*F_4(x))
Eq(F_136(x), F_128(x))
Eq(F_137(x), F_138(x) + F_143(x))
Eq(F_138(x), F_139(x))
Eq(F_139(x), F_140(x)*F_4(x))
Eq(F_140(x), F_141(x) + F_142(x))
Eq(F_141(x), F_124(x))
Eq(F_142(x), F_138(x))
Eq(F_143(x), F_144(x))
Eq(F_144(x), F_145(x)*F_4(x))
Eq(F_145(x), F_146(x) + F_147(x))
Eq(F_146(x), F_134(x))
Eq(F_147(x), F_148(x))
Eq(F_148(x), F_149(x)*F_4(x))
Eq(F_149(x), F_150(x) + F_151(x))
Eq(F_150(x), F_134(x))
Eq(F_151(x), F_148(x))
Eq(F_152(x), F_153(x)*F_4(x))
Eq(F_153(x), F_154(x))
Eq(F_154(x), F_155(x))
Eq(F_155(x), F_156(x)*F_4(x))
Eq(F_156(x), F_157(x) + F_159(x))
Eq(F_157(x), F_158(x) + F_35(x))
Eq(F_158(x), F_125(x))
Eq(F_159(x), F_154(x) + F_160(x))
Eq(F_160(x), F_152(x) + F_161(x) + 3*F_18(x))
Eq(F_161(x), F_162(x)*F_4(x))
Eq(F_162(x), F_163(x) + F_165(x))
Eq(F_163(x), F_158(x) + F_164(x))
Eq(F_164(x), F_135(x))
Eq(F_165(x), F_166(x) + F_171(x))
Eq(F_166(x), F_167(x))
Eq(F_167(x), F_168(x)*F_4(x))
Eq(F_168(x), F_169(x) + F_170(x))
Eq(F_169(x), F_158(x))
Eq(F_170(x), F_166(x))
Eq(F_171(x), F_172(x))
Eq(F_172(x), F_173(x)*F_4(x))
Eq(F_173(x), F_174(x) + F_175(x))
Eq(F_174(x), F_164(x))
Eq(F_175(x), F_176(x))
Eq(F_176(x), F_177(x)*F_4(x))
Eq(F_177(x), F_178(x) + F_179(x))
Eq(F_178(x), F_164(x))
Eq(F_179(x), F_176(x))
Eq(F_180(x), 3*F_18(x) + F_181(x) + F_202(x))
Eq(F_181(x), F_182(x)*F_4(x))
Eq(F_182(x), F_183(x) + F_187(x))
Eq(F_183(x), F_110(x) + F_184(x))
Eq(F_184(x), F_185(x))
Eq(F_185(x), F_186(x)*F_4(x))
Eq(F_186(x), F_114(x))
Eq(F_187(x), F_188(x) + F_193(x))
Eq(F_188(x), F_189(x))
Eq(F_189(x), F_190(x)*F_4(x))
Eq(F_190(x), F_191(x) + F_192(x))
Eq(F_191(x), F_110(x))
Eq(F_192(x), F_188(x))
Eq(F_193(x), F_194(x))
Eq(F_194(x), F_195(x)*F_4(x))
Eq(F_195(x), F_196(x) + F_197(x))
Eq(F_196(x), F_184(x))
Eq(F_197(x), F_198(x))
Eq(F_198(x), F_199(x)*F_4(x))
Eq(F_199(x), F_200(x) + F_201(x))
Eq(F_200(x), F_184(x))
Eq(F_201(x), F_198(x))
Eq(F_202(x), F_203(x)*F_4(x))
Eq(F_203(x), F_154(x))
Eq(F_204(x), F_205(x)*F_4(x))
Eq(F_205(x), F_206(x))
Eq(F_206(x), F_207(x) + F_80(x))
Eq(F_207(x), F_208(x))
Eq(F_208(x), F_209(x)*F_4(x))
Eq(F_209(x), F_210(x) + F_212(x))
Eq(F_210(x), F_211(x) + F_35(x))
Eq(F_211(x), F_111(x))
Eq(F_212(x), F_154(x) + F_213(x))
Eq(F_213(x), 3*F_18(x) + F_202(x) + F_214(x))
Eq(F_214(x), F_215(x)*F_4(x))
Eq(F_215(x), F_216(x) + F_218(x))
Eq(F_216(x), F_211(x) + F_217(x))
Eq(F_217(x), F_185(x))
Eq(F_218(x), F_219(x) + F_224(x))
Eq(F_219(x), F_220(x))
Eq(F_220(x), F_221(x)*F_4(x))
Eq(F_221(x), F_222(x) + F_223(x))
Eq(F_222(x), F_211(x))
Eq(F_223(x), F_219(x))
Eq(F_224(x), F_225(x))
Eq(F_225(x), F_226(x)*F_4(x))
Eq(F_226(x), F_227(x) + F_228(x))
Eq(F_227(x), F_217(x))
Eq(F_228(x), F_229(x))
Eq(F_229(x), F_230(x)*F_4(x))
Eq(F_230(x), F_231(x) + F_232(x))
Eq(F_231(x), F_217(x))
Eq(F_232(x), F_229(x))
Eq(F_233(x), F_234(x)*F_4(x))
Eq(F_234(x), F_235(x))
Eq(F_235(x), F_2(x) + F_236(x))
Eq(F_236(x), F_18(x) + F_237(x) + F_294(x))
Eq(F_237(x), F_238(x)*F_4(x))
Eq(F_238(x), F_239(x) + F_241(x))
Eq(F_239(x), F_11(x) + F_240(x))
Eq(F_240(x), F_18(x) + F_24(x) + F_32(x))
Eq(F_241(x), F_242(x) + F_80(x))
Eq(F_242(x), 2*F_18(x) + F_204(x) + F_243(x))
Eq(F_243(x), F_244(x)*F_4(x))
Eq(F_244(x), F_245(x) + F_247(x))
Eq(F_245(x), F_240(x) + F_246(x))
Eq(F_246(x), F_111(x) + F_117(x) + 2*F_18(x))
Eq(F_247(x), F_248(x) + F_274(x))
Eq(F_248(x), F_249(x))
Eq(F_249(x), F_250(x)*F_4(x))
Eq(F_250(x), F_251(x) + F_253(x))
Eq(F_251(x), F_240(x) + F_252(x))
Eq(F_252(x), F_125(x))
Eq(F_253(x), F_248(x) + F_254(x))
Eq(F_254(x), F_152(x) + 3*F_18(x) + F_255(x))
Eq(F_255(x), F_256(x)*F_4(x))
Eq(F_256(x), F_257(x) + F_259(x))
Eq(F_257(x), F_252(x) + F_258(x))
Eq(F_258(x), F_135(x))
Eq(F_259(x), F_260(x) + F_265(x))
Eq(F_260(x), F_261(x))
Eq(F_261(x), F_262(x)*F_4(x))
Eq(F_262(x), F_263(x) + F_264(x))
Eq(F_263(x), F_252(x))
Eq(F_264(x), F_260(x))
Eq(F_265(x), F_266(x))
Eq(F_266(x), F_267(x)*F_4(x))
Eq(F_267(x), F_268(x) + F_269(x))
Eq(F_268(x), F_258(x))
Eq(F_269(x), F_270(x))
Eq(F_270(x), F_271(x)*F_4(x))
Eq(F_271(x), F_272(x) + F_273(x))
Eq(F_272(x), F_258(x))
Eq(F_273(x), F_270(x))
Eq(F_274(x), 3*F_18(x) + F_202(x) + F_275(x))
Eq(F_275(x), F_276(x)*F_4(x))
Eq(F_276(x), F_277(x) + F_279(x))
Eq(F_277(x), F_246(x) + F_278(x))
Eq(F_278(x), F_185(x))
Eq(F_279(x), F_280(x) + F_285(x))
Eq(F_280(x), F_281(x))
Eq(F_281(x), F_282(x)*F_4(x))
Eq(F_282(x), F_283(x) + F_284(x))
Eq(F_283(x), F_246(x))
Eq(F_284(x), F_280(x))
Eq(F_285(x), F_286(x))
Eq(F_286(x), F_287(x)*F_4(x))
Eq(F_287(x), F_288(x) + F_289(x))
Eq(F_288(x), F_278(x))
Eq(F_289(x), F_290(x))
Eq(F_290(x), F_291(x)*F_4(x))
Eq(F_291(x), F_292(x) + F_293(x))
Eq(F_292(x), F_278(x))
Eq(F_293(x), F_290(x))
Eq(F_294(x), F_295(x)*F_4(x))
Eq(F_295(x), F_2(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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