0132_0231_0312_3120
Counting sequence:
1, 1, 2, 6, 20, 65, 197, 554, 1460, 3654, 8792, 20542, 46960, 105619, 234627, 516190, 1126800, 2443744, 5270398, 11311378, 24171980, 51455349, 109153409, 230821922, 486717020, 1023641114, 2147780420, 4496671446, 9395716856, 19596382999, 40802926591, 84826511350, 176094767832, 365073566260, 755915868522, 1563370044874, 3229817732788, 6665792005561, 13743898611389, 28312428255258, 58274120770244, 119846772675182, 246290610720176, 505775355837998, 1037938984767680, 2128654520743323, 4362862149755003, 8936830522837198, 18295873500178912, 37436171918410504, 76561193683310550, 156500087071486018, 319755573566265564, 653021945994552061, 1333065489730659577, 2720174174964251218, 5548434740956742636, 11313042263995164738, 23058430092182000540, 46981551312779081798, 95692484882423828168, 194843734278618617375, 396604997584823309303, 807045053224867861734, 1641760222560232697000, 3338860677341519705692, 6788401819125214794946, 13798164567134854066810, 28039050992038638338628, 56963545699615226511169, 115697978830306450959669, 234937732522765005625546, 476959014769834336819156, 968085128988277454064022, 1964504456873772610262792, 3985677311541980778984222, 8084691418672832842947344, 16396056428523408438810275, 33245460039402302582388851, 67397614443515576790375998, 136608617616453097066344752, 276844012691750081357882928, 560941580301187937441119790, 1136390270437751424630294130, 2301794760546253949077919980, 4661617960434010098137177221, 9439292799551024596610811249, 19110699356468057994297169282, 38685626227668133591178747004, 78299707484800302387992229994, 158456325028528675187754472948, 320626470174913491199586250934, 648680580585539264047903347608, 1312216441642503091393885903143, 2654143444227855309384591453167, 5367708010341408871963529692886, 10854258264454214250316509377912, 21946201016451221513412726871172, 44367771007988029052385732727322, 89686279966147230155892943842858, 181274035832636804414029825718996
Generating function in Maple syntax:
-(7*x^8-38*x^7+108*x^6-173*x^5+172*x^4-110*x^3+44*x^2-10*x+1)/(2*x-1)^2/(-1+x)^7
Generating function in latex syntax:
-\frac{7 x^{8}-38 x^{7}+108 x^{6}-173 x^{5}+172 x^{4}-110 x^{3}+44 x^{2}-10 x +1}{\left(2 x -1\right)^{2} \left(-1+x \right)^{7}}
Generating function in sympy syntax:
(-7*x**8 + 38*x**7 - 108*x**6 + 173*x**5 - 172*x**4 + 110*x**3 - 44*x**2 + 10*x - 1)/((x - 1)**7*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(-1+x)^7*F(x)+7*x^8-38*x^7+108*x^6-173*x^5+172*x^4-110*x^3+44*x^2-10*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(-1+x \right)^{7} F \! \left(x \right)+7 x^{8}-38 x^{7}+108 x^{6}-173 x^{5}+172 x^{4}-110 x^{3}+44 x^{2}-10 x +1 = 0
Explicit closed form in Maple syntax:
8+91/30*n-7*2^n-1/48*n^3+5/144*n^4-1/80*n^5+1/720*n^6+3*2^(-1+n)*n+347/360*n^2
Explicit closed form in latex syntax:
8+\frac{91 n}{30}-7 \,2^{n}-\frac{n^{3}}{48}+\frac{5 n^{4}}{144}-\frac{n^{5}}{80}+\frac{n^{6}}{720}+3 \,2^{-1+n} n +\frac{347 n^{2}}{360}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(6) = 197
a(7) = 554
a(8) = 1460
a(n+2) = 1/720*n^6-7/240*n^5+23/144*n^4-3/16*n^3+151/180*n^2-4*a(n)+4*a(n+1)-47/60*n+2, 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) = 20
a \! \left(5\right) = 65
a \! \left(6\right) = 197
a \! \left(7\right) = 554
a \! \left(8\right) = 1460
a \! \left(n +2\right) = \frac{n^{6}}{720}-\frac{7 n^{5}}{240}+\frac{23 n^{4}}{144}-\frac{3 n^{3}}{16}+\frac{151 n^{2}}{180}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{47 n}{60}+2, \quad n \geq 9
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/21210/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[18,x]*F[3,x]
F[3,x] = F[1,x]+F[228,x]+F[4,x]
F[4,x] = F[18,x]*F[5,x]
F[5,x] = F[1,x]+F[177,x]+F[222,x]+F[6,x]
F[6,x] = F[18,x]*F[7,x]
F[7,x] = F[1,x]+F[172,x]+F[173,x]+F[8,x]+F[95,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[18,x]*F[37,x]
F[10,x] = F[1,x]+F[11,x]+F[36,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[18,x]
F[13,x] = F[14,x]+F[31,x]
F[14,x] = F[15,x]*F[19,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[15,x]*F[18,x]
F[18,x] = x
F[19,x] = F[16,x]+F[20,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[18,x]*F[23,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[18,x]*F[24,x]
F[27,x] = F[16,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[18,x]*F[30,x]
F[30,x] = F[25,x]+F[28,x]
F[31,x] = F[24,x]*F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[18,x]*F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[13,x]*F[15,x]*F[18,x]
F[36,x] = F[10,x]*F[18,x]
F[37,x] = F[38,x]+F[58,x]
F[38,x] = F[20,x]+F[39,x]
F[39,x] = F[16,x]+F[40,x]
F[40,x] = F[41,x]+F[42,x]+F[44,x]
F[41,x] = 0
F[42,x] = F[18,x]*F[43,x]
F[43,x] = F[21,x]+F[40,x]
F[44,x] = F[18,x]*F[45,x]
F[45,x] = F[46,x]+F[51,x]
F[46,x] = F[16,x]+F[47,x]
F[47,x] = F[41,x]+F[48,x]+F[50,x]
F[48,x] = F[18,x]*F[49,x]
F[49,x] = F[25,x]+F[47,x]
F[50,x] = F[18,x]*F[46,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[53,x]
F[53,x] = F[18,x]*F[54,x]
F[54,x] = F[16,x]+F[52,x]
F[55,x] = F[56,x]
F[56,x] = F[18,x]*F[57,x]
F[57,x] = F[47,x]+F[55,x]
F[58,x] = F[59,x]+F[77,x]
F[59,x] = F[16,x]+F[60,x]
F[60,x] = F[41,x]+F[61,x]+F[64,x]
F[61,x] = F[18,x]*F[62,x]
F[62,x] = F[16,x]+F[63,x]
F[63,x] = F[41,x]+F[53,x]+F[61,x]
F[64,x] = F[18,x]*F[65,x]
F[65,x] = F[66,x]+F[73,x]
F[66,x] = F[16,x]+F[67,x]
F[67,x] = F[41,x]+F[68,x]+F[72,x]
F[68,x] = F[18,x]*F[69,x]
F[69,x] = F[18,x]+F[70,x]
F[70,x] = F[41,x]+F[68,x]+F[71,x]
F[71,x] = F[16,x]*F[18,x]
F[72,x] = F[18,x]*F[66,x]
F[73,x] = F[52,x]+F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[18,x]*F[76,x]
F[76,x] = F[67,x]+F[74,x]
F[77,x] = F[52,x]+F[78,x]
F[78,x] = 2*F[41,x]+F[79,x]+F[81,x]
F[79,x] = F[18,x]*F[80,x]
F[80,x] = F[60,x]+F[78,x]
F[81,x] = F[18,x]*F[82,x]
F[82,x] = F[83,x]+F[88,x]
F[83,x] = F[52,x]+F[84,x]
F[84,x] = 2*F[41,x]+F[85,x]+F[87,x]
F[85,x] = F[18,x]*F[86,x]
F[86,x] = F[67,x]+F[84,x]
F[87,x] = F[18,x]*F[83,x]
F[88,x] = F[89,x]+F[92,x]
F[89,x] = F[90,x]
F[90,x] = F[18,x]*F[91,x]
F[91,x] = F[52,x]+F[89,x]
F[92,x] = F[93,x]
F[93,x] = F[18,x]*F[94,x]
F[94,x] = F[84,x]+F[92,x]
F[95,x] = F[18,x]*F[96,x]
F[96,x] = F[37,x]+F[97,x]
F[97,x] = F[125,x]+F[98,x]
F[98,x] = F[110,x]+F[99,x]
F[99,x] = F[100,x]+F[16,x]
F[100,x] = F[101,x]
F[101,x] = F[102,x]*F[18,x]
F[102,x] = F[103,x]+F[106,x]
F[103,x] = F[104,x]+F[16,x]
F[104,x] = F[105,x]
F[105,x] = F[103,x]*F[18,x]
F[106,x] = F[107,x]+F[52,x]
F[107,x] = F[108,x]
F[108,x] = F[109,x]*F[18,x]
F[109,x] = F[104,x]+F[107,x]
F[110,x] = F[111,x]+F[52,x]
F[111,x] = 2*F[41,x]+F[112,x]+F[114,x]
F[112,x] = F[113,x]*F[18,x]
F[113,x] = F[100,x]+F[111,x]
F[114,x] = F[115,x]*F[18,x]
F[115,x] = F[116,x]+F[121,x]
F[116,x] = F[117,x]+F[52,x]
F[117,x] = 2*F[41,x]+F[118,x]+F[120,x]
F[118,x] = F[119,x]*F[18,x]
F[119,x] = F[104,x]+F[117,x]
F[120,x] = F[116,x]*F[18,x]
F[121,x] = F[122,x]+F[89,x]
F[122,x] = F[123,x]
F[123,x] = F[124,x]*F[18,x]
F[124,x] = F[117,x]+F[122,x]
F[125,x] = F[126,x]+F[154,x]
F[126,x] = F[127,x]+F[133,x]
F[127,x] = F[128,x]+F[131,x]+F[41,x]
F[128,x] = F[129,x]*F[18,x]
F[129,x] = F[130,x]+F[16,x]
F[130,x] = F[128,x]
F[131,x] = F[132,x]*F[18,x]
F[132,x] = F[127,x]+F[16,x]
F[133,x] = 2*F[41,x]+F[134,x]+F[140,x]
F[134,x] = F[135,x]*F[18,x]
F[135,x] = F[136,x]+F[52,x]
F[136,x] = 2*F[41,x]+F[134,x]+F[137,x]
F[137,x] = F[138,x]*F[18,x]
F[138,x] = F[127,x]+F[139,x]
F[139,x] = F[137,x]
F[140,x] = F[141,x]*F[18,x]
F[141,x] = F[142,x]+F[150,x]
F[142,x] = F[127,x]+F[143,x]
F[143,x] = 2*F[41,x]+F[144,x]+F[149,x]
F[144,x] = F[145,x]*F[18,x]
F[145,x] = F[146,x]+F[147,x]
F[146,x] = F[71,x]
F[147,x] = 2*F[41,x]+F[144,x]+F[148,x]
F[148,x] = F[127,x]*F[18,x]
F[149,x] = F[142,x]*F[18,x]
F[150,x] = F[139,x]+F[151,x]
F[151,x] = F[152,x]
F[152,x] = F[153,x]*F[18,x]
F[153,x] = F[143,x]+F[151,x]
F[154,x] = F[139,x]+F[155,x]
F[155,x] = 3*F[41,x]+F[156,x]+F[158,x]
F[156,x] = F[157,x]*F[18,x]
F[157,x] = F[133,x]+F[155,x]
F[158,x] = F[159,x]*F[18,x]
F[159,x] = F[160,x]+F[165,x]
F[160,x] = F[139,x]+F[161,x]
F[161,x] = 3*F[41,x]+F[162,x]+F[164,x]
F[162,x] = F[163,x]*F[18,x]
F[163,x] = F[143,x]+F[161,x]
F[164,x] = F[160,x]*F[18,x]
F[165,x] = F[166,x]+F[169,x]
F[166,x] = F[167,x]
F[167,x] = F[168,x]*F[18,x]
F[168,x] = F[139,x]+F[166,x]
F[169,x] = F[170,x]
F[170,x] = F[171,x]*F[18,x]
F[171,x] = F[161,x]+F[169,x]
F[172,x] = F[18,x]*F[38,x]
F[173,x] = F[174,x]
F[174,x] = F[15,x]*F[175,x]*F[18,x]
F[175,x] = F[176,x]+F[38,x]
F[176,x] = F[15,x]^2*F[18,x]
F[177,x] = F[178,x]*F[18,x]
F[178,x] = F[179,x]+F[38,x]
F[179,x] = F[180,x]+F[191,x]
F[180,x] = F[16,x]+F[181,x]
F[181,x] = F[182,x]+F[185,x]+F[41,x]
F[182,x] = F[18,x]*F[183,x]
F[183,x] = F[16,x]+F[184,x]
F[184,x] = F[131,x]+F[182,x]+F[41,x]
F[185,x] = F[18,x]*F[186,x]
F[186,x] = F[187,x]+F[66,x]
F[187,x] = F[127,x]+F[188,x]
F[188,x] = F[189,x]
F[189,x] = F[18,x]*F[190,x]
F[190,x] = F[188,x]+F[67,x]
F[191,x] = F[127,x]+F[192,x]
F[192,x] = F[193,x]+F[202,x]+F[204,x]+F[41,x]
F[193,x] = F[18,x]*F[194,x]
F[194,x] = F[195,x]+F[52,x]
F[195,x] = 2*F[41,x]+F[193,x]+F[196,x]
F[196,x] = F[18,x]*F[197,x]
F[197,x] = F[130,x]+F[198,x]
F[198,x] = 2*F[41,x]+F[196,x]+F[199,x]
F[199,x] = F[18,x]*F[200,x]
F[200,x] = F[201,x]+F[52,x]
F[201,x] = F[199,x]
F[202,x] = F[18,x]*F[203,x]
F[203,x] = F[181,x]+F[192,x]
F[204,x] = F[18,x]*F[205,x]
F[205,x] = F[206,x]+F[215,x]
F[206,x] = F[127,x]+F[207,x]
F[207,x] = F[208,x]+F[212,x]+F[214,x]+F[41,x]
F[208,x] = F[18,x]*F[209,x]
F[209,x] = F[146,x]+F[210,x]
F[210,x] = 2*F[41,x]+F[208,x]+F[211,x]
F[211,x] = F[130,x]*F[18,x]
F[212,x] = F[18,x]*F[213,x]
F[213,x] = F[207,x]+F[67,x]
F[214,x] = F[18,x]*F[206,x]
F[215,x] = F[216,x]+F[219,x]
F[216,x] = 2*F[41,x]+F[199,x]+F[217,x]
F[217,x] = F[18,x]*F[218,x]
F[218,x] = F[127,x]+F[216,x]
F[219,x] = F[220,x]
F[220,x] = F[18,x]*F[221,x]
F[221,x] = F[207,x]+F[219,x]
F[222,x] = F[223,x]
F[223,x] = F[18,x]*F[224,x]
F[224,x] = F[225,x]+F[226,x]+F[38,x]
F[225,x] = F[18,x]*F[224,x]
F[226,x] = F[15,x]*F[18,x]*F[227,x]
F[227,x] = F[132,x]+F[15,x]
F[228,x] = F[18,x]*F[229,x]
F[229,x] = F[1,x]+F[228,x]+F[230,x]+F[280,x]
F[230,x] = F[18,x]*F[231,x]
F[231,x] = F[1,x]+F[222,x]+F[232,x]+F[270,x]+F[272,x]
F[232,x] = F[18,x]*F[233,x]
F[233,x] = F[1,x]+F[173,x]+F[234,x]+F[252,x]+F[269,x]+F[270,x]
F[234,x] = F[235,x]
F[235,x] = F[10,x]*F[18,x]*F[236,x]
F[236,x] = F[237,x]+F[243,x]
F[237,x] = F[238,x]+F[240,x]
F[238,x] = F[15,x]+F[239,x]
F[239,x] = F[21,x]
F[240,x] = F[241,x]+F[242,x]
F[241,x] = F[16,x]+F[63,x]
F[242,x] = F[40,x]
F[243,x] = F[244,x]+F[246,x]
F[244,x] = F[245,x]+F[54,x]
F[245,x] = F[60,x]
F[246,x] = F[247,x]+F[251,x]
F[247,x] = F[248,x]+F[52,x]
F[248,x] = 2*F[41,x]+F[249,x]+F[90,x]
F[249,x] = F[18,x]*F[250,x]
F[250,x] = F[248,x]+F[52,x]
F[251,x] = F[78,x]
F[252,x] = F[18,x]*F[253,x]
F[253,x] = F[236,x]+F[254,x]
F[254,x] = F[255,x]+F[260,x]
F[255,x] = F[256,x]+F[258,x]
F[256,x] = F[257,x]+F[54,x]
F[257,x] = F[100,x]
F[258,x] = F[247,x]+F[259,x]
F[259,x] = F[111,x]
F[260,x] = F[261,x]+F[263,x]
F[261,x] = F[138,x]+F[262,x]
F[262,x] = F[133,x]
F[263,x] = F[264,x]+F[268,x]
F[264,x] = F[139,x]+F[265,x]
F[265,x] = 3*F[41,x]+F[167,x]+F[266,x]
F[266,x] = F[18,x]*F[267,x]
F[267,x] = F[139,x]+F[265,x]
F[268,x] = F[155,x]
F[269,x] = F[18,x]*F[237,x]
F[270,x] = F[271,x]
F[271,x] = F[15,x]^2*F[18,x]
F[272,x] = F[18,x]*F[273,x]
F[273,x] = F[237,x]+F[274,x]
F[274,x] = F[275,x]+F[277,x]
F[275,x] = F[276,x]+F[54,x]
F[276,x] = F[181,x]
F[277,x] = F[278,x]+F[279,x]
F[278,x] = F[127,x]+F[136,x]
F[279,x] = F[192,x]
F[280,x] = F[18,x]*F[281,x]
F[281,x] = F[15,x]+F[282,x]
F[282,x] = F[16,x]+F[184,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_{18}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{228}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{18}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{177}\! \left(x \right)+F_{222}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{18}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{172}\! \left(x \right)+F_{173}\! \left(x \right)+F_{8}\! \left(x \right)+F_{95}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right) F_{37}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{36}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{31}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = x
F_{19}\! \left(x \right) = F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{18}\! \left(x \right) F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{16}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{24}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{58}\! \left(x \right)
F_{38}\! \left(x \right) = F_{20}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{16}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = 0
F_{42}\! \left(x \right) = F_{18}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{21}\! \left(x \right)+F_{40}\! \left(x \right)
F_{44}\! \left(x \right) = F_{18}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{51}\! \left(x \right)
F_{46}\! \left(x \right) = F_{16}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{41}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{48}\! \left(x \right) = F_{18}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{25}\! \left(x \right)+F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{18}\! \left(x \right) F_{46}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{18}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{16}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{18}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{47}\! \left(x \right)+F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{77}\! \left(x \right)
F_{59}\! \left(x \right) = F_{16}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{41}\! \left(x \right)+F_{61}\! \left(x \right)+F_{64}\! \left(x \right)
F_{61}\! \left(x \right) = F_{18}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{16}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{41}\! \left(x \right)+F_{53}\! \left(x \right)+F_{61}\! \left(x \right)
F_{64}\! \left(x \right) = F_{18}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{73}\! \left(x \right)
F_{66}\! \left(x \right) = F_{16}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{41}\! \left(x \right)+F_{68}\! \left(x \right)+F_{72}\! \left(x \right)
F_{68}\! \left(x \right) = F_{18}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{18}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{41}\! \left(x \right)+F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{16}\! \left(x \right) F_{18}\! \left(x \right)
F_{72}\! \left(x \right) = F_{18}\! \left(x \right) F_{66}\! \left(x \right)
F_{73}\! \left(x \right) = F_{52}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{18}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{67}\! \left(x \right)+F_{74}\! \left(x \right)
F_{77}\! \left(x \right) = F_{52}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{79}\! \left(x \right)+F_{81}\! \left(x \right)
F_{79}\! \left(x \right) = F_{18}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{60}\! \left(x \right)+F_{78}\! \left(x \right)
F_{81}\! \left(x \right) = F_{18}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{88}\! \left(x \right)
F_{83}\! \left(x \right) = F_{52}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{85}\! \left(x \right)+F_{87}\! \left(x \right)
F_{85}\! \left(x \right) = F_{18}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{67}\! \left(x \right)+F_{84}\! \left(x \right)
F_{87}\! \left(x \right) = F_{18}\! \left(x \right) F_{83}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{92}\! \left(x \right)
F_{89}\! \left(x \right) = F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{18}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{52}\! \left(x \right)+F_{89}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{18}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{84}\! \left(x \right)+F_{92}\! \left(x \right)
F_{95}\! \left(x \right) = F_{18}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{37}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{125}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{110}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{16}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{18}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{106}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{16}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)
F_{105}\! \left(x \right) = F_{103}\! \left(x \right) F_{18}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{52}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{18}\! \left(x \right)
F_{109}\! \left(x \right) = F_{104}\! \left(x \right)+F_{107}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{52}\! \left(x \right)
F_{111}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{112}\! \left(x \right)+F_{114}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{18}\! \left(x \right)
F_{113}\! \left(x \right) = F_{100}\! \left(x \right)+F_{111}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right) F_{18}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{121}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)+F_{52}\! \left(x \right)
F_{117}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{118}\! \left(x \right)+F_{120}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right) F_{18}\! \left(x \right)
F_{119}\! \left(x \right) = F_{104}\! \left(x \right)+F_{117}\! \left(x \right)
F_{120}\! \left(x \right) = F_{116}\! \left(x \right) F_{18}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)+F_{89}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right) F_{18}\! \left(x \right)
F_{124}\! \left(x \right) = F_{117}\! \left(x \right)+F_{122}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)+F_{154}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)+F_{133}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)+F_{131}\! \left(x \right)+F_{41}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right) F_{18}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)+F_{16}\! \left(x \right)
F_{130}\! \left(x \right) = F_{128}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right) F_{18}\! \left(x \right)
F_{132}\! \left(x \right) = F_{127}\! \left(x \right)+F_{16}\! \left(x \right)
F_{133}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{134}\! \left(x \right)+F_{140}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right) F_{18}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right)+F_{52}\! \left(x \right)
F_{136}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{134}\! \left(x \right)+F_{137}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right) F_{18}\! \left(x \right)
F_{138}\! \left(x \right) = F_{127}\! \left(x \right)+F_{139}\! \left(x \right)
F_{139}\! \left(x \right) = F_{137}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right) F_{18}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right)+F_{150}\! \left(x \right)
F_{142}\! \left(x \right) = F_{127}\! \left(x \right)+F_{143}\! \left(x \right)
F_{143}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{144}\! \left(x \right)+F_{149}\! \left(x \right)
F_{144}\! \left(x \right) = F_{145}\! \left(x \right) F_{18}\! \left(x \right)
F_{145}\! \left(x \right) = F_{146}\! \left(x \right)+F_{147}\! \left(x \right)
F_{146}\! \left(x \right) = F_{71}\! \left(x \right)
F_{147}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{144}\! \left(x \right)+F_{148}\! \left(x \right)
F_{148}\! \left(x \right) = F_{127}\! \left(x \right) F_{18}\! \left(x \right)
F_{149}\! \left(x \right) = F_{142}\! \left(x \right) F_{18}\! \left(x \right)
F_{150}\! \left(x \right) = F_{139}\! \left(x \right)+F_{151}\! \left(x \right)
F_{151}\! \left(x \right) = F_{152}\! \left(x \right)
F_{152}\! \left(x \right) = F_{153}\! \left(x \right) F_{18}\! \left(x \right)
F_{153}\! \left(x \right) = F_{143}\! \left(x \right)+F_{151}\! \left(x \right)
F_{154}\! \left(x \right) = F_{139}\! \left(x \right)+F_{155}\! \left(x \right)
F_{155}\! \left(x \right) = 3 F_{41}\! \left(x \right)+F_{156}\! \left(x \right)+F_{158}\! \left(x \right)
F_{156}\! \left(x \right) = F_{157}\! \left(x \right) F_{18}\! \left(x \right)
F_{157}\! \left(x \right) = F_{133}\! \left(x \right)+F_{155}\! \left(x \right)
F_{158}\! \left(x \right) = F_{159}\! \left(x \right) F_{18}\! \left(x \right)
F_{159}\! \left(x \right) = F_{160}\! \left(x \right)+F_{165}\! \left(x \right)
F_{160}\! \left(x \right) = F_{139}\! \left(x \right)+F_{161}\! \left(x \right)
F_{161}\! \left(x \right) = 3 F_{41}\! \left(x \right)+F_{162}\! \left(x \right)+F_{164}\! \left(x \right)
F_{162}\! \left(x \right) = F_{163}\! \left(x \right) F_{18}\! \left(x \right)
F_{163}\! \left(x \right) = F_{143}\! \left(x \right)+F_{161}\! \left(x \right)
F_{164}\! \left(x \right) = F_{160}\! \left(x \right) F_{18}\! \left(x \right)
F_{165}\! \left(x \right) = F_{166}\! \left(x \right)+F_{169}\! \left(x \right)
F_{166}\! \left(x \right) = F_{167}\! \left(x \right)
F_{167}\! \left(x \right) = F_{168}\! \left(x \right) F_{18}\! \left(x \right)
F_{168}\! \left(x \right) = F_{139}\! \left(x \right)+F_{166}\! \left(x \right)
F_{169}\! \left(x \right) = F_{170}\! \left(x \right)
F_{170}\! \left(x \right) = F_{171}\! \left(x \right) F_{18}\! \left(x \right)
F_{171}\! \left(x \right) = F_{161}\! \left(x \right)+F_{169}\! \left(x \right)
F_{172}\! \left(x \right) = F_{18}\! \left(x \right) F_{38}\! \left(x \right)
F_{173}\! \left(x \right) = F_{174}\! \left(x \right)
F_{174}\! \left(x \right) = F_{15}\! \left(x \right) F_{175}\! \left(x \right) F_{18}\! \left(x \right)
F_{175}\! \left(x \right) = F_{176}\! \left(x \right)+F_{38}\! \left(x \right)
F_{176}\! \left(x \right) = F_{15} \left(x \right)^{2} F_{18}\! \left(x \right)
F_{177}\! \left(x \right) = F_{178}\! \left(x \right) F_{18}\! \left(x \right)
F_{178}\! \left(x \right) = F_{179}\! \left(x \right)+F_{38}\! \left(x \right)
F_{179}\! \left(x \right) = F_{180}\! \left(x \right)+F_{191}\! \left(x \right)
F_{180}\! \left(x \right) = F_{16}\! \left(x \right)+F_{181}\! \left(x \right)
F_{181}\! \left(x \right) = F_{182}\! \left(x \right)+F_{185}\! \left(x \right)+F_{41}\! \left(x \right)
F_{182}\! \left(x \right) = F_{18}\! \left(x \right) F_{183}\! \left(x \right)
F_{183}\! \left(x \right) = F_{16}\! \left(x \right)+F_{184}\! \left(x \right)
F_{184}\! \left(x \right) = F_{131}\! \left(x \right)+F_{182}\! \left(x \right)+F_{41}\! \left(x \right)
F_{185}\! \left(x \right) = F_{18}\! \left(x \right) F_{186}\! \left(x \right)
F_{186}\! \left(x \right) = F_{187}\! \left(x \right)+F_{66}\! \left(x \right)
F_{187}\! \left(x \right) = F_{127}\! \left(x \right)+F_{188}\! \left(x \right)
F_{188}\! \left(x \right) = F_{189}\! \left(x \right)
F_{189}\! \left(x \right) = F_{18}\! \left(x \right) F_{190}\! \left(x \right)
F_{190}\! \left(x \right) = F_{188}\! \left(x \right)+F_{67}\! \left(x \right)
F_{191}\! \left(x \right) = F_{127}\! \left(x \right)+F_{192}\! \left(x \right)
F_{192}\! \left(x \right) = F_{193}\! \left(x \right)+F_{202}\! \left(x \right)+F_{204}\! \left(x \right)+F_{41}\! \left(x \right)
F_{193}\! \left(x \right) = F_{18}\! \left(x \right) F_{194}\! \left(x \right)
F_{194}\! \left(x \right) = F_{195}\! \left(x \right)+F_{52}\! \left(x \right)
F_{195}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{193}\! \left(x \right)+F_{196}\! \left(x \right)
F_{196}\! \left(x \right) = F_{18}\! \left(x \right) F_{197}\! \left(x \right)
F_{197}\! \left(x \right) = F_{130}\! \left(x \right)+F_{198}\! \left(x \right)
F_{198}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{196}\! \left(x \right)+F_{199}\! \left(x \right)
F_{199}\! \left(x \right) = F_{18}\! \left(x \right) F_{200}\! \left(x \right)
F_{200}\! \left(x \right) = F_{201}\! \left(x \right)+F_{52}\! \left(x \right)
F_{201}\! \left(x \right) = F_{199}\! \left(x \right)
F_{202}\! \left(x \right) = F_{18}\! \left(x \right) F_{203}\! \left(x \right)
F_{203}\! \left(x \right) = F_{181}\! \left(x \right)+F_{192}\! \left(x \right)
F_{204}\! \left(x \right) = F_{18}\! \left(x \right) F_{205}\! \left(x \right)
F_{205}\! \left(x \right) = F_{206}\! \left(x \right)+F_{215}\! \left(x \right)
F_{206}\! \left(x \right) = F_{127}\! \left(x \right)+F_{207}\! \left(x \right)
F_{207}\! \left(x \right) = F_{208}\! \left(x \right)+F_{212}\! \left(x \right)+F_{214}\! \left(x \right)+F_{41}\! \left(x \right)
F_{208}\! \left(x \right) = F_{18}\! \left(x \right) F_{209}\! \left(x \right)
F_{209}\! \left(x \right) = F_{146}\! \left(x \right)+F_{210}\! \left(x \right)
F_{210}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{208}\! \left(x \right)+F_{211}\! \left(x \right)
F_{211}\! \left(x \right) = F_{130}\! \left(x \right) F_{18}\! \left(x \right)
F_{212}\! \left(x \right) = F_{18}\! \left(x \right) F_{213}\! \left(x \right)
F_{213}\! \left(x \right) = F_{207}\! \left(x \right)+F_{67}\! \left(x \right)
F_{214}\! \left(x \right) = F_{18}\! \left(x \right) F_{206}\! \left(x \right)
F_{215}\! \left(x \right) = F_{216}\! \left(x \right)+F_{219}\! \left(x \right)
F_{216}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{199}\! \left(x \right)+F_{217}\! \left(x \right)
F_{217}\! \left(x \right) = F_{18}\! \left(x \right) F_{218}\! \left(x \right)
F_{218}\! \left(x \right) = F_{127}\! \left(x \right)+F_{216}\! \left(x \right)
F_{219}\! \left(x \right) = F_{220}\! \left(x \right)
F_{220}\! \left(x \right) = F_{18}\! \left(x \right) F_{221}\! \left(x \right)
F_{221}\! \left(x \right) = F_{207}\! \left(x \right)+F_{219}\! \left(x \right)
F_{222}\! \left(x \right) = F_{223}\! \left(x \right)
F_{223}\! \left(x \right) = F_{18}\! \left(x \right) F_{224}\! \left(x \right)
F_{224}\! \left(x \right) = F_{225}\! \left(x \right)+F_{226}\! \left(x \right)+F_{38}\! \left(x \right)
F_{225}\! \left(x \right) = F_{18}\! \left(x \right) F_{224}\! \left(x \right)
F_{226}\! \left(x \right) = F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{227}\! \left(x \right)
F_{227}\! \left(x \right) = F_{132}\! \left(x \right)+F_{15}\! \left(x \right)
F_{228}\! \left(x \right) = F_{18}\! \left(x \right) F_{229}\! \left(x \right)
F_{229}\! \left(x \right) = F_{1}\! \left(x \right)+F_{228}\! \left(x \right)+F_{230}\! \left(x \right)+F_{280}\! \left(x \right)
F_{230}\! \left(x \right) = F_{18}\! \left(x \right) F_{231}\! \left(x \right)
F_{231}\! \left(x \right) = F_{1}\! \left(x \right)+F_{222}\! \left(x \right)+F_{232}\! \left(x \right)+F_{270}\! \left(x \right)+F_{272}\! \left(x \right)
F_{232}\! \left(x \right) = F_{18}\! \left(x \right) F_{233}\! \left(x \right)
F_{233}\! \left(x \right) = F_{1}\! \left(x \right)+F_{173}\! \left(x \right)+F_{234}\! \left(x \right)+F_{252}\! \left(x \right)+F_{269}\! \left(x \right)+F_{270}\! \left(x \right)
F_{234}\! \left(x \right) = F_{235}\! \left(x \right)
F_{235}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right) F_{236}\! \left(x \right)
F_{236}\! \left(x \right) = F_{237}\! \left(x \right)+F_{243}\! \left(x \right)
F_{237}\! \left(x \right) = F_{238}\! \left(x \right)+F_{240}\! \left(x \right)
F_{238}\! \left(x \right) = F_{15}\! \left(x \right)+F_{239}\! \left(x \right)
F_{239}\! \left(x \right) = F_{21}\! \left(x \right)
F_{240}\! \left(x \right) = F_{241}\! \left(x \right)+F_{242}\! \left(x \right)
F_{241}\! \left(x \right) = F_{16}\! \left(x \right)+F_{63}\! \left(x \right)
F_{242}\! \left(x \right) = F_{40}\! \left(x \right)
F_{243}\! \left(x \right) = F_{244}\! \left(x \right)+F_{246}\! \left(x \right)
F_{244}\! \left(x \right) = F_{245}\! \left(x \right)+F_{54}\! \left(x \right)
F_{245}\! \left(x \right) = F_{60}\! \left(x \right)
F_{246}\! \left(x \right) = F_{247}\! \left(x \right)+F_{251}\! \left(x \right)
F_{247}\! \left(x \right) = F_{248}\! \left(x \right)+F_{52}\! \left(x \right)
F_{248}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{249}\! \left(x \right)+F_{90}\! \left(x \right)
F_{249}\! \left(x \right) = F_{18}\! \left(x \right) F_{250}\! \left(x \right)
F_{250}\! \left(x \right) = F_{248}\! \left(x \right)+F_{52}\! \left(x \right)
F_{251}\! \left(x \right) = F_{78}\! \left(x \right)
F_{252}\! \left(x \right) = F_{18}\! \left(x \right) F_{253}\! \left(x \right)
F_{253}\! \left(x \right) = F_{236}\! \left(x \right)+F_{254}\! \left(x \right)
F_{254}\! \left(x \right) = F_{255}\! \left(x \right)+F_{260}\! \left(x \right)
F_{255}\! \left(x \right) = F_{256}\! \left(x \right)+F_{258}\! \left(x \right)
F_{256}\! \left(x \right) = F_{257}\! \left(x \right)+F_{54}\! \left(x \right)
F_{257}\! \left(x \right) = F_{100}\! \left(x \right)
F_{258}\! \left(x \right) = F_{247}\! \left(x \right)+F_{259}\! \left(x \right)
F_{259}\! \left(x \right) = F_{111}\! \left(x \right)
F_{260}\! \left(x \right) = F_{261}\! \left(x \right)+F_{263}\! \left(x \right)
F_{261}\! \left(x \right) = F_{138}\! \left(x \right)+F_{262}\! \left(x \right)
F_{262}\! \left(x \right) = F_{133}\! \left(x \right)
F_{263}\! \left(x \right) = F_{264}\! \left(x \right)+F_{268}\! \left(x \right)
F_{264}\! \left(x \right) = F_{139}\! \left(x \right)+F_{265}\! \left(x \right)
F_{265}\! \left(x \right) = 3 F_{41}\! \left(x \right)+F_{167}\! \left(x \right)+F_{266}\! \left(x \right)
F_{266}\! \left(x \right) = F_{18}\! \left(x \right) F_{267}\! \left(x \right)
F_{267}\! \left(x \right) = F_{139}\! \left(x \right)+F_{265}\! \left(x \right)
F_{268}\! \left(x \right) = F_{155}\! \left(x \right)
F_{269}\! \left(x \right) = F_{18}\! \left(x \right) F_{237}\! \left(x \right)
F_{270}\! \left(x \right) = F_{271}\! \left(x \right)
F_{271}\! \left(x \right) = F_{15} \left(x \right)^{2} F_{18}\! \left(x \right)
F_{272}\! \left(x \right) = F_{18}\! \left(x \right) F_{273}\! \left(x \right)
F_{273}\! \left(x \right) = F_{237}\! \left(x \right)+F_{274}\! \left(x \right)
F_{274}\! \left(x \right) = F_{275}\! \left(x \right)+F_{277}\! \left(x \right)
F_{275}\! \left(x \right) = F_{276}\! \left(x \right)+F_{54}\! \left(x \right)
F_{276}\! \left(x \right) = F_{181}\! \left(x \right)
F_{277}\! \left(x \right) = F_{278}\! \left(x \right)+F_{279}\! \left(x \right)
F_{278}\! \left(x \right) = F_{127}\! \left(x \right)+F_{136}\! \left(x \right)
F_{279}\! \left(x \right) = F_{192}\! \left(x \right)
F_{280}\! \left(x \right) = F_{18}\! \left(x \right) F_{281}\! \left(x \right)
F_{281}\! \left(x \right) = F_{15}\! \left(x \right)+F_{282}\! \left(x \right)
F_{282}\! \left(x \right) = F_{16}\! \left(x \right)+F_{184}\! \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_18(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_228(x) + F_4(x))
Eq(F_4(x), F_18(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_177(x) + F_222(x) + F_6(x))
Eq(F_6(x), F_18(x)*F_7(x))
Eq(F_7(x), F_1(x) + F_172(x) + F_173(x) + F_8(x) + F_95(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_18(x)*F_37(x))
Eq(F_10(x), F_1(x) + F_11(x) + F_36(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_18(x))
Eq(F_13(x), F_14(x) + F_31(x))
Eq(F_14(x), F_15(x)*F_19(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_15(x)*F_18(x))
Eq(F_18(x), x)
Eq(F_19(x), F_16(x) + F_20(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_18(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_18(x)*F_24(x))
Eq(F_27(x), F_16(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_18(x)*F_30(x))
Eq(F_30(x), F_25(x) + F_28(x))
Eq(F_31(x), F_24(x)*F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_18(x)*F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_13(x)*F_15(x)*F_18(x))
Eq(F_36(x), F_10(x)*F_18(x))
Eq(F_37(x), F_38(x) + F_58(x))
Eq(F_38(x), F_20(x) + F_39(x))
Eq(F_39(x), F_16(x) + F_40(x))
Eq(F_40(x), F_41(x) + F_42(x) + F_44(x))
Eq(F_41(x), 0)
Eq(F_42(x), F_18(x)*F_43(x))
Eq(F_43(x), F_21(x) + F_40(x))
Eq(F_44(x), F_18(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_51(x))
Eq(F_46(x), F_16(x) + F_47(x))
Eq(F_47(x), F_41(x) + F_48(x) + F_50(x))
Eq(F_48(x), F_18(x)*F_49(x))
Eq(F_49(x), F_25(x) + F_47(x))
Eq(F_50(x), F_18(x)*F_46(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_18(x)*F_54(x))
Eq(F_54(x), F_16(x) + F_52(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_18(x)*F_57(x))
Eq(F_57(x), F_47(x) + F_55(x))
Eq(F_58(x), F_59(x) + F_77(x))
Eq(F_59(x), F_16(x) + F_60(x))
Eq(F_60(x), F_41(x) + F_61(x) + F_64(x))
Eq(F_61(x), F_18(x)*F_62(x))
Eq(F_62(x), F_16(x) + F_63(x))
Eq(F_63(x), F_41(x) + F_53(x) + F_61(x))
Eq(F_64(x), F_18(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_73(x))
Eq(F_66(x), F_16(x) + F_67(x))
Eq(F_67(x), F_41(x) + F_68(x) + F_72(x))
Eq(F_68(x), F_18(x)*F_69(x))
Eq(F_69(x), F_18(x) + F_70(x))
Eq(F_70(x), F_41(x) + F_68(x) + F_71(x))
Eq(F_71(x), F_16(x)*F_18(x))
Eq(F_72(x), F_18(x)*F_66(x))
Eq(F_73(x), F_52(x) + F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_18(x)*F_76(x))
Eq(F_76(x), F_67(x) + F_74(x))
Eq(F_77(x), F_52(x) + F_78(x))
Eq(F_78(x), 2*F_41(x) + F_79(x) + F_81(x))
Eq(F_79(x), F_18(x)*F_80(x))
Eq(F_80(x), F_60(x) + F_78(x))
Eq(F_81(x), F_18(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_88(x))
Eq(F_83(x), F_52(x) + F_84(x))
Eq(F_84(x), 2*F_41(x) + F_85(x) + F_87(x))
Eq(F_85(x), F_18(x)*F_86(x))
Eq(F_86(x), F_67(x) + F_84(x))
Eq(F_87(x), F_18(x)*F_83(x))
Eq(F_88(x), F_89(x) + F_92(x))
Eq(F_89(x), F_90(x))
Eq(F_90(x), F_18(x)*F_91(x))
Eq(F_91(x), F_52(x) + F_89(x))
Eq(F_92(x), F_93(x))
Eq(F_93(x), F_18(x)*F_94(x))
Eq(F_94(x), F_84(x) + F_92(x))
Eq(F_95(x), F_18(x)*F_96(x))
Eq(F_96(x), F_37(x) + F_97(x))
Eq(F_97(x), F_125(x) + F_98(x))
Eq(F_98(x), F_110(x) + F_99(x))
Eq(F_99(x), F_100(x) + F_16(x))
Eq(F_100(x), F_101(x))
Eq(F_101(x), F_102(x)*F_18(x))
Eq(F_102(x), F_103(x) + F_106(x))
Eq(F_103(x), F_104(x) + F_16(x))
Eq(F_104(x), F_105(x))
Eq(F_105(x), F_103(x)*F_18(x))
Eq(F_106(x), F_107(x) + F_52(x))
Eq(F_107(x), F_108(x))
Eq(F_108(x), F_109(x)*F_18(x))
Eq(F_109(x), F_104(x) + F_107(x))
Eq(F_110(x), F_111(x) + F_52(x))
Eq(F_111(x), F_112(x) + F_114(x) + 2*F_41(x))
Eq(F_112(x), F_113(x)*F_18(x))
Eq(F_113(x), F_100(x) + F_111(x))
Eq(F_114(x), F_115(x)*F_18(x))
Eq(F_115(x), F_116(x) + F_121(x))
Eq(F_116(x), F_117(x) + F_52(x))
Eq(F_117(x), F_118(x) + F_120(x) + 2*F_41(x))
Eq(F_118(x), F_119(x)*F_18(x))
Eq(F_119(x), F_104(x) + F_117(x))
Eq(F_120(x), F_116(x)*F_18(x))
Eq(F_121(x), F_122(x) + F_89(x))
Eq(F_122(x), F_123(x))
Eq(F_123(x), F_124(x)*F_18(x))
Eq(F_124(x), F_117(x) + F_122(x))
Eq(F_125(x), F_126(x) + F_154(x))
Eq(F_126(x), F_127(x) + F_133(x))
Eq(F_127(x), F_128(x) + F_131(x) + F_41(x))
Eq(F_128(x), F_129(x)*F_18(x))
Eq(F_129(x), F_130(x) + F_16(x))
Eq(F_130(x), F_128(x))
Eq(F_131(x), F_132(x)*F_18(x))
Eq(F_132(x), F_127(x) + F_16(x))
Eq(F_133(x), F_134(x) + F_140(x) + 2*F_41(x))
Eq(F_134(x), F_135(x)*F_18(x))
Eq(F_135(x), F_136(x) + F_52(x))
Eq(F_136(x), F_134(x) + F_137(x) + 2*F_41(x))
Eq(F_137(x), F_138(x)*F_18(x))
Eq(F_138(x), F_127(x) + F_139(x))
Eq(F_139(x), F_137(x))
Eq(F_140(x), F_141(x)*F_18(x))
Eq(F_141(x), F_142(x) + F_150(x))
Eq(F_142(x), F_127(x) + F_143(x))
Eq(F_143(x), F_144(x) + F_149(x) + 2*F_41(x))
Eq(F_144(x), F_145(x)*F_18(x))
Eq(F_145(x), F_146(x) + F_147(x))
Eq(F_146(x), F_71(x))
Eq(F_147(x), F_144(x) + F_148(x) + 2*F_41(x))
Eq(F_148(x), F_127(x)*F_18(x))
Eq(F_149(x), F_142(x)*F_18(x))
Eq(F_150(x), F_139(x) + F_151(x))
Eq(F_151(x), F_152(x))
Eq(F_152(x), F_153(x)*F_18(x))
Eq(F_153(x), F_143(x) + F_151(x))
Eq(F_154(x), F_139(x) + F_155(x))
Eq(F_155(x), F_156(x) + F_158(x) + 3*F_41(x))
Eq(F_156(x), F_157(x)*F_18(x))
Eq(F_157(x), F_133(x) + F_155(x))
Eq(F_158(x), F_159(x)*F_18(x))
Eq(F_159(x), F_160(x) + F_165(x))
Eq(F_160(x), F_139(x) + F_161(x))
Eq(F_161(x), F_162(x) + F_164(x) + 3*F_41(x))
Eq(F_162(x), F_163(x)*F_18(x))
Eq(F_163(x), F_143(x) + F_161(x))
Eq(F_164(x), F_160(x)*F_18(x))
Eq(F_165(x), F_166(x) + F_169(x))
Eq(F_166(x), F_167(x))
Eq(F_167(x), F_168(x)*F_18(x))
Eq(F_168(x), F_139(x) + F_166(x))
Eq(F_169(x), F_170(x))
Eq(F_170(x), F_171(x)*F_18(x))
Eq(F_171(x), F_161(x) + F_169(x))
Eq(F_172(x), F_18(x)*F_38(x))
Eq(F_173(x), F_174(x))
Eq(F_174(x), F_15(x)*F_175(x)*F_18(x))
Eq(F_175(x), F_176(x) + F_38(x))
Eq(F_176(x), F_15(x)**2*F_18(x))
Eq(F_177(x), F_178(x)*F_18(x))
Eq(F_178(x), F_179(x) + F_38(x))
Eq(F_179(x), F_180(x) + F_191(x))
Eq(F_180(x), F_16(x) + F_181(x))
Eq(F_181(x), F_182(x) + F_185(x) + F_41(x))
Eq(F_182(x), F_18(x)*F_183(x))
Eq(F_183(x), F_16(x) + F_184(x))
Eq(F_184(x), F_131(x) + F_182(x) + F_41(x))
Eq(F_185(x), F_18(x)*F_186(x))
Eq(F_186(x), F_187(x) + F_66(x))
Eq(F_187(x), F_127(x) + F_188(x))
Eq(F_188(x), F_189(x))
Eq(F_189(x), F_18(x)*F_190(x))
Eq(F_190(x), F_188(x) + F_67(x))
Eq(F_191(x), F_127(x) + F_192(x))
Eq(F_192(x), F_193(x) + F_202(x) + F_204(x) + F_41(x))
Eq(F_193(x), F_18(x)*F_194(x))
Eq(F_194(x), F_195(x) + F_52(x))
Eq(F_195(x), F_193(x) + F_196(x) + 2*F_41(x))
Eq(F_196(x), F_18(x)*F_197(x))
Eq(F_197(x), F_130(x) + F_198(x))
Eq(F_198(x), F_196(x) + F_199(x) + 2*F_41(x))
Eq(F_199(x), F_18(x)*F_200(x))
Eq(F_200(x), F_201(x) + F_52(x))
Eq(F_201(x), F_199(x))
Eq(F_202(x), F_18(x)*F_203(x))
Eq(F_203(x), F_181(x) + F_192(x))
Eq(F_204(x), F_18(x)*F_205(x))
Eq(F_205(x), F_206(x) + F_215(x))
Eq(F_206(x), F_127(x) + F_207(x))
Eq(F_207(x), F_208(x) + F_212(x) + F_214(x) + F_41(x))
Eq(F_208(x), F_18(x)*F_209(x))
Eq(F_209(x), F_146(x) + F_210(x))
Eq(F_210(x), F_208(x) + F_211(x) + 2*F_41(x))
Eq(F_211(x), F_130(x)*F_18(x))
Eq(F_212(x), F_18(x)*F_213(x))
Eq(F_213(x), F_207(x) + F_67(x))
Eq(F_214(x), F_18(x)*F_206(x))
Eq(F_215(x), F_216(x) + F_219(x))
Eq(F_216(x), F_199(x) + F_217(x) + 2*F_41(x))
Eq(F_217(x), F_18(x)*F_218(x))
Eq(F_218(x), F_127(x) + F_216(x))
Eq(F_219(x), F_220(x))
Eq(F_220(x), F_18(x)*F_221(x))
Eq(F_221(x), F_207(x) + F_219(x))
Eq(F_222(x), F_223(x))
Eq(F_223(x), F_18(x)*F_224(x))
Eq(F_224(x), F_225(x) + F_226(x) + F_38(x))
Eq(F_225(x), F_18(x)*F_224(x))
Eq(F_226(x), F_15(x)*F_18(x)*F_227(x))
Eq(F_227(x), F_132(x) + F_15(x))
Eq(F_228(x), F_18(x)*F_229(x))
Eq(F_229(x), F_1(x) + F_228(x) + F_230(x) + F_280(x))
Eq(F_230(x), F_18(x)*F_231(x))
Eq(F_231(x), F_1(x) + F_222(x) + F_232(x) + F_270(x) + F_272(x))
Eq(F_232(x), F_18(x)*F_233(x))
Eq(F_233(x), F_1(x) + F_173(x) + F_234(x) + F_252(x) + F_269(x) + F_270(x))
Eq(F_234(x), F_235(x))
Eq(F_235(x), F_10(x)*F_18(x)*F_236(x))
Eq(F_236(x), F_237(x) + F_243(x))
Eq(F_237(x), F_238(x) + F_240(x))
Eq(F_238(x), F_15(x) + F_239(x))
Eq(F_239(x), F_21(x))
Eq(F_240(x), F_241(x) + F_242(x))
Eq(F_241(x), F_16(x) + F_63(x))
Eq(F_242(x), F_40(x))
Eq(F_243(x), F_244(x) + F_246(x))
Eq(F_244(x), F_245(x) + F_54(x))
Eq(F_245(x), F_60(x))
Eq(F_246(x), F_247(x) + F_251(x))
Eq(F_247(x), F_248(x) + F_52(x))
Eq(F_248(x), F_249(x) + 2*F_41(x) + F_90(x))
Eq(F_249(x), F_18(x)*F_250(x))
Eq(F_250(x), F_248(x) + F_52(x))
Eq(F_251(x), F_78(x))
Eq(F_252(x), F_18(x)*F_253(x))
Eq(F_253(x), F_236(x) + F_254(x))
Eq(F_254(x), F_255(x) + F_260(x))
Eq(F_255(x), F_256(x) + F_258(x))
Eq(F_256(x), F_257(x) + F_54(x))
Eq(F_257(x), F_100(x))
Eq(F_258(x), F_247(x) + F_259(x))
Eq(F_259(x), F_111(x))
Eq(F_260(x), F_261(x) + F_263(x))
Eq(F_261(x), F_138(x) + F_262(x))
Eq(F_262(x), F_133(x))
Eq(F_263(x), F_264(x) + F_268(x))
Eq(F_264(x), F_139(x) + F_265(x))
Eq(F_265(x), F_167(x) + F_266(x) + 3*F_41(x))
Eq(F_266(x), F_18(x)*F_267(x))
Eq(F_267(x), F_139(x) + F_265(x))
Eq(F_268(x), F_155(x))
Eq(F_269(x), F_18(x)*F_237(x))
Eq(F_270(x), F_271(x))
Eq(F_271(x), F_15(x)**2*F_18(x))
Eq(F_272(x), F_18(x)*F_273(x))
Eq(F_273(x), F_237(x) + F_274(x))
Eq(F_274(x), F_275(x) + F_277(x))
Eq(F_275(x), F_276(x) + F_54(x))
Eq(F_276(x), F_181(x))
Eq(F_277(x), F_278(x) + F_279(x))
Eq(F_278(x), F_127(x) + F_136(x))
Eq(F_279(x), F_192(x))
Eq(F_280(x), F_18(x)*F_281(x))
Eq(F_281(x), F_15(x) + F_282(x))
Eq(F_282(x), F_16(x) + F_184(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "row_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 3, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 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