0231_2103_3021
Counting sequence:
1, 1, 2, 6, 21, 72, 228, 670, 1864, 5000, 13099, 33789, 86239, 218432, 550107, 1379348, 3446817, 8590103, 21362792, 53037845, 131501095, 325690131, 805941635, 1992965105, 4925514322, 12167639830, 30047049122, 74176872249, 183075826494, 451761419037, 1114599813984, 2749624947004, 6782395588375, 16728478286366, 41257252496038, 101746690219860, 250911703062682, 618736658247686, 1525731267552645, 3762183046591067, 9276698140205217, 22873895764245038, 56400290763169709, 139065077186953442, 342887139294467851, 845437237672147533, 2084534874056178770, 5139667502305515203, 12672412905073877089, 31245129837169002721, 77037879542688238157, 189943952125059703431, 468323489148164896260, 1154691212295146311500, 2846986015214380932060, 7019471977625168972383, 17307058492911974388136, 42671886172360723934243, 105210776244149587606882, 259405065766312767706122, 639582472707149532162073, 1576937755420611508075268, 3888055584827893584881304, 9586284439094479216177850, 23635680016029311105614428, 58275478458186584378378036, 143682394739999330198302991, 354259282057384553346821993, 873451700584565757788218963, 2153557797632267139528296460, 5309750976500251198241798063, 13091570905037081906748846752, 32278203901431304130488243621, 79584217551815422735168047667, 196220570498766140312945335916, 483795820107380808576341479905, 1192833108925337698164014667915, 2941015103329142965148929847007, 7251282442348821384376155619719, 17878553806199174983994987507917, 44080848776387204265031901991910, 108684474308293771004253018319922, 267969316583225317065781287906310, 660697444747371484880843290123125, 1628996626251795584895000824161090, 4016407245527110701748680806225177, 9902738214105799007913349167416340, 24415906565571468619097195496922760, 60199157044395090225072479341081419, 148425310224154172194552788925550426, 365953175835583158242157948574738298, 902283624283202072135120493868792400, 2224644550484265072841421011726819726, 5485019611741619523272696247984908562, 13523706574822399181001323064306662345, 33343661906889574797590637351272060711, 82211173618381722798023149425088642549, 202697504722967336632841633332072220890, 499765136639471700705419204646977783009, 1232206544035879775594331112983447810894, 3038093007590431583834339354256382697967
Generating function in Maple syntax:
-(9*x^6-35*x^5+54*x^4-49*x^3+27*x^2-8*x+1)/(2*x-1)/(3*x^3-5*x^2+4*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{9 x^{6}-35 x^{5}+54 x^{4}-49 x^{3}+27 x^{2}-8 x +1}{\left(2 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-9*x**6 + 35*x**5 - 54*x**4 + 49*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**3*(2*x - 1)*(3*x**3 - 5*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(3*x^3-5*x^2+4*x-1)*(x-1)^3*F(x)+9*x^6-35*x^5+54*x^4-49*x^3+27*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+9 x^{6}-35 x^{5}+54 x^{4}-49 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
1/90024*(-3069*2^(2/3)*((-67/93*I+67/279*3^(1/2))*31^(1/2)+I*3^(1/2)-1)*(47+9*31^(1/2)*3^(1/2))^(1/3)-1953*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)*((-29/651*I-29/1953*3^(1/2))*31^(1/2)+I*3^(1/2)+1))*(47/8712*((I-9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)+1)*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/90024*(3069*((-67/93*I-67/279*3^(1/2))*31^(1/2)+I*3^(1/2)+1)*2^(2/3)*(47+9*31^(1/2)*3^(1/2))^(1/3)+1953*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)*((-29/651*I+29/1953*3^(1/2))*31^(1/2)+I*3^(1/2)-1))*(-47/8712*2^(1/3)*((I+9/47*31^(1/2))*3^(1/2)-27/47*I*31^(1/2)-1)*(47+9*31^(1/2)*3^(1/2))^(2/3)+1/36*I*3^(1/2)*(188+36*31^(1/2)*3^(1/2))^(1/3)+1/36*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)+1/90024*((1474*2^(2/3)*31^(1/2)*3^(1/2)-6138*2^(2/3))*(47+9*31^(1/2)*3^(1/2))^(1/3)-58*2^(1/3)*(47+9*31^(1/2)*3^(1/2))^(2/3)*(31^(1/2)*3^(1/2)-1953/29))*(1/4356*2^(1/3)*(9*31^(1/2)*3^(1/2)-47)*(47+9*31^(1/2)*3^(1/2))^(2/3)-1/18*(188+36*31^(1/2)*3^(1/2))^(1/3)+5/9)^(-n)-1/2*n^2-1/2*n-3*2^n+4
Explicit closed form in latex syntax:
\frac{\left(-3069 \,2^{\frac{2}{3}} \left(\left(-\frac{67 \,\mathrm{I}}{93}+\frac{67 \sqrt{3}}{279}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-1953 \,2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\left(-\frac{29 \,\mathrm{I}}{651}-\frac{29 \sqrt{3}}{1953}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right)\right) \left(\frac{47 \left(\left(\mathrm{I}-\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}+1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}-\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{90024}+\frac{\left(3069 \left(\left(-\frac{67 \,\mathrm{I}}{93}-\frac{67 \sqrt{3}}{279}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+1953 \,2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\left(-\frac{29 \,\mathrm{I}}{651}+\frac{29 \sqrt{3}}{1953}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right)\right) \left(-\frac{47 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}+\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{90024}+\frac{\left(\left(1474 \,2^{\frac{2}{3}} \sqrt{31}\, \sqrt{3}-6138 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-58 \,2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\sqrt{31}\, \sqrt{3}-\frac{1953}{29}\right)\right) \left(\frac{2^{\frac{1}{3}} \left(9 \sqrt{31}\, \sqrt{3}-47\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{4356}-\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n}}{90024}-\frac{n^{2}}{2}-\frac{n}{2}-3 \,2^{n}+4
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 72
a(6) = 228
a(n+4) = -1/2*n^2-6*a(n)+13*a(n+1)-13*a(n+2)+6*a(n+3)+1/2*n+4, n >= 7
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) = 21
a \! \left(5\right) = 72
a \! \left(6\right) = 228
a \! \left(n +4\right) = -\frac{n^{2}}{2}-6 a \! \left(n \right)+13 a \! \left(n +1\right)-13 a \! \left(n +2\right)+6 a \! \left(n +3\right)+\frac{n}{2}+4, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14778/
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[30,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[6,x]
F[10,x] = F[11,x]+F[24,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[4,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[15,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[21,x]*F[4,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]+F[7,x]
F[28,x] = F[29,x]
F[29,x] = F[27,x]*F[4,x]
F[30,x] = F[31,x]+F[46,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[14,x]+F[34,x]
F[34,x] = F[31,x]+F[35,x]
F[35,x] = F[36,x]+F[37,x]+F[45,x]
F[36,x] = 0
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[21,x]+F[39,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[31,x]+F[40,x]
F[43,x] = F[44,x]
F[44,x] = F[39,x]*F[4,x]
F[45,x] = F[34,x]*F[4,x]
F[46,x] = F[113,x]+F[36,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[69,x]
F[49,x] = F[50,x]+F[7,x]
F[50,x] = F[36,x]+F[51,x]+F[52,x]
F[51,x] = F[4,x]*F[49,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[15,x]+F[28,x]
F[55,x] = F[56,x]+F[65,x]
F[56,x] = F[23,x]+F[36,x]+F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[15,x]+F[22,x]
F[60,x] = F[56,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[22,x]+F[64,x]
F[64,x] = F[62,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[28,x]+F[68,x]
F[68,x] = F[66,x]
F[69,x] = F[70,x]+F[91,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]+F[74,x]
F[73,x] = F[31,x]+F[70,x]
F[74,x] = F[75,x]+F[85,x]
F[75,x] = F[76,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[42,x]+F[78,x]
F[78,x] = F[75,x]+F[79,x]
F[79,x] = F[80,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[82,x]
F[82,x] = F[40,x]+F[83,x]
F[83,x] = F[84,x]
F[84,x] = F[4,x]*F[82,x]
F[85,x] = F[86,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[88,x]
F[88,x] = F[70,x]+F[89,x]
F[89,x] = F[90,x]
F[90,x] = F[4,x]*F[88,x]
F[91,x] = 2*F[36,x]+F[92,x]+F[93,x]
F[92,x] = F[4,x]*F[69,x]
F[93,x] = F[4,x]*F[94,x]
F[94,x] = F[95,x]+F[99,x]
F[95,x] = F[35,x]+F[96,x]
F[96,x] = F[97,x]
F[97,x] = F[4,x]*F[98,x]
F[98,x] = F[70,x]+F[96,x]
F[99,x] = F[100,x]+F[109,x]
F[100,x] = 2*F[36,x]+F[101,x]+F[44,x]
F[101,x] = F[102,x]*F[4,x]
F[102,x] = F[103,x]+F[104,x]
F[103,x] = F[35,x]+F[43,x]
F[104,x] = F[100,x]+F[105,x]
F[105,x] = F[106,x]
F[106,x] = F[107,x]*F[4,x]
F[107,x] = F[108,x]+F[83,x]
F[108,x] = F[106,x]
F[109,x] = F[110,x]
F[110,x] = F[111,x]*F[4,x]
F[111,x] = F[112,x]+F[89,x]
F[112,x] = F[110,x]
F[113,x] = F[114,x]*F[4,x]
F[114,x] = F[115,x]+F[30,x]
F[115,x] = F[116,x]+F[125,x]
F[116,x] = F[117,x]+F[36,x]+F[37,x]
F[117,x] = F[118,x]*F[4,x]
F[118,x] = F[119,x]+F[34,x]
F[119,x] = F[116,x]+F[120,x]
F[120,x] = 2*F[36,x]+F[121,x]+F[123,x]
F[121,x] = F[122,x]*F[4,x]
F[122,x] = F[107,x]+F[63,x]
F[123,x] = F[124,x]*F[4,x]
F[124,x] = F[39,x]
F[125,x] = 2*F[36,x]+F[126,x]+F[134,x]
F[126,x] = F[127,x]*F[4,x]
F[127,x] = F[128,x]+F[131,x]
F[128,x] = F[129,x]+F[28,x]
F[129,x] = F[130,x]
F[130,x] = F[128,x]*F[4,x]
F[131,x] = F[132,x]+F[89,x]
F[132,x] = F[133,x]
F[133,x] = F[131,x]*F[4,x]
F[134,x] = F[135,x]*F[4,x]
F[135,x] = F[98,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_{30}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \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_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{46}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{14}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{45}\! \left(x \right)
F_{36}\! \left(x \right) = 0
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{21}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \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_{31}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = F_{113}\! \left(x \right)+F_{36}\! \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_{69}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{7}\! \left(x \right)
F_{50}\! \left(x \right) = F_{36}\! \left(x \right)+F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{65}\! \left(x \right)
F_{56}\! \left(x \right) = F_{23}\! \left(x \right)+F_{36}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{15}\! \left(x \right)+F_{22}\! \left(x \right)
F_{60}\! \left(x \right) = F_{56}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{22}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{28}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{91}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{74}\! \left(x \right)
F_{73}\! \left(x \right) = F_{31}\! \left(x \right)+F_{70}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{85}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{42}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{75}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \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_{82}\! \left(x \right) = F_{40}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{70}\! \left(x \right)+F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{91}\! \left(x \right) = 2 F_{36}\! \left(x \right)+F_{92}\! \left(x \right)+F_{93}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \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_{99}\! \left(x \right)
F_{95}\! \left(x \right) = F_{35}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{4}\! \left(x \right) F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{70}\! \left(x \right)+F_{96}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{109}\! \left(x \right)
F_{100}\! \left(x \right) = 2 F_{36}\! \left(x \right)+F_{101}\! \left(x \right)+F_{44}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{4}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{104}\! \left(x \right)
F_{103}\! \left(x \right) = F_{35}\! \left(x \right)+F_{43}\! \left(x \right)
F_{104}\! \left(x \right) = F_{100}\! \left(x \right)+F_{105}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right) F_{4}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)+F_{83}\! \left(x \right)
F_{108}\! \left(x \right) = F_{106}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right) F_{4}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)+F_{89}\! \left(x \right)
F_{112}\! \left(x \right) = F_{110}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right) F_{4}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{30}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{125}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)+F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right) F_{4}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right)+F_{34}\! \left(x \right)
F_{119}\! \left(x \right) = F_{116}\! \left(x \right)+F_{120}\! \left(x \right)
F_{120}\! \left(x \right) = 2 F_{36}\! \left(x \right)+F_{121}\! \left(x \right)+F_{123}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right) F_{4}\! \left(x \right)
F_{122}\! \left(x \right) = F_{107}\! \left(x \right)+F_{63}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right) F_{4}\! \left(x \right)
F_{124}\! \left(x \right) = F_{39}\! \left(x \right)
F_{125}\! \left(x \right) = 2 F_{36}\! \left(x \right)+F_{126}\! \left(x \right)+F_{134}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right) F_{4}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)+F_{131}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right)+F_{28}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)
F_{130}\! \left(x \right) = F_{128}\! \left(x \right) F_{4}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)+F_{89}\! \left(x \right)
F_{132}\! \left(x \right) = F_{133}\! \left(x \right)
F_{133}\! \left(x \right) = F_{131}\! \left(x \right) F_{4}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right) F_{4}\! \left(x \right)
F_{135}\! \left(x \right) = F_{98}\! \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_30(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_6(x))
Eq(F_10(x), F_11(x) + F_24(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_4(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_15(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_21(x)*F_4(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x) + F_7(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_27(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_46(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_14(x) + F_34(x))
Eq(F_34(x), F_31(x) + F_35(x))
Eq(F_35(x), F_36(x) + F_37(x) + F_45(x))
Eq(F_36(x), 0)
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_21(x) + F_39(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_31(x) + F_40(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_39(x)*F_4(x))
Eq(F_45(x), F_34(x)*F_4(x))
Eq(F_46(x), F_113(x) + F_36(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_69(x))
Eq(F_49(x), F_50(x) + F_7(x))
Eq(F_50(x), F_36(x) + F_51(x) + F_52(x))
Eq(F_51(x), F_4(x)*F_49(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_15(x) + F_28(x))
Eq(F_55(x), F_56(x) + F_65(x))
Eq(F_56(x), F_23(x) + F_36(x) + F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_15(x) + F_22(x))
Eq(F_60(x), F_56(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_22(x) + F_64(x))
Eq(F_64(x), F_62(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_28(x) + F_68(x))
Eq(F_68(x), F_66(x))
Eq(F_69(x), F_70(x) + F_91(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_74(x))
Eq(F_73(x), F_31(x) + F_70(x))
Eq(F_74(x), F_75(x) + F_85(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_42(x) + F_78(x))
Eq(F_78(x), F_75(x) + F_79(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_40(x) + F_83(x))
Eq(F_83(x), F_84(x))
Eq(F_84(x), F_4(x)*F_82(x))
Eq(F_85(x), F_86(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_88(x))
Eq(F_88(x), F_70(x) + F_89(x))
Eq(F_89(x), F_90(x))
Eq(F_90(x), F_4(x)*F_88(x))
Eq(F_91(x), 2*F_36(x) + F_92(x) + F_93(x))
Eq(F_92(x), F_4(x)*F_69(x))
Eq(F_93(x), F_4(x)*F_94(x))
Eq(F_94(x), F_95(x) + F_99(x))
Eq(F_95(x), F_35(x) + F_96(x))
Eq(F_96(x), F_97(x))
Eq(F_97(x), F_4(x)*F_98(x))
Eq(F_98(x), F_70(x) + F_96(x))
Eq(F_99(x), F_100(x) + F_109(x))
Eq(F_100(x), F_101(x) + 2*F_36(x) + F_44(x))
Eq(F_101(x), F_102(x)*F_4(x))
Eq(F_102(x), F_103(x) + F_104(x))
Eq(F_103(x), F_35(x) + F_43(x))
Eq(F_104(x), F_100(x) + F_105(x))
Eq(F_105(x), F_106(x))
Eq(F_106(x), F_107(x)*F_4(x))
Eq(F_107(x), F_108(x) + F_83(x))
Eq(F_108(x), F_106(x))
Eq(F_109(x), F_110(x))
Eq(F_110(x), F_111(x)*F_4(x))
Eq(F_111(x), F_112(x) + F_89(x))
Eq(F_112(x), F_110(x))
Eq(F_113(x), F_114(x)*F_4(x))
Eq(F_114(x), F_115(x) + F_30(x))
Eq(F_115(x), F_116(x) + F_125(x))
Eq(F_116(x), F_117(x) + F_36(x) + F_37(x))
Eq(F_117(x), F_118(x)*F_4(x))
Eq(F_118(x), F_119(x) + F_34(x))
Eq(F_119(x), F_116(x) + F_120(x))
Eq(F_120(x), F_121(x) + F_123(x) + 2*F_36(x))
Eq(F_121(x), F_122(x)*F_4(x))
Eq(F_122(x), F_107(x) + F_63(x))
Eq(F_123(x), F_124(x)*F_4(x))
Eq(F_124(x), F_39(x))
Eq(F_125(x), F_126(x) + F_134(x) + 2*F_36(x))
Eq(F_126(x), F_127(x)*F_4(x))
Eq(F_127(x), F_128(x) + F_131(x))
Eq(F_128(x), F_129(x) + F_28(x))
Eq(F_129(x), F_130(x))
Eq(F_130(x), F_128(x)*F_4(x))
Eq(F_131(x), F_132(x) + F_89(x))
Eq(F_132(x), F_133(x))
Eq(F_133(x), F_131(x)*F_4(x))
Eq(F_134(x), F_135(x)*F_4(x))
Eq(F_135(x), F_98(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, 2, 3, 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, 2, 1], "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, 2, 3, 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, 2, 1], "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, 2, 3, 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, 2, 1], "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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, 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