0132_0231_0312_2130
Counting sequence:
1, 1, 2, 6, 20, 64, 195, 578, 1691, 4917, 14257, 41287, 119500, 345801, 1000564, 2894992, 8376130, 24234658, 70117977, 202871700, 586966625, 1698264335, 4913570007, 14216379185, 41132096490, 119007050595, 344321813986, 996222584690, 2882360041648, 8339501168628, 24128588634047, 69810984817270, 201983368155511, 584396296917881, 1690827492233229, 4892052916093419, 14154123849883416, 40951973618023949, 118485902836375968, 342813982590871524, 991859992172287278, 2869737799598743062, 8302981371805685813, 24022926300162889832, 69505273127880470749, 201098855827089895027, 581837146953714995027, 1683423131289816032629, 4870629958569912894038, 14092141038327352640711, 40772639418991777094574, 117967037135787804432358, 341312754065026341485244, 987516503897274774213896, 2857170831898373590022587, 8266621499927898678825834, 23917726675679871994884243, 69200900190908464485873661, 200218216896901318820390057, 579289203851155235965066703, 1676051195038381318046559140, 4849300815057791355815392657, 14030429657837303221329754124, 40594090548530880944266989944, 117450443617870922323102003034, 339818099620756492467698529482, 983192036341460316968527567441, 2844658896639366303084088181116, 8230420852818151577219134037145, 23812987734498009062510369067543, 68897860143954608216480149530127, 199341434403002118741368347599609, 576752418534047766132645631406338, 1668711541487050396089528845682283, 4828065074733107659618677600842426, 13968988519780368115031795219660442, 40416323567582097477655765749506312, 116936112332432139637172857230546268, 338329990469275057663926396381456375, 978886506210557228309917863412838558, 2832201752827260906540172908318154991, 8194378733209775606638453654304542401, 23708707459223180369995023781421152645, 68596147150114287316426963506595228915, 198468491457539232792562774046973622384, 574226742140949771801251406197167947093, 1661404029265538657244045371557720680376, 4806922328570744665516010256569941342540, 13907816440728594882906488210546439245254, 40239335052146117297691545118184817757886, 116424033372851246341200167657147896717485, 336848397947864811491721934274351750889168, 974599830575035422729348725402276991469973, 2819799160522943949378724577246775016959995, 8158494446889526048390268874886751561192715, 23604883841295705041250779498218033649562557, 68295755398043476974119518774260899421419502, 197599371246613777977888314574338167737086031, 571712126024389267042046983945189671088968454, 1654128517622640977096491674809752644372663182, 4785872169336761220203204669632744597503198100
Generating function in Maple syntax:
-(3*x^5-7*x^4+8*x^3-9*x^2+5*x-1)/(2*x^3-x^2+3*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{3 x^{5}-7 x^{4}+8 x^{3}-9 x^{2}+5 x -1}{\left(2 x^{3}-x^{2}+3 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-3*x**5 + 7*x**4 - 8*x**3 + 9*x**2 - 5*x + 1)/((x - 1)**3*(2*x**3 - x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x^3-x^2+3*x-1)*(x-1)^3*F(x)+3*x^5-7*x^4+8*x^3-9*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{3}-x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+3 x^{5}-7 x^{4}+8 x^{3}-9 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
1/19757196*(((-43809*I-14603*3^(1/2))*211^(1/2)+104023*I*3^(1/2)+104023)*(28+3*211^(1/2)*3^(1/2))^(1/3)+975664+((-1005*I+335*3^(1/2))*211^(1/2)+85877*I*3^(1/2)-85877)*(28+3*211^(1/2)*3^(1/2))^(2/3))*(1/3468*((28*I+3*211^(1/2))*3^(1/2)-9*I*211^(1/2)-28)*(28+3*211^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(28+3*211^(1/2)*3^(1/2))^(1/3)-1/12*(28+3*211^(1/2)*3^(1/2))^(1/3)+1/6)^(-n)+1/19757196*(((43809*I-14603*3^(1/2))*211^(1/2)-104023*I*3^(1/2)+104023)*(28+3*211^(1/2)*3^(1/2))^(1/3)+975664+((1005*I+335*3^(1/2))*211^(1/2)-85877*I*3^(1/2)-85877)*(28+3*211^(1/2)*3^(1/2))^(2/3))*(1/3468*((-28*I+3*211^(1/2))*3^(1/2)+9*I*211^(1/2)-28)*(28+3*211^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(28+3*211^(1/2)*3^(1/2))^(1/3)-1/12*(28+3*211^(1/2)*3^(1/2))^(1/3)+1/6)^(-n)+1/19757196*((29206*211^(1/2)*3^(1/2)-208046)*(28+3*211^(1/2)*3^(1/2))^(1/3)-670*(28+3*211^(1/2)*3^(1/2))^(2/3)*211^(1/2)*3^(1/2)+171754*(28+3*211^(1/2)*3^(1/2))^(2/3)+975664)*(1/6*(28+3*211^(1/2)*3^(1/2))^(1/3)+1/6+14/867*(28+3*211^(1/2)*3^(1/2))^(2/3)-1/578*(28+3*211^(1/2)*3^(1/2))^(2/3)*211^(1/2)*3^(1/2))^(-n)-1/6*n^2-11/18*n+23/27
Explicit closed form in latex syntax:
\frac{\left(\left(\left(-43809 \,\mathrm{I}-14603 \sqrt{3}\right) \sqrt{211}+104023 \,\mathrm{I} \sqrt{3}+104023\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}+975664+\left(\left(-1005 \,\mathrm{I}+335 \sqrt{3}\right) \sqrt{211}+85877 \,\mathrm{I} \sqrt{3}-85877\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(28 \,\mathrm{I}+3 \sqrt{211}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{211}-28\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{3468}-\frac{\mathrm{I} \sqrt{3}\, \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{6}\right)^{-n}}{19757196}+\frac{\left(\left(\left(43809 \,\mathrm{I}-14603 \sqrt{3}\right) \sqrt{211}-104023 \,\mathrm{I} \sqrt{3}+104023\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}+975664+\left(\left(1005 \,\mathrm{I}+335 \sqrt{3}\right) \sqrt{211}-85877 \,\mathrm{I} \sqrt{3}-85877\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(-28 \,\mathrm{I}+3 \sqrt{211}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{211}-28\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{3468}+\frac{\mathrm{I} \sqrt{3}\, \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{6}\right)^{-n}}{19757196}+\frac{\left(\left(29206 \sqrt{211}\, \sqrt{3}-208046\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}-670 \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{211}\, \sqrt{3}+171754 \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}+975664\right) \left(\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{6}+\frac{14 \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{867}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{211}\, \sqrt{3}}{578}\right)^{-n}}{19757196}-\frac{n^{2}}{6}-\frac{11 n}{18}+\frac{23}{27}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(n+3) = 1/2*n^2+2*a(n)-a(n+1)+3*a(n+2)+5/2*n-1, 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) = 64
a \! \left(n +3\right) = \frac{n^{2}}{2}+2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{5 n}{2}-1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15308/
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[20,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[19,x]
F[19,x] = F[10,x]+F[17,x]
F[20,x] = F[21,x]+F[40,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[29,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[9,x]
F[28,x] = F[14,x]
F[29,x] = F[21,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[21,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[12,x]*F[33,x]
F[36,x] = F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[21,x]+F[37,x]
F[40,x] = F[41,x]+F[42,x]+F[93,x]
F[41,x] = 0
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[61,x]
F[44,x] = F[45,x]+F[6,x]
F[45,x] = F[41,x]+F[46,x]+F[52,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[10,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[48,x]
F[51,x] = F[17,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[13,x]+F[54,x]
F[54,x] = F[55,x]+F[58,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[14,x]+F[55,x]
F[58,x] = F[59,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[17,x]+F[58,x]
F[61,x] = F[40,x]+F[62,x]
F[62,x] = 2*F[41,x]+F[63,x]+F[84,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[65,x]+F[83,x]
F[65,x] = F[66,x]+F[81,x]
F[66,x] = F[41,x]+F[67,x]+F[79,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[69,x]+F[72,x]
F[69,x] = F[10,x]+F[70,x]
F[70,x] = F[41,x]+F[46,x]+F[71,x]
F[71,x] = F[12,x]*F[13,x]
F[72,x] = F[66,x]+F[73,x]
F[73,x] = 2*F[41,x]+F[63,x]+F[74,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[37,x]+F[76,x]
F[76,x] = F[77,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[66,x]+F[76,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[21,x]+F[66,x]
F[81,x] = F[82,x]
F[82,x] = F[12,x]*F[65,x]
F[83,x] = F[76,x]
F[84,x] = F[12,x]*F[85,x]
F[85,x] = F[75,x]+F[86,x]
F[86,x] = F[87,x]+F[90,x]
F[87,x] = F[88,x]
F[88,x] = F[12,x]*F[89,x]
F[89,x] = F[37,x]+F[87,x]
F[90,x] = F[91,x]
F[91,x] = F[12,x]*F[92,x]
F[92,x] = F[76,x]+F[90,x]
F[93,x] = F[12,x]*F[94,x]
F[94,x] = F[80,x]+F[95,x]
F[95,x] = F[120,x]+F[96,x]
F[96,x] = F[118,x]+F[41,x]+F[97,x]
F[97,x] = F[12,x]*F[98,x]
F[98,x] = F[106,x]+F[99,x]
F[99,x] = F[100,x]+F[14,x]
F[100,x] = F[101,x]+F[41,x]+F[56,x]
F[101,x] = F[102,x]*F[12,x]
F[102,x] = F[103,x]
F[103,x] = F[104,x]+F[12,x]
F[104,x] = F[105,x]
F[105,x] = F[103,x]*F[12,x]
F[106,x] = F[107,x]+F[37,x]
F[107,x] = 2*F[41,x]+F[108,x]+F[115,x]
F[108,x] = F[109,x]*F[12,x]
F[109,x] = F[110,x]
F[110,x] = F[111,x]+F[113,x]
F[111,x] = F[112,x]
F[112,x] = F[12,x]*F[21,x]
F[113,x] = F[114,x]
F[114,x] = F[110,x]*F[12,x]
F[115,x] = F[116,x]*F[12,x]
F[116,x] = F[117,x]+F[37,x]
F[117,x] = F[115,x]
F[118,x] = F[119,x]*F[12,x]
F[119,x] = F[21,x]+F[96,x]
F[120,x] = 2*F[41,x]+F[121,x]+F[144,x]
F[121,x] = F[12,x]*F[122,x]
F[122,x] = F[123,x]+F[132,x]
F[123,x] = F[124,x]+F[17,x]
F[124,x] = 2*F[41,x]+F[125,x]+F[59,x]
F[125,x] = F[12,x]*F[126,x]
F[126,x] = F[127,x]
F[127,x] = F[128,x]+F[130,x]
F[128,x] = F[129,x]
F[129,x] = F[10,x]*F[12,x]
F[130,x] = F[131,x]
F[131,x] = F[12,x]*F[127,x]
F[132,x] = F[133,x]+F[76,x]
F[133,x] = 3*F[41,x]+F[134,x]+F[141,x]
F[134,x] = F[12,x]*F[135,x]
F[135,x] = F[136,x]
F[136,x] = F[137,x]+F[139,x]
F[137,x] = F[138,x]
F[138,x] = F[12,x]*F[66,x]
F[139,x] = F[140,x]
F[140,x] = F[12,x]*F[136,x]
F[141,x] = F[12,x]*F[142,x]
F[142,x] = F[143,x]+F[76,x]
F[143,x] = F[141,x]
F[144,x] = F[12,x]*F[145,x]
F[145,x] = F[120,x]+F[66,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_{20}\! \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_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{40}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{29}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{9}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{21}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{21}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{93}\! \left(x \right)
F_{41}\! \left(x \right) = 0
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_{61}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{6}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)+F_{52}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{17}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{13}\! \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_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{14}\! \left(x \right)+F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{17}\! \left(x \right)+F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{40}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{63}\! \left(x \right)+F_{84}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{83}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{81}\! \left(x \right)
F_{66}\! \left(x \right) = F_{41}\! \left(x \right)+F_{67}\! \left(x \right)+F_{79}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{72}\! \left(x \right)
F_{69}\! \left(x \right) = F_{10}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{72}\! \left(x \right) = F_{66}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{63}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{37}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{66}\! \left(x \right)+F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{21}\! \left(x \right)+F_{66}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{83}\! \left(x \right) = F_{76}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{75}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)+F_{90}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{12}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{37}\! \left(x \right)+F_{87}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{12}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{76}\! \left(x \right)+F_{90}\! \left(x \right)
F_{93}\! \left(x \right) = F_{12}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{80}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{120}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{118}\! \left(x \right)+F_{41}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{12}\! \left(x \right) F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{106}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{14}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{41}\! \left(x \right)+F_{56}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{12}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{12}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)
F_{105}\! \left(x \right) = F_{103}\! \left(x \right) F_{12}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{37}\! \left(x \right)
F_{107}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{108}\! \left(x \right)+F_{115}\! \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_{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_{21}\! \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_{12}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)+F_{37}\! \left(x \right)
F_{117}\! \left(x \right) = F_{115}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right) F_{12}\! \left(x \right)
F_{119}\! \left(x \right) = F_{21}\! \left(x \right)+F_{96}\! \left(x \right)
F_{120}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{121}\! \left(x \right)+F_{144}\! \left(x \right)
F_{121}\! \left(x \right) = F_{12}\! \left(x \right) F_{122}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)+F_{132}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)+F_{17}\! \left(x \right)
F_{124}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{125}\! \left(x \right)+F_{59}\! \left(x \right)
F_{125}\! \left(x \right) = F_{12}\! \left(x \right) F_{126}\! \left(x \right)
F_{126}\! \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_{129}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right)
F_{131}\! \left(x \right) = F_{12}\! \left(x \right) F_{127}\! \left(x \right)
F_{132}\! \left(x \right) = F_{133}\! \left(x \right)+F_{76}\! \left(x \right)
F_{133}\! \left(x \right) = 3 F_{41}\! \left(x \right)+F_{134}\! \left(x \right)+F_{141}\! \left(x \right)
F_{134}\! \left(x \right) = F_{12}\! \left(x \right) F_{135}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right)
F_{136}\! \left(x \right) = F_{137}\! \left(x \right)+F_{139}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right)
F_{138}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{139}\! \left(x \right) = F_{140}\! \left(x \right)
F_{140}\! \left(x \right) = F_{12}\! \left(x \right) F_{136}\! \left(x \right)
F_{141}\! \left(x \right) = F_{12}\! \left(x \right) F_{142}\! \left(x \right)
F_{142}\! \left(x \right) = F_{143}\! \left(x \right)+F_{76}\! \left(x \right)
F_{143}\! \left(x \right) = F_{141}\! \left(x \right)
F_{144}\! \left(x \right) = F_{12}\! \left(x \right) F_{145}\! \left(x \right)
F_{145}\! \left(x \right) = F_{120}\! \left(x \right)+F_{66}\! \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_20(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_19(x))
Eq(F_19(x), F_10(x) + F_17(x))
Eq(F_20(x), F_21(x) + F_40(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_29(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_9(x))
Eq(F_28(x), F_14(x))
Eq(F_29(x), F_21(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_21(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_12(x)*F_33(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_21(x) + F_37(x))
Eq(F_40(x), F_41(x) + F_42(x) + F_93(x))
Eq(F_41(x), 0)
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_61(x))
Eq(F_44(x), F_45(x) + F_6(x))
Eq(F_45(x), F_41(x) + F_46(x) + F_52(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_10(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_48(x))
Eq(F_51(x), F_17(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_13(x) + F_54(x))
Eq(F_54(x), F_55(x) + F_58(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_14(x) + F_55(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_17(x) + F_58(x))
Eq(F_61(x), F_40(x) + F_62(x))
Eq(F_62(x), 2*F_41(x) + F_63(x) + F_84(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_83(x))
Eq(F_65(x), F_66(x) + F_81(x))
Eq(F_66(x), F_41(x) + F_67(x) + F_79(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_72(x))
Eq(F_69(x), F_10(x) + F_70(x))
Eq(F_70(x), F_41(x) + F_46(x) + F_71(x))
Eq(F_71(x), F_12(x)*F_13(x))
Eq(F_72(x), F_66(x) + F_73(x))
Eq(F_73(x), 2*F_41(x) + F_63(x) + F_74(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_37(x) + F_76(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_12(x)*F_78(x))
Eq(F_78(x), F_66(x) + F_76(x))
Eq(F_79(x), F_12(x)*F_80(x))
Eq(F_80(x), F_21(x) + F_66(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_12(x)*F_65(x))
Eq(F_83(x), F_76(x))
Eq(F_84(x), F_12(x)*F_85(x))
Eq(F_85(x), F_75(x) + F_86(x))
Eq(F_86(x), F_87(x) + F_90(x))
Eq(F_87(x), F_88(x))
Eq(F_88(x), F_12(x)*F_89(x))
Eq(F_89(x), F_37(x) + F_87(x))
Eq(F_90(x), F_91(x))
Eq(F_91(x), F_12(x)*F_92(x))
Eq(F_92(x), F_76(x) + F_90(x))
Eq(F_93(x), F_12(x)*F_94(x))
Eq(F_94(x), F_80(x) + F_95(x))
Eq(F_95(x), F_120(x) + F_96(x))
Eq(F_96(x), F_118(x) + F_41(x) + F_97(x))
Eq(F_97(x), F_12(x)*F_98(x))
Eq(F_98(x), F_106(x) + F_99(x))
Eq(F_99(x), F_100(x) + F_14(x))
Eq(F_100(x), F_101(x) + F_41(x) + F_56(x))
Eq(F_101(x), F_102(x)*F_12(x))
Eq(F_102(x), F_103(x))
Eq(F_103(x), F_104(x) + F_12(x))
Eq(F_104(x), F_105(x))
Eq(F_105(x), F_103(x)*F_12(x))
Eq(F_106(x), F_107(x) + F_37(x))
Eq(F_107(x), F_108(x) + F_115(x) + 2*F_41(x))
Eq(F_108(x), F_109(x)*F_12(x))
Eq(F_109(x), F_110(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_21(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_12(x))
Eq(F_116(x), F_117(x) + F_37(x))
Eq(F_117(x), F_115(x))
Eq(F_118(x), F_119(x)*F_12(x))
Eq(F_119(x), F_21(x) + F_96(x))
Eq(F_120(x), F_121(x) + F_144(x) + 2*F_41(x))
Eq(F_121(x), F_12(x)*F_122(x))
Eq(F_122(x), F_123(x) + F_132(x))
Eq(F_123(x), F_124(x) + F_17(x))
Eq(F_124(x), F_125(x) + 2*F_41(x) + F_59(x))
Eq(F_125(x), F_12(x)*F_126(x))
Eq(F_126(x), F_127(x))
Eq(F_127(x), F_128(x) + F_130(x))
Eq(F_128(x), F_129(x))
Eq(F_129(x), F_10(x)*F_12(x))
Eq(F_130(x), F_131(x))
Eq(F_131(x), F_12(x)*F_127(x))
Eq(F_132(x), F_133(x) + F_76(x))
Eq(F_133(x), F_134(x) + F_141(x) + 3*F_41(x))
Eq(F_134(x), F_12(x)*F_135(x))
Eq(F_135(x), F_136(x))
Eq(F_136(x), F_137(x) + F_139(x))
Eq(F_137(x), F_138(x))
Eq(F_138(x), F_12(x)*F_66(x))
Eq(F_139(x), F_140(x))
Eq(F_140(x), F_12(x)*F_136(x))
Eq(F_141(x), F_12(x)*F_142(x))
Eq(F_142(x), F_143(x) + F_76(x))
Eq(F_143(x), F_141(x))
Eq(F_144(x), F_12(x)*F_145(x))
Eq(F_145(x), F_120(x) + F_66(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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "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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