0132_0213_0231_2130
Counting sequence:
1, 1, 2, 6, 20, 65, 197, 562, 1541, 4130, 10934, 28767, 75454, 197628, 517298, 1353707, 3542224, 9268897, 24254604, 63471521, 166105046, 434715364, 1137742227, 2977815905, 7794088030, 20400687147, 53399229067, 139776670838, 365883522576, 957764027418, 2507153142764, 6563101627101, 17180771380389, 44976003568975, 117739779432428, 308225992576563, 806897882683599, 2112373933040985, 5530006038213439, 14477137676172565, 37900229508040581, 99220813533783701, 259755847609912492, 680031935991824666, 1780305570105802345, 4660804826671162535, 12201923054159812307, 31944532273613733242, 83630669343939087136, 218945140758617833595, 573199324716447386164, 1500640214320752327967, 3928691982467241963869, 10285367535669747624788, 26927252084795486360973, 70496020158521052049477, 184559951592262494717609, 483181842803395711157352, 1264980946410127166643099, 3311750232034829639255486, 8670244725518617525845951, 22698925770370305089366814, 59426397347099476496541117, 155579951879575352135664865, 407312727420403187759184230, 1066356531312176757971544442, 2791752916658848738682359977, 7308893036360225929845679668, 19134904846154497707940940246, 50095771878052281874222540883, 131152295426079911113060497019, 343360846216254780356555848851, 898929619770680304914975509179, 2353426563745716664678616067710, 6161346702136336858966468833028, 16130605709921030236135456522608, 42230452218710159011782952111559, 110560708615614056808258817890940, 289451575221436767439358319127334, 757793788280884698733162807399830, 1983929257800577786386485889736486, 5193992748787857882885033703542226, 13598046114437491067615466840658932, 35600138912993442948710799780930526, 93202355091867339476530266991805995, 244006890253518920522767990358274455, 638818231725240281450160435557517492, 1672447609777358313517110866899867372, 4378524143950112454494479343113628806, 11463123767449050516125840326658962135, 30010844706693854291264364324159576719, 78569404653114092812885501557244371992, 205697356002875606584443754871472644359, 538522632553527928538545431493317975648, 1409870470051780999406606564780540907848, 3691088611138230310048368616016093887197, 9663394976382025212687053960863133322944, 25299095418386433513499682618195965122882, 66233889187411224560742646946569743157044, 173402567282011026778494993354398282155401, 453973801356223905323668121761212853535306
Generating function in Maple syntax:
(6*x^6-28*x^5+44*x^4-44*x^3+26*x^2-8*x+1)/(x^2-3*x+1)/(x^3-2*x^2+3*x-1)/(x-1)^3
Generating function in latex syntax:
\frac{6 x^{6}-28 x^{5}+44 x^{4}-44 x^{3}+26 x^{2}-8 x +1}{\left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(6*x**6 - 28*x**5 + 44*x**4 - 44*x**3 + 26*x**2 - 8*x + 1)/((x - 1)**3*(x**2 - 3*x + 1)*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x^3-2*x^2+3*x-1)*(x-1)^3*F(x)-6*x^6+28*x^5-44*x^4+44*x^3-26*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)-6 x^{6}+28 x^{5}-44 x^{4}+44 x^{3}-26 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
1/6900*(575*((-3/23*I+1/23*3^(1/2))*23^(1/2)+I*3^(1/2)-1)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)+2300-23*((-87/23*I-29/23*3^(1/2))*23^(1/2)+I*3^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(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)+1/6900*(-575*2^(2/3)*((-3/23*I-1/23*3^(1/2))*23^(1/2)+I*3^(1/2)+1)*(11+3*23^(1/2)*3^(1/2))^(1/3)+2300+23*((-87/23*I+29/23*3^(1/2))*23^(1/2)+I*3^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-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)+1/6900*((-50*2^(2/3)*23^(1/2)*3^(1/2)+1150*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)+2300+(-58*23^(1/2)*3^(1/2)*2^(1/3)+46*2^(1/3))*(11+3*23^(1/2)*3^(1/2))^(2/3))*(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)+1/6900*(690*5^(1/2)+3450)*(3/2-1/2*5^(1/2))^(-n)+1/6900*(-690*5^(1/2)+3450)*(3/2+1/2*5^(1/2))^(-n)-3/2*n^2+5/2*n-1
Explicit closed form in latex syntax:
\frac{\left(575 \left(\left(-\frac{3 \,\mathrm{I}}{23}+\frac{\sqrt{3}}{23}\right) \sqrt{23}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+2300-23 \left(\left(-\frac{87 \,\mathrm{I}}{23}-\frac{29 \sqrt{3}}{23}\right) \sqrt{23}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}{6900}+\frac{\left(-575 \,2^{\frac{2}{3}} \left(\left(-\frac{3 \,\mathrm{I}}{23}-\frac{\sqrt{3}}{23}\right) \sqrt{23}+\mathrm{I} \sqrt{3}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+2300+23 \left(\left(-\frac{87 \,\mathrm{I}}{23}+\frac{29 \sqrt{3}}{23}\right) \sqrt{23}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}{6900}+\frac{\left(\left(-50 \,2^{\frac{2}{3}} \sqrt{23}\, \sqrt{3}+1150 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+2300+\left(-58 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}+46 \,2^{\frac{1}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \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}}{6900}+\frac{\left(690 \sqrt{5}+3450\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{6900}+\frac{\left(-690 \sqrt{5}+3450\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{6900}-\frac{3 n^{2}}{2}+\frac{5 n}{2}-1
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(6) = 197
a(n+5) = -3/2*n^2+a(n)-5*a(n+1)+10*a(n+2)-12*a(n+3)+6*a(n+4)-7/2*n+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) = 65
a \! \left(6\right) = 197
a \! \left(n +5\right) = -\frac{3 n^{2}}{2}+a \! \left(n \right)-5 a \! \left(n +1\right)+10 a \! \left(n +2\right)-12 a \! \left(n +3\right)+6 a \! \left(n +4\right)-\frac{7 n}{2}+1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15247/
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[12,x]*F[4,x]
F[4,x] = F[19,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[13,x]
F[19,x] = F[20,x]+F[36,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[5,x]
F[23,x] = F[20,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[20,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[27,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[20,x]+F[31,x]
F[34,x] = F[35,x]
F[35,x] = F[12,x]*F[30,x]
F[36,x] = F[37,x]+F[38,x]+F[88,x]
F[37,x] = 0
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[40,x]+F[58,x]
F[40,x] = F[41,x]+F[6,x]
F[41,x] = F[37,x]+F[42,x]+F[50,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[10,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[12,x]*F[44,x]
F[47,x] = F[17,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[47,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[13,x]+F[52,x]
F[52,x] = F[53,x]+F[56,x]
F[53,x] = F[54,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[14,x]+F[53,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[52,x]
F[58,x] = F[36,x]+F[59,x]
F[59,x] = 2*F[37,x]+F[60,x]+F[80,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[62,x]+F[77,x]
F[62,x] = F[63,x]+F[75,x]
F[63,x] = F[37,x]+F[64,x]+F[73,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[66,x]+F[68,x]
F[66,x] = F[10,x]+F[67,x]
F[67,x] = F[18,x]+F[37,x]+F[42,x]
F[68,x] = F[63,x]+F[69,x]
F[69,x] = 2*F[37,x]+F[60,x]+F[70,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[31,x]+F[72,x]
F[72,x] = F[70,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[20,x]+F[63,x]
F[75,x] = F[76,x]
F[76,x] = F[12,x]*F[62,x]
F[77,x] = F[72,x]+F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[12,x]*F[77,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[71,x]+F[82,x]
F[82,x] = F[83,x]+F[86,x]
F[83,x] = F[84,x]
F[84,x] = F[12,x]*F[85,x]
F[85,x] = F[31,x]+F[83,x]
F[86,x] = F[87,x]
F[87,x] = F[12,x]*F[82,x]
F[88,x] = F[12,x]*F[89,x]
F[89,x] = F[74,x]+F[90,x]
F[90,x] = F[125,x]+F[91,x]
F[91,x] = F[123,x]+F[37,x]+F[92,x]
F[92,x] = F[12,x]*F[93,x]
F[93,x] = F[106,x]+F[94,x]
F[94,x] = F[14,x]+F[95,x]
F[95,x] = F[37,x]+F[54,x]+F[96,x]
F[96,x] = F[12,x]*F[97,x]
F[97,x] = F[101,x]+F[98,x]
F[98,x] = F[12,x]+F[99,x]
F[99,x] = F[100,x]
F[100,x] = F[12,x]*F[98,x]
F[101,x] = F[102,x]+F[104,x]
F[102,x] = F[103,x]
F[103,x] = F[12,x]*F[14,x]
F[104,x] = F[105,x]
F[105,x] = F[101,x]*F[12,x]
F[106,x] = F[107,x]+F[31,x]
F[107,x] = 2*F[37,x]+F[108,x]+F[120,x]
F[108,x] = F[109,x]*F[12,x]
F[109,x] = F[110,x]+F[115,x]
F[110,x] = F[111,x]+F[113,x]
F[111,x] = F[112,x]
F[112,x] = F[12,x]*F[20,x]
F[113,x] = F[114,x]
F[114,x] = F[110,x]*F[12,x]
F[115,x] = F[116,x]+F[118,x]
F[116,x] = F[117,x]
F[117,x] = F[12,x]*F[31,x]
F[118,x] = F[119,x]
F[119,x] = F[115,x]*F[12,x]
F[120,x] = F[12,x]*F[121,x]
F[121,x] = F[122,x]+F[31,x]
F[122,x] = F[120,x]
F[123,x] = F[12,x]*F[124,x]
F[124,x] = F[20,x]+F[91,x]
F[125,x] = 2*F[37,x]+F[126,x]+F[135,x]
F[126,x] = F[12,x]*F[127,x]
F[127,x] = F[128,x]+F[130,x]
F[128,x] = F[129,x]+F[17,x]
F[129,x] = F[57,x]
F[130,x] = F[131,x]+F[72,x]
F[131,x] = F[132,x]
F[132,x] = F[12,x]*F[133,x]
F[133,x] = F[122,x]+F[134,x]
F[134,x] = F[132,x]
F[135,x] = F[12,x]*F[90,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{19}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{36}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{5}\! \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_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \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_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{20}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{88}\! \left(x \right)
F_{37}\! \left(x \right) = 0
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{58}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{6}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)+F_{42}\! \left(x \right)+F_{50}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{17}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{56}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{14}\! \left(x \right)+F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{58}\! \left(x \right) = F_{36}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{37}\! \left(x \right)+F_{60}\! \left(x \right)+F_{80}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{77}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{75}\! \left(x \right)
F_{63}\! \left(x \right) = F_{37}\! \left(x \right)+F_{64}\! \left(x \right)+F_{73}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{68}\! \left(x \right)
F_{66}\! \left(x \right) = F_{10}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{18}\! \left(x \right)+F_{37}\! \left(x \right)+F_{42}\! \left(x \right)
F_{68}\! \left(x \right) = F_{63}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{37}\! \left(x \right)+F_{60}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{31}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{70}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{20}\! \left(x \right)+F_{63}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{77}\! \left(x \right) = F_{72}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{71}\! \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_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{31}\! \left(x \right)+F_{83}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{88}\! \left(x \right) = F_{12}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{74}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{125}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{123}\! \left(x \right)+F_{37}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{12}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{106}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{14}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{37}\! \left(x \right)+F_{54}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{12}\! \left(x \right) F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{101}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{12}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)
F_{100}\! \left(x \right) = F_{12}\! \left(x \right) F_{98}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{104}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)
F_{103}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)
F_{105}\! \left(x \right) = F_{101}\! \left(x \right) F_{12}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{31}\! \left(x \right)
F_{107}\! \left(x \right) = 2 F_{37}\! \left(x \right)+F_{108}\! \left(x \right)+F_{120}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{12}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)+F_{115}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{113}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)
F_{114}\! \left(x \right) = F_{110}\! \left(x \right) F_{12}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{118}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)
F_{117}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right)
F_{119}\! \left(x \right) = F_{115}\! \left(x \right) F_{12}\! \left(x \right)
F_{120}\! \left(x \right) = F_{12}\! \left(x \right) F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)+F_{31}\! \left(x \right)
F_{122}\! \left(x \right) = F_{120}\! \left(x \right)
F_{123}\! \left(x \right) = F_{12}\! \left(x \right) F_{124}\! \left(x \right)
F_{124}\! \left(x \right) = F_{20}\! \left(x \right)+F_{91}\! \left(x \right)
F_{125}\! \left(x \right) = 2 F_{37}\! \left(x \right)+F_{126}\! \left(x \right)+F_{135}\! \left(x \right)
F_{126}\! \left(x \right) = F_{12}\! \left(x \right) F_{127}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)+F_{130}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right)+F_{17}\! \left(x \right)
F_{129}\! \left(x \right) = F_{57}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right)+F_{72}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)
F_{132}\! \left(x \right) = F_{12}\! \left(x \right) F_{133}\! \left(x \right)
F_{133}\! \left(x \right) = F_{122}\! \left(x \right)+F_{134}\! \left(x \right)
F_{134}\! \left(x \right) = F_{132}\! \left(x \right)
F_{135}\! \left(x \right) = F_{12}\! \left(x \right) F_{90}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_19(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_13(x))
Eq(F_19(x), F_20(x) + F_36(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_5(x))
Eq(F_23(x), F_20(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_20(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_27(x))
Eq(F_30(x), F_31(x) + F_34(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_20(x) + F_31(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_12(x)*F_30(x))
Eq(F_36(x), F_37(x) + F_38(x) + F_88(x))
Eq(F_37(x), 0)
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_58(x))
Eq(F_40(x), F_41(x) + F_6(x))
Eq(F_41(x), F_37(x) + F_42(x) + F_50(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_10(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_12(x)*F_44(x))
Eq(F_47(x), F_17(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_47(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_13(x) + F_52(x))
Eq(F_52(x), F_53(x) + F_56(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_14(x) + F_53(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_52(x))
Eq(F_58(x), F_36(x) + F_59(x))
Eq(F_59(x), 2*F_37(x) + F_60(x) + F_80(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_77(x))
Eq(F_62(x), F_63(x) + F_75(x))
Eq(F_63(x), F_37(x) + F_64(x) + F_73(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_68(x))
Eq(F_66(x), F_10(x) + F_67(x))
Eq(F_67(x), F_18(x) + F_37(x) + F_42(x))
Eq(F_68(x), F_63(x) + F_69(x))
Eq(F_69(x), 2*F_37(x) + F_60(x) + F_70(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_31(x) + F_72(x))
Eq(F_72(x), F_70(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_20(x) + F_63(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_12(x)*F_62(x))
Eq(F_77(x), F_72(x) + F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_12(x)*F_77(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_71(x) + F_82(x))
Eq(F_82(x), F_83(x) + F_86(x))
Eq(F_83(x), F_84(x))
Eq(F_84(x), F_12(x)*F_85(x))
Eq(F_85(x), F_31(x) + F_83(x))
Eq(F_86(x), F_87(x))
Eq(F_87(x), F_12(x)*F_82(x))
Eq(F_88(x), F_12(x)*F_89(x))
Eq(F_89(x), F_74(x) + F_90(x))
Eq(F_90(x), F_125(x) + F_91(x))
Eq(F_91(x), F_123(x) + F_37(x) + F_92(x))
Eq(F_92(x), F_12(x)*F_93(x))
Eq(F_93(x), F_106(x) + F_94(x))
Eq(F_94(x), F_14(x) + F_95(x))
Eq(F_95(x), F_37(x) + F_54(x) + F_96(x))
Eq(F_96(x), F_12(x)*F_97(x))
Eq(F_97(x), F_101(x) + F_98(x))
Eq(F_98(x), F_12(x) + F_99(x))
Eq(F_99(x), F_100(x))
Eq(F_100(x), F_12(x)*F_98(x))
Eq(F_101(x), F_102(x) + F_104(x))
Eq(F_102(x), F_103(x))
Eq(F_103(x), F_12(x)*F_14(x))
Eq(F_104(x), F_105(x))
Eq(F_105(x), F_101(x)*F_12(x))
Eq(F_106(x), F_107(x) + F_31(x))
Eq(F_107(x), F_108(x) + F_120(x) + 2*F_37(x))
Eq(F_108(x), F_109(x)*F_12(x))
Eq(F_109(x), F_110(x) + F_115(x))
Eq(F_110(x), F_111(x) + F_113(x))
Eq(F_111(x), F_112(x))
Eq(F_112(x), F_12(x)*F_20(x))
Eq(F_113(x), F_114(x))
Eq(F_114(x), F_110(x)*F_12(x))
Eq(F_115(x), F_116(x) + F_118(x))
Eq(F_116(x), F_117(x))
Eq(F_117(x), F_12(x)*F_31(x))
Eq(F_118(x), F_119(x))
Eq(F_119(x), F_115(x)*F_12(x))
Eq(F_120(x), F_12(x)*F_121(x))
Eq(F_121(x), F_122(x) + F_31(x))
Eq(F_122(x), F_120(x))
Eq(F_123(x), F_12(x)*F_124(x))
Eq(F_124(x), F_20(x) + F_91(x))
Eq(F_125(x), F_126(x) + F_135(x) + 2*F_37(x))
Eq(F_126(x), F_12(x)*F_127(x))
Eq(F_127(x), F_128(x) + F_130(x))
Eq(F_128(x), F_129(x) + F_17(x))
Eq(F_129(x), F_57(x))
Eq(F_130(x), F_131(x) + F_72(x))
Eq(F_131(x), F_132(x))
Eq(F_132(x), F_12(x)*F_133(x))
Eq(F_133(x), F_122(x) + F_134(x))
Eq(F_134(x), F_132(x))
Eq(F_135(x), F_12(x)*F_90(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": [0, 2, 1, 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": [2, 1, 3, 0], "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, 1, 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": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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