0132_1302_3021_3102
Counting sequence:
1, 1, 2, 6, 20, 66, 213, 674, 2098, 6443, 19570, 58910, 176023, 522718, 1544202, 4541663, 13306722, 38859286, 113154079, 328663462, 952508146, 2755072175, 7954986842, 22933501086, 66023275903, 189837470558, 545229899034, 1564366222831, 4484363989426, 12844132166694, 36760705308351, 105140039704086, 300527759238530, 858533741562479, 2451366209805770, 6996110541947630, 19958158062082815, 56913537100826126, 162239467790499626, 462334615937079151, 1317125667398997794, 3751282566207536630, 10681297100833874623, 30406670869701167686, 86541144123352144914, 246259377566277536239, 700625862645653635514, 1993008099617852013374, 5668487729047728828415, 16120030450036311938238, 45836327678240148442682, 130317725949035642197487, 370468198122579002487698, 1053066614890502110299334, 2993102787349984428808895, 8506510020383901046262646, 24173975296438891278993314, 68693277148185777787030383, 195187645721030671531563690, 554580469234817479758300302, 1575625563129923831740658431, 4476303767007141758549333102, 12716450339154066646161967690, 36123819070389572144607087215, 102613450840406820323494171906, 291473551247186645515042124950, 827903041852876488764300511423, 2351507746869697625263535613350, 6678839568775053628176671303730, 18968991427201239751010569999087, 53873723448852932236201298006170, 153003050732509191918302724415198, 434524521992675669899647622974463, 1234014578440963916684258052490526, 3504440836762299640165253976277850, 9951996359280486901007423507006703, 28261503583479926609399421116238706, 80255425297888314644601034488601958, 227901971422367145802266681912478911, 647167552475827080897095040055904470, 1837725965396299458951970944525651394, 5218437605475721168626769702500261999, 14818228907529950005618127831424455562, 42077358854842243916440869842194373166, 119480572905438197927134251041004878591, 339268106017323726243207515799418689230, 963354048251629051718178261149102573930, 2735433717031425039427964135432593294703, 7767192095909410977725993022157285672674, 22054624435998613238155962897518582894390, 62622908057614313690757845763850714769087, 177813569643882176094878140655942508487942, 504887736850675678509666936810762230302802, 1433583958877623091962273747342754576069615, 4070520607567608368348118305579819288220538, 11557806335042235094684913581793903714469502, 32817055047224526623290398243074243184738303, 93179984455995760821051982711654065035065726, 264572431261992767356198917760493904235293818, 751217251796775782544582987393895202807974895, 2132974213529316331492100843371488593868568402
Generating function in Maple syntax:
(x^6-4*x^5+13*x^4-21*x^3+18*x^2-7*x+1)/(x-1)/(2*x^3-4*x^2+4*x-1)/(x^2-3*x+1)
Generating function in latex syntax:
\frac{x^{6}-4 x^{5}+13 x^{4}-21 x^{3}+18 x^{2}-7 x +1}{\left(x -1\right) \left(2 x^{3}-4 x^{2}+4 x -1\right) \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(x**6 - 4*x**5 + 13*x**4 - 21*x**3 + 18*x**2 - 7*x + 1)/((x - 1)*(x**2 - 3*x + 1)*(2*x**3 - 4*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x^3-4*x^2+4*x-1)*(x^2-3*x+1)*F(x)-x^6+4*x^5-13*x^4+21*x^3-18*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x^{3}-4 x^{2}+4 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)-x^{6}+4 x^{5}-13 x^{4}+21 x^{3}-18 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/4224*(-99*2^(2/3)*((I-13/99*11^(1/2))*3^(1/2)-13/33*I*11^(1/2)+1)*(13+3*11^(1/2)*3^(1/2))^(2/3)+2112+96*(I-1/3*3^(1/2))*2^(1/3)*11^(1/2)*(13+3*11^(1/2)*3^(1/2))^(1/3))*(13/384*2^(2/3)*((I-3/13*11^(1/2))*3^(1/2)-9/13*I*11^(1/2)+1)*(13+3*11^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(26+6*11^(1/2)*3^(1/2))^(1/3)+1/12*(26+6*11^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/4224*(-96*2^(1/3)*11^(1/2)*(I+1/3*3^(1/2))*(13+3*11^(1/2)*3^(1/2))^(1/3)+2112+99*((I+13/99*11^(1/2))*3^(1/2)-13/33*I*11^(1/2)-1)*2^(2/3)*(13+3*11^(1/2)*3^(1/2))^(2/3))*(-13/384*((I+3/13*11^(1/2))*3^(1/2)-9/13*I*11^(1/2)-1)*2^(2/3)*(13+3*11^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(26+6*11^(1/2)*3^(1/2))^(1/3)+1/12*(26+6*11^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/4224*((-26*11^(1/2)*3^(1/2)*2^(2/3)+198*2^(2/3))*(13+3*11^(1/2)*3^(1/2))^(2/3)+64*11^(1/2)*3^(1/2)*2^(1/3)*(13+3*11^(1/2)*3^(1/2))^(1/3)+2112)*(1/192*2^(2/3)*(3*11^(1/2)*3^(1/2)-13)*(13+3*11^(1/2)*3^(1/2))^(2/3)-1/6*(26+6*11^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)-(3/2-1/2*5^(1/2))^(-n)-(3/2+1/2*5^(1/2))^(-n)+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-99 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{13 \sqrt{11}}{99}\right) \sqrt{3}-\frac{13 \,\mathrm{I} \sqrt{11}}{33}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+2112+96 \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) 2^{\frac{1}{3}} \sqrt{11}\, \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{13 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{11}}{13}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{11}}{13}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}\, \left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{4224}\\+\\\frac{\left(-96 \,2^{\frac{1}{3}} \sqrt{11}\, \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+2112+99 \left(\left(\mathrm{I}+\frac{13 \sqrt{11}}{99}\right) \sqrt{3}-\frac{13 \,\mathrm{I} \sqrt{11}}{33}-1\right) 2^{\frac{2}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{13 \left(\left(\mathrm{I}+\frac{3 \sqrt{11}}{13}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{11}}{13}-1\right) 2^{\frac{2}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}+\frac{\mathrm{I} \sqrt{3}\, \left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{4224}\\+\\\frac{\left(\left(-26 \sqrt{11}\, \sqrt{3}\, 2^{\frac{2}{3}}+198 \,2^{\frac{2}{3}}\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+64 \sqrt{11}\, \sqrt{3}\, 2^{\frac{1}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+2112\right) \left(\frac{2^{\frac{2}{3}} \left(3 \sqrt{11}\, \sqrt{3}-13\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{192}-\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{4224}\\-\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}-\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}+1 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(6) = 213
a(n+5) = 2*a(n)-10*a(n+1)+18*a(n+2)-17*a(n+3)+7*a(n+4)+1, 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) = 20
a \! \left(5\right) = 66
a \! \left(6\right) = 213
a \! \left(n +5\right) = 2 a \! \left(n \right)-10 a \! \left(n +1\right)+18 a \! \left(n +2\right)-17 a \! \left(n +3\right)+7 a \! \left(n +4\right)+1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15667/
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[31,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[20,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[14,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[21,x]+F[22,x]+F[30,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[24,x]*F[4,x]
F[27,x] = F[20,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[27,x]*F[4,x]
F[30,x] = F[13,x]*F[4,x]
F[31,x] = F[14,x]+F[32,x]
F[32,x] = F[21,x]+F[33,x]+F[99,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[62,x]
F[35,x] = F[36,x]+F[7,x]
F[36,x] = F[21,x]+F[26,x]+F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[42,x]
F[39,x] = F[11,x]+F[40,x]
F[40,x] = F[21,x]+F[26,x]+F[41,x]
F[41,x] = F[39,x]*F[4,x]
F[42,x] = F[43,x]+F[52,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[11,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[46,x]
F[49,x] = F[43,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[49,x]
F[52,x] = 2*F[21,x]+F[53,x]+F[61,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[58,x]
F[55,x] = F[40,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[55,x]
F[58,x] = F[52,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[58,x]
F[61,x] = F[4,x]*F[42,x]
F[62,x] = F[32,x]+F[63,x]
F[63,x] = 2*F[21,x]+F[64,x]+F[77,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[76,x]
F[66,x] = F[21,x]+F[67,x]+F[74,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]+F[70,x]
F[69,x] = F[11,x]+F[40,x]
F[70,x] = F[66,x]+F[71,x]
F[71,x] = 2*F[21,x]+F[64,x]+F[72,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[18,x]+F[71,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[14,x]+F[66,x]
F[76,x] = F[64,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[73,x]+F[79,x]
F[79,x] = F[80,x]+F[89,x]
F[80,x] = F[81,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[83,x]+F[86,x]
F[83,x] = F[18,x]+F[84,x]
F[84,x] = F[85,x]
F[85,x] = F[4,x]*F[83,x]
F[86,x] = F[80,x]+F[87,x]
F[87,x] = F[88,x]
F[88,x] = F[4,x]*F[86,x]
F[89,x] = 3*F[21,x]+F[90,x]+F[98,x]
F[90,x] = F[4,x]*F[91,x]
F[91,x] = F[92,x]+F[95,x]
F[92,x] = F[71,x]+F[93,x]
F[93,x] = F[94,x]
F[94,x] = F[4,x]*F[92,x]
F[95,x] = F[89,x]+F[96,x]
F[96,x] = F[97,x]
F[97,x] = F[4,x]*F[95,x]
F[98,x] = F[4,x]*F[79,x]
F[99,x] = F[100,x]*F[4,x]
F[100,x] = F[101,x]+F[75,x]
F[101,x] = F[102,x]+F[108,x]
F[102,x] = F[103,x]
F[103,x] = F[104,x]*F[4,x]
F[104,x] = F[105,x]+F[17,x]
F[105,x] = F[102,x]+F[106,x]
F[106,x] = F[107,x]
F[107,x] = F[105,x]*F[4,x]
F[108,x] = 2*F[21,x]+F[109,x]+F[114,x]
F[109,x] = F[110,x]*F[4,x]
F[110,x] = F[111,x]+F[65,x]
F[111,x] = F[108,x]+F[112,x]
F[112,x] = F[113,x]
F[113,x] = F[111,x]*F[4,x]
F[114,x] = F[101,x]*F[4,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_{31}\! \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_{20}\! \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_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{30}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{21}\! \left(x \right)+F_{33}\! \left(x \right)+F_{99}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{62}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{7}\! \left(x \right)
F_{36}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \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_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{52}\! \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_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{11}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{43}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{53}\! \left(x \right)+F_{61}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{58}\! \left(x \right)
F_{55}\! \left(x \right) = F_{40}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{52}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{62}\! \left(x \right) = F_{32}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{64}\! \left(x \right)+F_{77}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{76}\! \left(x \right)
F_{66}\! \left(x \right) = F_{21}\! \left(x \right)+F_{67}\! \left(x \right)+F_{74}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{70}\! \left(x \right)
F_{69}\! \left(x \right) = F_{11}\! \left(x \right)+F_{40}\! \left(x \right)
F_{70}\! \left(x \right) = F_{66}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{64}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{18}\! \left(x \right)+F_{71}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{14}\! \left(x \right)+F_{66}\! \left(x \right)
F_{76}\! \left(x \right) = F_{64}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{73}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{89}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{86}\! \left(x \right)
F_{83}\! \left(x \right) = F_{18}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{86}\! \left(x \right) = F_{80}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{89}\! \left(x \right) = 3 F_{21}\! \left(x \right)+F_{90}\! \left(x \right)+F_{98}\! \left(x \right)
F_{90}\! \left(x \right) = F_{4}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)+F_{95}\! \left(x \right)
F_{92}\! \left(x \right) = F_{71}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{4}\! \left(x \right) F_{92}\! \left(x \right)
F_{95}\! \left(x \right) = F_{89}\! \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_{95}\! \left(x \right)
F_{98}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \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_{75}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{108}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{4}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{17}\! \left(x \right)
F_{105}\! \left(x \right) = F_{102}\! \left(x \right)+F_{106}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)
F_{107}\! \left(x \right) = F_{105}\! \left(x \right) F_{4}\! \left(x \right)
F_{108}\! \left(x \right) = 2 F_{21}\! \left(x \right)+F_{109}\! \left(x \right)+F_{114}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right) F_{4}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{65}\! \left(x \right)
F_{111}\! \left(x \right) = F_{108}\! \left(x \right)+F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)
F_{113}\! \left(x \right) = F_{111}\! \left(x \right) F_{4}\! \left(x \right)
F_{114}\! \left(x \right) = F_{101}\! \left(x \right) F_{4}\! \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_31(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_20(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_14(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_30(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_24(x)*F_4(x))
Eq(F_27(x), F_20(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_27(x)*F_4(x))
Eq(F_30(x), F_13(x)*F_4(x))
Eq(F_31(x), F_14(x) + F_32(x))
Eq(F_32(x), F_21(x) + F_33(x) + F_99(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_62(x))
Eq(F_35(x), F_36(x) + F_7(x))
Eq(F_36(x), F_21(x) + F_26(x) + F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_42(x))
Eq(F_39(x), F_11(x) + F_40(x))
Eq(F_40(x), F_21(x) + F_26(x) + F_41(x))
Eq(F_41(x), F_39(x)*F_4(x))
Eq(F_42(x), F_43(x) + F_52(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_11(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_46(x))
Eq(F_49(x), F_43(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_49(x))
Eq(F_52(x), 2*F_21(x) + F_53(x) + F_61(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_58(x))
Eq(F_55(x), F_40(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_55(x))
Eq(F_58(x), F_52(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_58(x))
Eq(F_61(x), F_4(x)*F_42(x))
Eq(F_62(x), F_32(x) + F_63(x))
Eq(F_63(x), 2*F_21(x) + F_64(x) + F_77(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_76(x))
Eq(F_66(x), F_21(x) + F_67(x) + F_74(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_70(x))
Eq(F_69(x), F_11(x) + F_40(x))
Eq(F_70(x), F_66(x) + F_71(x))
Eq(F_71(x), 2*F_21(x) + F_64(x) + F_72(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_18(x) + F_71(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_14(x) + F_66(x))
Eq(F_76(x), F_64(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_73(x) + F_79(x))
Eq(F_79(x), F_80(x) + F_89(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_86(x))
Eq(F_83(x), F_18(x) + F_84(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), F_4(x)*F_83(x))
Eq(F_86(x), F_80(x) + F_87(x))
Eq(F_87(x), F_88(x))
Eq(F_88(x), F_4(x)*F_86(x))
Eq(F_89(x), 3*F_21(x) + F_90(x) + F_98(x))
Eq(F_90(x), F_4(x)*F_91(x))
Eq(F_91(x), F_92(x) + F_95(x))
Eq(F_92(x), F_71(x) + F_93(x))
Eq(F_93(x), F_94(x))
Eq(F_94(x), F_4(x)*F_92(x))
Eq(F_95(x), F_89(x) + F_96(x))
Eq(F_96(x), F_97(x))
Eq(F_97(x), F_4(x)*F_95(x))
Eq(F_98(x), F_4(x)*F_79(x))
Eq(F_99(x), F_100(x)*F_4(x))
Eq(F_100(x), F_101(x) + F_75(x))
Eq(F_101(x), F_102(x) + F_108(x))
Eq(F_102(x), F_103(x))
Eq(F_103(x), F_104(x)*F_4(x))
Eq(F_104(x), F_105(x) + F_17(x))
Eq(F_105(x), F_102(x) + F_106(x))
Eq(F_106(x), F_107(x))
Eq(F_107(x), F_105(x)*F_4(x))
Eq(F_108(x), F_109(x) + F_114(x) + 2*F_21(x))
Eq(F_109(x), F_110(x)*F_4(x))
Eq(F_110(x), F_111(x) + F_65(x))
Eq(F_111(x), F_108(x) + F_112(x))
Eq(F_112(x), F_113(x))
Eq(F_113(x), F_111(x)*F_4(x))
Eq(F_114(x), F_101(x)*F_4(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, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": 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