0132_0231_1302_3012
Counting sequence:
1, 1, 2, 6, 20, 65, 206, 647, 2030, 6376, 20047, 63070, 198488, 624754, 1966578, 6190484, 19486891, 61342613, 193100173, 607859640, 1913480535, 6023443258, 18961190915, 59687914251, 187891527328, 591463557265, 1861867561097, 5860971099669, 18449745272375, 58077935351291, 182823476687384, 575509845967688, 1811646889159514, 5702881495421971, 17952095160176596, 56511383043599521, 177892128189289772, 559986458786536594, 1762780833621876021, 5549056086317183480, 17467868303189641822, 54987085787428892204, 173093794326463826128, 544881788246758268982, 1715232855795070488969, 5399379852766470974239, 16996702631927430698171, 53503903825205380358034, 168424887257805700310917, 530184540187534090866748, 1668967402801924642891241, 5253740877902901999274557, 16538245843393368159290126, 52060728142666691282226899, 163881915918389562650310551, 515883725088951544913850359, 1623949880743304182986695357, 5112030263624751659855539525, 16092155137358045635375920958, 50656479639301231744923759130, 159461483408755541733422811632, 501968649854476517334295888501, 1580146628830940675881838876586, 4974142049169088693257890650763, 15658096960038009262720341860026, 49290108321475711646989534362572, 155160284454957907493055002519083, 488428909815442504293971748846690, 1537524894218372006343711522190068, 4839973131882118419985985844737206, 15235746754688800165206780561223166, 47960592517327217407212876232868904, 150975102937124587553217149092541047, 475254380951205498156436748232778297, 1496052807510775447373308421886356537, 4709423190126609882577140591939629820, 14824788718925318686345073850332225411, 46666938112832220545022182047396366094, 146902809484679536003630647230528415263, 462435212319146801065801116630256785407, 1455699358935380816797334706876838259084, 4582394608268762399995566124847111092453, 14424915568588053950656061495388186374965, 45408177808481309419434882313888250478505, 142940359136429838225766910894235553878419, 449961818688863478595419058130665650449887, 1416434375154645709123810382153844190639476, 4458792403688422238934751265117615365083260, 14035828307978615306962992972139594538861014, 44183370396003840220171528475359376250830183, 139084789063767937198731886250136336751817296, 437824873375038843441375952546606812793635909, 1378228496704855402389479679911377617169918104, 4338524155758073016432522748255826367050578070, 13657236006292764702003027905835779429486189873, 42991600054601695161104023058487016233962939292, 135333216355286559554839531264696553014443761434, 426015301263633917438049083789160942785372413456, 1341053156043277686248781764421041128332121304444, 4221499936737495552235819337581378237986811734378, 13288855580083783067634981140851550509736255433077
Generating function in Maple syntax:
-(3*x^5-7*x^4+13*x^3-13*x^2+6*x-1)/(x^3-3*x^2+4*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{3 x^{5}-7 x^{4}+13 x^{3}-13 x^{2}+6 x -1}{\left(x^{3}-3 x^{2}+4 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-3*x**5 + 7*x**4 - 13*x**3 + 13*x**2 - 6*x + 1)/((x - 1)**3*(x**3 - 3*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3-3*x^2+4*x-1)*(x-1)^3*F(x)+3*x^5-7*x^4+13*x^3-13*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-3 x^{2}+4 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+3 x^{5}-7 x^{4}+13 x^{3}-13 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
1/2232*(96*((I*3^(1/3)-1/3*3^(5/6))*31^(1/2)-31/16*I*3^(5/6)+31/16*3^(1/3))*2^(2/3)*(9+31^(1/2)*3^(1/2))^(1/3)-2976+51*((I*3^(2/3)+3^(1/6))*31^(1/2)-217/17*I*3^(1/6)-217/51*3^(2/3))*2^(1/3)*(9+31^(1/2)*3^(1/2))^(2/3))*(1/12*(108+12*31^(1/2)*3^(1/2))^(1/3)-1/12*I*(36+4*31^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1+1/24*((-I*31^(1/2)+3)*18^(1/3)+9*I*2^(1/3)*3^(1/6)-31^(1/2)*3^(1/6)*2^(1/3))*(9+31^(1/2)*3^(1/2))^(2/3))^(-n)+1/2232*(-96*2^(2/3)*((I*3^(1/3)+1/3*3^(5/6))*31^(1/2)-31/16*I*3^(5/6)-31/16*3^(1/3))*(9+31^(1/2)*3^(1/2))^(1/3)-2976-51*((I*3^(2/3)-3^(1/6))*31^(1/2)-217/17*I*3^(1/6)+217/51*3^(2/3))*2^(1/3)*(9+31^(1/2)*3^(1/2))^(2/3))*(1/12*(108+12*31^(1/2)*3^(1/2))^(1/3)+1/12*I*(36+4*31^(1/2)*3^(1/2))^(1/3)*3^(5/6)+1+1/24*((I*31^(1/2)+3)*18^(1/3)-9*I*2^(1/3)*3^(1/6)-31^(1/2)*3^(1/6)*2^(1/3))*(9+31^(1/2)*3^(1/2))^(2/3))^(-n)+1/2232*(64*2^(2/3)*(31^(1/2)*3^(5/6)-93/16*3^(1/3))*(9+31^(1/2)*3^(1/2))^(1/3)-2976-102*(3^(1/6)*31^(1/2)-217/51*3^(2/3))*2^(1/3)*(9+31^(1/2)*3^(1/2))^(2/3))*(-1/6*(108+12*31^(1/2)*3^(1/2))^(1/3)+1+1/12*(31^(1/2)*3^(1/6)*2^(1/3)-3*18^(1/3))*(9+31^(1/2)*3^(1/2))^(2/3))^(-n)+1/2*n^2-7/2*n+5
Explicit closed form in latex syntax:
\frac{\left(96 \left(\left(\mathrm{I} \,3^{\frac{1}{3}}-\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{31}-\frac{31 \,\mathrm{I} \,3^{\frac{5}{6}}}{16}+\frac{31 \,3^{\frac{1}{3}}}{16}\right) 2^{\frac{2}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-2976+51 \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{31}-\frac{217 \,\mathrm{I} \,3^{\frac{1}{6}}}{17}-\frac{217 \,3^{\frac{2}{3}}}{51}\right) 2^{\frac{1}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(108+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+1+\frac{\left(\left(-\mathrm{I} \sqrt{31}+3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}-\sqrt{31}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}\right)^{-n}}{2232}+\frac{\left(-96 \,2^{\frac{2}{3}} \left(\left(\mathrm{I} \,3^{\frac{1}{3}}+\frac{3^{\frac{5}{6}}}{3}\right) \sqrt{31}-\frac{31 \,\mathrm{I} \,3^{\frac{5}{6}}}{16}-\frac{31 \,3^{\frac{1}{3}}}{16}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-2976-51 \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \sqrt{31}-\frac{217 \,\mathrm{I} \,3^{\frac{1}{6}}}{17}+\frac{217 \,3^{\frac{2}{3}}}{51}\right) 2^{\frac{1}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(108+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \left(36+4 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+1+\frac{\left(\left(\mathrm{I} \sqrt{31}+3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}-\sqrt{31}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}\right)^{-n}}{2232}+\frac{\left(64 \,2^{\frac{2}{3}} \left(\sqrt{31}\, 3^{\frac{5}{6}}-\frac{93 \,3^{\frac{1}{3}}}{16}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-2976-102 \left(3^{\frac{1}{6}} \sqrt{31}-\frac{217 \,3^{\frac{2}{3}}}{51}\right) 2^{\frac{1}{3}} \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{\left(108+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+1+\frac{\left(\sqrt{31}\, 3^{\frac{1}{6}} 2^{\frac{1}{3}}-3 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}\right)^{-n}}{2232}+\frac{n^{2}}{2}-\frac{7 n}{2}+5
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(n+3) = a(n)-3*a(n+1)+4*a(n+2)-1/2*n*(n-3), n >= 6
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(n +3\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)-\frac{n \left(n -3\right)}{2}, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21241/
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[17,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[17,x]*F[7,x]
F[7,x] = F[8,x]+F[89,x]
F[8,x] = F[45,x]+F[9,x]
F[9,x] = F[10,x]+F[26,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[17,x]
F[13,x] = F[14,x]+F[18,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[17,x]
F[17,x] = x
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[25,x]
F[20,x] = 0
F[21,x] = F[17,x]*F[22,x]
F[22,x] = F[23,x]+F[24,x]
F[23,x] = F[15,x]
F[24,x] = F[19,x]
F[25,x] = F[17,x]*F[18,x]
F[26,x] = F[15,x]+F[27,x]
F[27,x] = F[20,x]+F[28,x]+F[33,x]
F[28,x] = F[17,x]*F[29,x]
F[29,x] = F[15,x]+F[30,x]
F[30,x] = F[20,x]+F[28,x]+F[31,x]
F[31,x] = F[17,x]*F[32,x]
F[32,x] = F[15,x]+F[30,x]
F[33,x] = F[17,x]*F[34,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[15,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[17,x]*F[35,x]
F[38,x] = F[27,x]+F[39,x]
F[39,x] = 2*F[20,x]+F[40,x]+F[44,x]
F[40,x] = F[17,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[36,x]
F[43,x] = F[39,x]
F[44,x] = F[17,x]*F[38,x]
F[45,x] = F[46,x]+F[72,x]
F[46,x] = F[15,x]+F[47,x]
F[47,x] = F[20,x]+F[48,x]+F[64,x]
F[48,x] = F[17,x]*F[49,x]
F[49,x] = F[50,x]+F[59,x]
F[50,x] = F[51,x]
F[51,x] = F[17,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[1,x]+F[17,x]
F[54,x] = F[15,x]+F[55,x]
F[55,x] = F[20,x]+F[56,x]+F[58,x]
F[56,x] = F[17,x]*F[57,x]
F[57,x] = F[17,x]+F[55,x]
F[58,x] = F[15,x]*F[17,x]
F[59,x] = F[20,x]+F[48,x]+F[60,x]
F[60,x] = F[17,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[15,x]
F[63,x] = F[30,x]
F[64,x] = F[17,x]*F[65,x]
F[65,x] = F[35,x]+F[66,x]
F[66,x] = F[47,x]+F[67,x]
F[67,x] = 2*F[20,x]+F[68,x]+F[71,x]
F[68,x] = F[17,x]*F[69,x]
F[69,x] = F[42,x]+F[70,x]
F[70,x] = F[67,x]
F[71,x] = F[17,x]*F[66,x]
F[72,x] = F[73,x]+F[76,x]
F[73,x] = F[74,x]
F[74,x] = F[17,x]*F[75,x]
F[75,x] = F[15,x]+F[73,x]
F[76,x] = F[77,x]
F[77,x] = F[17,x]*F[78,x]
F[78,x] = F[79,x]+F[82,x]
F[79,x] = F[73,x]+F[80,x]
F[80,x] = F[81,x]
F[81,x] = F[17,x]*F[79,x]
F[82,x] = F[76,x]+F[83,x]
F[83,x] = 3*F[20,x]+F[84,x]+F[88,x]
F[84,x] = F[17,x]*F[85,x]
F[85,x] = F[86,x]+F[87,x]
F[86,x] = F[80,x]
F[87,x] = F[83,x]
F[88,x] = F[17,x]*F[82,x]
F[89,x] = F[2,x]*F[90,x]
F[90,x] = F[9,x]+F[91,x]
F[91,x] = F[26,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_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{17}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)+F_{89}\! \left(x \right)
F_{8}\! \left(x \right) = F_{45}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{26}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \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_{17}\! \left(x \right)
F_{17}\! \left(x \right) = x
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{17}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)
F_{23}\! \left(x \right) = F_{15}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right)
F_{25}\! \left(x \right) = F_{17}\! \left(x \right) F_{18}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)+F_{33}\! \left(x \right)
F_{28}\! \left(x \right) = F_{17}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{17}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{17}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{17}\! \left(x \right) F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{27}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{17}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{36}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = F_{17}\! \left(x \right) F_{38}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{72}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{20}\! \left(x \right)+F_{48}\! \left(x \right)+F_{64}\! \left(x \right)
F_{48}\! \left(x \right) = F_{17}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{59}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{17}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{54}\! \left(x \right) = F_{15}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{20}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{17}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{17}\! \left(x \right)+F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right)
F_{59}\! \left(x \right) = F_{20}\! \left(x \right)+F_{48}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{17}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{15}\! \left(x \right)
F_{63}\! \left(x \right) = F_{30}\! \left(x \right)
F_{64}\! \left(x \right) = F_{17}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{35}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{47}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{68}\! \left(x \right) = F_{17}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{42}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{67}\! \left(x \right)
F_{71}\! \left(x \right) = F_{17}\! \left(x \right) F_{66}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{76}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{17}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{15}\! \left(x \right)+F_{73}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{17}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)+F_{82}\! \left(x \right)
F_{79}\! \left(x \right) = F_{73}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{17}\! \left(x \right) F_{79}\! \left(x \right)
F_{82}\! \left(x \right) = F_{76}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = 3 F_{20}\! \left(x \right)+F_{84}\! \left(x \right)+F_{88}\! \left(x \right)
F_{84}\! \left(x \right) = F_{17}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)+F_{87}\! \left(x \right)
F_{86}\! \left(x \right) = F_{80}\! \left(x \right)
F_{87}\! \left(x \right) = F_{83}\! \left(x \right)
F_{88}\! \left(x \right) = F_{17}\! \left(x \right) F_{82}\! \left(x \right)
F_{89}\! \left(x \right) = F_{2}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{9}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{26}\! \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_17(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_17(x)*F_7(x))
Eq(F_7(x), F_8(x) + F_89(x))
Eq(F_8(x), F_45(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_26(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_17(x))
Eq(F_13(x), F_14(x) + F_18(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_17(x))
Eq(F_17(x), x)
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_25(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_17(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_24(x))
Eq(F_23(x), F_15(x))
Eq(F_24(x), F_19(x))
Eq(F_25(x), F_17(x)*F_18(x))
Eq(F_26(x), F_15(x) + F_27(x))
Eq(F_27(x), F_20(x) + F_28(x) + F_33(x))
Eq(F_28(x), F_17(x)*F_29(x))
Eq(F_29(x), F_15(x) + F_30(x))
Eq(F_30(x), F_20(x) + F_28(x) + F_31(x))
Eq(F_31(x), F_17(x)*F_32(x))
Eq(F_32(x), F_15(x) + F_30(x))
Eq(F_33(x), F_17(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_15(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_17(x)*F_35(x))
Eq(F_38(x), F_27(x) + F_39(x))
Eq(F_39(x), 2*F_20(x) + F_40(x) + F_44(x))
Eq(F_40(x), F_17(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_36(x))
Eq(F_43(x), F_39(x))
Eq(F_44(x), F_17(x)*F_38(x))
Eq(F_45(x), F_46(x) + F_72(x))
Eq(F_46(x), F_15(x) + F_47(x))
Eq(F_47(x), F_20(x) + F_48(x) + F_64(x))
Eq(F_48(x), F_17(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_59(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_17(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_1(x) + F_17(x))
Eq(F_54(x), F_15(x) + F_55(x))
Eq(F_55(x), F_20(x) + F_56(x) + F_58(x))
Eq(F_56(x), F_17(x)*F_57(x))
Eq(F_57(x), F_17(x) + F_55(x))
Eq(F_58(x), F_15(x)*F_17(x))
Eq(F_59(x), F_20(x) + F_48(x) + F_60(x))
Eq(F_60(x), F_17(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_15(x))
Eq(F_63(x), F_30(x))
Eq(F_64(x), F_17(x)*F_65(x))
Eq(F_65(x), F_35(x) + F_66(x))
Eq(F_66(x), F_47(x) + F_67(x))
Eq(F_67(x), 2*F_20(x) + F_68(x) + F_71(x))
Eq(F_68(x), F_17(x)*F_69(x))
Eq(F_69(x), F_42(x) + F_70(x))
Eq(F_70(x), F_67(x))
Eq(F_71(x), F_17(x)*F_66(x))
Eq(F_72(x), F_73(x) + F_76(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_17(x)*F_75(x))
Eq(F_75(x), F_15(x) + F_73(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_17(x)*F_78(x))
Eq(F_78(x), F_79(x) + F_82(x))
Eq(F_79(x), F_73(x) + F_80(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_17(x)*F_79(x))
Eq(F_82(x), F_76(x) + F_83(x))
Eq(F_83(x), 3*F_20(x) + F_84(x) + F_88(x))
Eq(F_84(x), F_17(x)*F_85(x))
Eq(F_85(x), F_86(x) + F_87(x))
Eq(F_86(x), F_80(x))
Eq(F_87(x), F_83(x))
Eq(F_88(x), F_17(x)*F_82(x))
Eq(F_89(x), F_2(x)*F_90(x))
Eq(F_90(x), F_9(x) + F_91(x))
Eq(F_91(x), F_26(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": 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:
{"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": [0, 2, 3, 1], "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, 3, 2], "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, 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, 3, 2], "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, 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": false, "strategy_class": "RequirementInsertionStrategy"}}, {"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, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "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, 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": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[1, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0], "pos": [[3, 1]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1], "pos": [[2, 0], [2, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [2, 0], [3, 0]]}, {"patt": [0, 1, 2], "pos": [[2, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2], "pos": [[3, 0], [3, 0], [3, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [2, 0], [3, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [3, 0], [3, 0]]}, {"patt": [1, 0, 2], "pos": [[2, 0], [3, 0], [3, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [1, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], 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