0123_0231_1032_1302_3012
Counting sequence:
1, 1, 2, 6, 19, 52, 136, 351, 898, 2279, 5743, 14385, 35845, 88918, 219701, 540945, 1327756, 3249880, 7934499, 19327574, 46981796, 113986579, 276068936, 667545609, 1611747429, 3886090581, 9357740595, 22506563098, 54070658197, 129765535535, 311121748966, 745246987418, 1783578615267, 4265086055032, 10191226149856, 24333590383607, 58060887171998, 138444021066411, 329907362639291, 785688237088697, 1870084266815745, 4448731343331758, 10577569481328357, 25137392218681501, 59710191779023024, 141768359966732332, 336450304027863411, 798143453901149146, 1892633892182090588, 4486262483997779659, 10630202868044860548, 25179277723980525805, 59620488425024710225, 141124927565878890173, 333943491369039891951, 789965499556536127090, 1868158619241632770021, 4416662500698714868635, 10438874255990660911754, 24665847738588450365294, 58267328427874895416051, 137608233519891616778684, 324905748720860982964536, 766950411694202840614479, 1809994238492121086438394, 4270611691794138315657711, 10074158932529177280903175, 23759431585162484700940993, 56024040586348279488222077, 132076767590718756098305990, 311310983317452348423283541, 733636197620943897895377513, 1728572310413483409886502932, 4072086114706005736873966912, 9591156616155967923926715011, 22586637741016152361250814302, 53181506744984900616986324628, 125198734359855906313542337091, 294693952835408723216285417216, 693547427703870508205795416017, 1631985787298951971163868087485, 3839665548979481732307135071013, 9032515949185053376067989964587, 21245347380505459237202982260746, 49964332514330359384602130391029, 117489442659699053542304198684487, 276236713515744519772224021224686, 649395106161710593367754372104066, 1526451106930904742486595373454403, 3587601806783439460828065252977664, 8430904239829844076853587324515728, 19810423797613190957038059051919975, 46544045604681375623557472367150230, 109341814870462102144098263663098803, 256839143250403815489165060325818451, 603239681828358851067723241000614985, 1416684971805182098580293278263095161, 3326689582484572234545877574917727390, 7811020188410218918740763185121797509, 18338364103281988492607606100117686389, 43049846191154306898636796364016624056
Generating function in Maple syntax:
-(2*x^7-4*x^6+6*x^5-5*x^4+7*x^3-9*x^2+5*x-1)/(x^3-2*x^2+3*x-1)^2
Generating function in latex syntax:
-\frac{2 x^{7}-4 x^{6}+6 x^{5}-5 x^{4}+7 x^{3}-9 x^{2}+5 x -1}{\left(x^{3}-2 x^{2}+3 x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**7 + 4*x**6 - 6*x**5 + 5*x**4 - 7*x**3 + 9*x**2 - 5*x + 1)/(x**3 - 2*x**2 + 3*x - 1)**2
Implicit equation for the generating function in Maple syntax:
(x^3-2*x^2+3*x-1)^2*F(x)+2*x^7-4*x^6+6*x^5-5*x^4+7*x^3-9*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-2 x^{2}+3 x -1\right)^{2} F \! \left(x \right)+2 x^{7}-4 x^{6}+6 x^{5}-5 x^{4}+7 x^{3}-9 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/317400*(-1357*(((-7/59*n+1157/1357)*23^(1/2)+I*n-583/59*I)*3^(1/2)+(-21/59*I*n+3471/1357*I)*23^(1/2)+n-583/59)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1840*2^(2/3)*(((-1/8*n+275/184)*23^(1/2)+I*n-5*I)*3^(1/2)+(3/8*I*n-825/184*I)*23^(1/2)-n+5)*(11+3*23^(1/2)*3^(1/2))^(1/3)-9200*n+529000)*(11/600*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/317400*(-1840*2^(2/3)*(((1/8*n-275/184)*23^(1/2)+I*n-5*I)*3^(1/2)+(3/8*I*n-825/184*I)*23^(1/2)+n-5)*(11+3*23^(1/2)*3^(1/2))^(1/3)+1357*2^(1/3)*(((7/59*n-1157/1357)*23^(1/2)+I*n-583/59*I)*3^(1/2)+(-21/59*I*n+3471/1357*I)*23^(1/2)-n+583/59)*(11+3*23^(1/2)*3^(1/2))^(2/3)-9200*n+529000)*(-11/600*2^(1/3)*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)-7/6900*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)*(-10/7*2^(2/3)*(3^(1/2)*(n-275/23)*23^(1/2)+8*n-40)*(11+3*23^(1/2)*3^(1/2))^(1/3)+(3^(1/2)*(n-1157/161)*23^(1/2)-59/7*n+583/7)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+200/7*n-11500/7))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(-1357 \left(\left(\left(-\frac{7 n}{59}+\frac{1157}{1357}\right) \sqrt{23}+\mathrm{I} n -\frac{583 \,\mathrm{I}}{59}\right) \sqrt{3}+\left(-\frac{21 \,\mathrm{I} n}{59}+\frac{3471 \,\mathrm{I}}{1357}\right) \sqrt{23}+n -\frac{583}{59}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+1840 \,2^{\frac{2}{3}} \left(\left(\left(-\frac{n}{8}+\frac{275}{184}\right) \sqrt{23}+\mathrm{I} n -5 \,\mathrm{I}\right) \sqrt{3}+\left(\frac{3 \,\mathrm{I} n}{8}-\frac{825 \,\mathrm{I}}{184}\right) \sqrt{23}-n +5\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-9200 n +529000\right) \left(\frac{11 \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\+\\\frac{\left(-1840 \,2^{\frac{2}{3}} \left(\left(\left(\frac{n}{8}-\frac{275}{184}\right) \sqrt{23}+\mathrm{I} n -5 \,\mathrm{I}\right) \sqrt{3}+\left(\frac{3 \,\mathrm{I} n}{8}-\frac{825 \,\mathrm{I}}{184}\right) \sqrt{23}+n -5\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+1357 \,2^{\frac{1}{3}} \left(\left(\left(\frac{7 n}{59}-\frac{1157}{1357}\right) \sqrt{23}+\mathrm{I} n -\frac{583 \,\mathrm{I}}{59}\right) \sqrt{3}+\left(-\frac{21 \,\mathrm{I} n}{59}+\frac{3471 \,\mathrm{I}}{1357}\right) \sqrt{23}-n +\frac{583}{59}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}-9200 n +529000\right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\-\\\frac{7 \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n} \left(-\frac{10 \,2^{\frac{2}{3}} \left(\sqrt{3}\, \left(n -\frac{275}{23}\right) \sqrt{23}+8 n -40\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{7}+\left(\sqrt{3}\, \left(n -\frac{1157}{161}\right) \sqrt{23}-\frac{59 n}{7}+\frac{583}{7}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{200 n}{7}-\frac{11500}{7}\right)}{6900} & \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) = 52
a(6) = 136
a(7) = 351
a(n+6) = -a(n)+4*a(n+1)-10*a(n+2)+14*a(n+3)-13*a(n+4)+6*a(n+5), n >= 8
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 19
a \! \left(5\right) = 52
a \! \left(6\right) = 136
a \! \left(7\right) = 351
a \! \left(n +6\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17234/
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[19,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[4,x]
F[18,x] = F[15,x]
F[19,x] = F[20,x]+F[39,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[10,x]+F[23,x]
F[23,x] = F[20,x]+F[24,x]
F[24,x] = F[25,x]+F[26,x]+F[37,x]
F[25,x] = 0
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[29,x]+F[4,x]
F[29,x] = F[30,x]
F[30,x] = F[11,x]*F[4,x]
F[31,x] = F[32,x]+F[34,x]
F[32,x] = F[25,x]+F[26,x]+F[33,x]
F[33,x] = F[20,x]*F[4,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[20,x]+F[36,x]
F[39,x] = F[25,x]+F[40,x]+F[89,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[52,x]
F[42,x] = F[43,x]+F[46,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[1,x]+F[4,x]
F[46,x] = F[25,x]+F[47,x]+F[50,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[4,x]+F[49,x]
F[49,x] = F[47,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[11,x]+F[49,x]
F[52,x] = F[53,x]+F[78,x]
F[53,x] = F[25,x]+F[40,x]+F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[20,x]+F[56,x]
F[56,x] = F[25,x]+F[33,x]+F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[61,x]
F[59,x] = F[4,x]+F[60,x]
F[60,x] = F[25,x]+F[30,x]+F[47,x]
F[61,x] = F[56,x]+F[62,x]
F[62,x] = F[25,x]+F[35,x]+F[63,x]+F[76,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[70,x]
F[65,x] = F[66,x]+F[68,x]
F[66,x] = F[67,x]
F[67,x] = x^2
F[68,x] = F[69,x]
F[69,x] = F[4,x]*F[49,x]
F[70,x] = F[71,x]+F[73,x]
F[71,x] = 2*F[25,x]+F[63,x]+F[72,x]
F[72,x] = F[4,x]*F[56,x]
F[73,x] = F[74,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[76,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[56,x]+F[75,x]
F[78,x] = F[25,x]+F[76,x]+F[79,x]+F[87,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[81,x]+F[84,x]
F[81,x] = F[82,x]+F[83,x]
F[82,x] = F[67,x]
F[83,x] = F[69,x]
F[84,x] = F[85,x]+F[86,x]
F[85,x] = 2*F[25,x]+F[72,x]+F[79,x]
F[86,x] = F[74,x]
F[87,x] = F[4,x]*F[88,x]
F[88,x] = F[36,x]+F[75,x]
F[89,x] = F[4,x]*F[90,x]
F[90,x] = F[91,x]+F[93,x]
F[91,x] = F[20,x]+F[92,x]
F[92,x] = F[25,x]+F[37,x]+F[57,x]
F[93,x] = F[103,x]+F[94,x]
F[94,x] = F[95,x]
F[95,x] = F[4,x]*F[96,x]
F[96,x] = F[38,x]+F[97,x]
F[97,x] = F[94,x]+F[98,x]
F[98,x] = F[99,x]
F[99,x] = F[100,x]*F[4,x]
F[100,x] = F[101,x]+F[102,x]
F[101,x] = F[33,x]
F[102,x] = F[99,x]
F[103,x] = F[104,x]
F[104,x] = F[105,x]*F[4,x]
F[105,x] = F[106,x]+F[107,x]
F[106,x] = F[56,x]
F[107,x] = F[104,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_{19}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{39}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{37}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \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_{30}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{32}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{20}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{25}\! \left(x \right)+F_{40}\! \left(x \right)+F_{89}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{52}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{46}\! \left(x \right) = F_{25}\! \left(x \right)+F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{11}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{78}\! \left(x \right)
F_{53}\! \left(x \right) = F_{25}\! \left(x \right)+F_{40}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{20}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{25}\! \left(x \right)+F_{33}\! \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_{61}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)+F_{47}\! \left(x \right)
F_{61}\! \left(x \right) = F_{56}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{25}\! \left(x \right)+F_{35}\! \left(x \right)+F_{63}\! \left(x \right)+F_{76}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{70}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{68}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = x^{2}
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{73}\! \left(x \right)
F_{71}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{63}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \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_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{56}\! \left(x \right)+F_{75}\! \left(x \right)
F_{78}\! \left(x \right) = F_{25}\! \left(x \right)+F_{76}\! \left(x \right)+F_{79}\! \left(x \right)+F_{87}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{84}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{83}\! \left(x \right)
F_{82}\! \left(x \right) = F_{67}\! \left(x \right)
F_{83}\! \left(x \right) = F_{69}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)+F_{86}\! \left(x \right)
F_{85}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{72}\! \left(x \right)+F_{79}\! \left(x \right)
F_{86}\! \left(x \right) = F_{74}\! \left(x \right)
F_{87}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{36}\! \left(x \right)+F_{75}\! \left(x \right)
F_{89}\! \left(x \right) = F_{4}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)+F_{93}\! \left(x \right)
F_{91}\! \left(x \right) = F_{20}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{25}\! \left(x \right)+F_{37}\! \left(x \right)+F_{57}\! \left(x \right)
F_{93}\! \left(x \right) = F_{103}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{4}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{38}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{94}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right) F_{4}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{102}\! \left(x \right)
F_{101}\! \left(x \right) = F_{33}\! \left(x \right)
F_{102}\! \left(x \right) = F_{99}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right) F_{4}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{107}\! \left(x \right)
F_{106}\! \left(x \right) = F_{56}\! \left(x \right)
F_{107}\! \left(x \right) = F_{104}\! \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_19(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_15(x))
Eq(F_19(x), F_20(x) + F_39(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_10(x) + F_23(x))
Eq(F_23(x), F_20(x) + F_24(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_37(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_29(x) + F_4(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_11(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_34(x))
Eq(F_32(x), F_25(x) + F_26(x) + F_33(x))
Eq(F_33(x), F_20(x)*F_4(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_20(x) + F_36(x))
Eq(F_39(x), F_25(x) + F_40(x) + F_89(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_52(x))
Eq(F_42(x), F_43(x) + F_46(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_1(x) + F_4(x))
Eq(F_46(x), F_25(x) + F_47(x) + F_50(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_4(x) + F_49(x))
Eq(F_49(x), F_47(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_11(x) + F_49(x))
Eq(F_52(x), F_53(x) + F_78(x))
Eq(F_53(x), F_25(x) + F_40(x) + F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_20(x) + F_56(x))
Eq(F_56(x), F_25(x) + F_33(x) + F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_61(x))
Eq(F_59(x), F_4(x) + F_60(x))
Eq(F_60(x), F_25(x) + F_30(x) + F_47(x))
Eq(F_61(x), F_56(x) + F_62(x))
Eq(F_62(x), F_25(x) + F_35(x) + F_63(x) + F_76(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_70(x))
Eq(F_65(x), F_66(x) + F_68(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), x**2)
Eq(F_68(x), F_69(x))
Eq(F_69(x), F_4(x)*F_49(x))
Eq(F_70(x), F_71(x) + F_73(x))
Eq(F_71(x), 2*F_25(x) + F_63(x) + F_72(x))
Eq(F_72(x), F_4(x)*F_56(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))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_56(x) + F_75(x))
Eq(F_78(x), F_25(x) + F_76(x) + F_79(x) + F_87(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_81(x) + F_84(x))
Eq(F_81(x), F_82(x) + F_83(x))
Eq(F_82(x), F_67(x))
Eq(F_83(x), F_69(x))
Eq(F_84(x), F_85(x) + F_86(x))
Eq(F_85(x), 2*F_25(x) + F_72(x) + F_79(x))
Eq(F_86(x), F_74(x))
Eq(F_87(x), F_4(x)*F_88(x))
Eq(F_88(x), F_36(x) + F_75(x))
Eq(F_89(x), F_4(x)*F_90(x))
Eq(F_90(x), F_91(x) + F_93(x))
Eq(F_91(x), F_20(x) + F_92(x))
Eq(F_92(x), F_25(x) + F_37(x) + F_57(x))
Eq(F_93(x), F_103(x) + F_94(x))
Eq(F_94(x), F_95(x))
Eq(F_95(x), F_4(x)*F_96(x))
Eq(F_96(x), F_38(x) + F_97(x))
Eq(F_97(x), F_94(x) + F_98(x))
Eq(F_98(x), F_99(x))
Eq(F_99(x), F_100(x)*F_4(x))
Eq(F_100(x), F_101(x) + F_102(x))
Eq(F_101(x), F_33(x))
Eq(F_102(x), F_99(x))
Eq(F_103(x), F_104(x))
Eq(F_104(x), F_105(x)*F_4(x))
Eq(F_105(x), F_106(x) + F_107(x))
Eq(F_106(x), F_56(x))
Eq(F_107(x), F_104(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": 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