0132_0231_3012_3102
Counting sequence:
1, 1, 2, 6, 20, 63, 186, 531, 1492, 4160, 11556, 32046, 88798, 245971, 681240, 1886639, 5224754, 14468984, 40068974, 110962846, 307288720, 850972479, 2356591774, 6526091795, 18072656344, 50048469288, 138598843616, 383820718230, 1062911780714, 2943513467907, 8151449341644, 22573746338175, 62513303141254, 173117612427728, 479413280480634, 1327635532155990, 3676610928415660, 10181610533572255, 28195856204457762, 78082569008217395, 216233461353767420, 598813671257543408, 1658290121427081020, 4592290154377467934, 12717394012962867094, 35218182005938215315, 97529442159229247776, 270087538484712396143, 747951355300846157050, 2071295969577055323912, 5736024091915032212006, 15884727661468887470382, 43989454862068685524328, 121819661016652201315935, 337354255849356805895238, 934232561393486901895539, 2587163089347756101108800, 7164610962499138207327640, 19840902359543145422771256, 54945259205478054162096582, 152159486219390155270847842, 421374101812235557073219971, 1166908078422794570110910484, 3231509619675538308530361215, 8948994882416055912553395086, 24782383105995423999240736544, 68629664055245754393699177842, 190055603942157895094410194214, 526319530877223354888772143556, 1457532652744757923965605416575, 4036334031299208312102454302730, 11177788972030090367230529637587, 30954565537535820713554740029092, 85722241671876580085536144754976, 237389948450124009910284223876052, 657402169216031270358871266904974, 1820538800869846381251865721596622, 5041604181843631883307010121763795, 13961675913877080539028035910602664, 38663962360657456115028963331643823, 107071815349938919689365864206095682, 296513159603036383492096665197249048, 821131625819827686901953094717298526, 2273953533206385596903128483160745534, 6297242133402365487299529019962190528, 17438904492820538551897411671443128063, 48293424878265635765295834477527942766, 133738612275378734231289620781102894419, 370361316440691105480470439537223872296, 1025638761924960217975417532308096669064, 2840293581609568282677071778168169033232, 7865603299344435735536268241733634306934, 21782155078348699141907150479340830560730, 60321155517311229972862254974457026412675, 167046914772929426512215882685763882988508, 462601743879825748705692543562175453117951, 1281079472383859049577724002975219502782486, 3547683588044680826539695348049246938222576, 9824573035630009178747054584734696765007850, 27207114991229205759279192409130062859588214, 75344455526102288925630217990704738751984220
Generating function in Maple syntax:
(x^7-x^6-2*x^5+5*x^4-7*x^3+9*x^2-5*x+1)/(x^3-x^2+3*x-1)/(x-1)^3
Generating function in latex syntax:
\frac{x^{7}-x^{6}-2 x^{5}+5 x^{4}-7 x^{3}+9 x^{2}-5 x +1}{\left(x^{3}-x^{2}+3 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**7 - x**6 - 2*x**5 + 5*x**4 - 7*x**3 + 9*x**2 - 5*x + 1)/((x - 1)**3*(x**3 - x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^3-x^2+3*x-1)*(x-1)^3*F(x)+x^7-x^6-2*x^5+5*x^4-7*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{3}-x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{7}-x^{6}-2 x^{5}+5 x^{4}-7 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/29184*(-960*(I+1/3*3^(1/2))*19^(1/2)*(1+3*19^(1/2)*3^(1/2))^(1/3)-14592+((855*I+5*19^(1/2))*3^(1/2)-15*I*19^(1/2)-855)*(1+3*19^(1/2)*3^(1/2))^(2/3))*(1/384*((I+3*19^(1/2))*3^(1/2)-9*I*19^(1/2)-1)*(1+3*19^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(1+3*19^(1/2)*3^(1/2))^(1/3)-1/6*(1+3*19^(1/2)*3^(1/2))^(1/3)+1/3)^(-n)+1/29184*(960*(I-1/3*3^(1/2))*19^(1/2)*(1+3*19^(1/2)*3^(1/2))^(1/3)-14592+((-855*I+5*19^(1/2))*3^(1/2)+15*I*19^(1/2)-855)*(1+3*19^(1/2)*3^(1/2))^(2/3))*(1/384*((-I+3*19^(1/2))*3^(1/2)+9*I*19^(1/2)-1)*(1+3*19^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(1+3*19^(1/2)*3^(1/2))^(1/3)-1/6*(1+3*19^(1/2)*3^(1/2))^(1/3)+1/3)^(-n)+1/29184*(-10*(1+3*19^(1/2)*3^(1/2))^(2/3)*19^(1/2)*3^(1/2)+1710*(1+3*19^(1/2)*3^(1/2))^(2/3)+640*(1+3*19^(1/2)*3^(1/2))^(1/3)*19^(1/2)*3^(1/2)-14592)*(1/3*(1+3*19^(1/2)*3^(1/2))^(1/3)+1/3+1/192*(1+3*19^(1/2)*3^(1/2))^(2/3)-1/64*(1+3*19^(1/2)*3^(1/2))^(2/3)*19^(1/2)*3^(1/2))^(-n)-1/4*n^2-1/4*n-1/2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(-960 \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \sqrt{19}\, \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}-14592+\left(\left(855 \,\mathrm{I}+5 \sqrt{19}\right) \sqrt{3}-15 \,\mathrm{I} \sqrt{19}-855\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(\mathrm{I}+3 \sqrt{19}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{19}-1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}\, \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}\right)^{-n}}{29184}\\+\\\frac{\left(960 \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \sqrt{19}\, \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}-14592+\left(\left(-855 \,\mathrm{I}+5 \sqrt{19}\right) \sqrt{3}+15 \,\mathrm{I} \sqrt{19}-855\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(-\mathrm{I}+3 \sqrt{19}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{19}-1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}+\frac{\mathrm{I} \sqrt{3}\, \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}\right)^{-n}}{29184}\\+\\\frac{\left(-10 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{19}\, \sqrt{3}+1710 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}+640 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{19}\, \sqrt{3}-14592\right) \left(\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}+\frac{1}{3}+\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{192}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{19}\, \sqrt{3}}{64}\right)^{-n}}{29184}\\-\frac{n^{2}}{4}-\frac{n}{4}-\frac{1}{2} & \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) = 63
a(6) = 186
a(7) = 531
a(n+3) = 1/2*n^2+a(n)-a(n+1)+3*a(n+2)+3/2*n+2, n >= 8
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 20
a \! \left(5\right) = 63
a \! \left(6\right) = 186
a \! \left(7\right) = 531
a \! \left(n +3\right) = \frac{n^{2}}{2}+a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{3 n}{2}+2, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15407/
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[26,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[19,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[4,x]
F[18,x] = F[14,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[11,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[11,x]*F[4,x]
F[25,x] = F[19,x]
F[26,x] = F[27,x]+F[36,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[1,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[30,x]*F[4,x]
F[33,x] = F[27,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[33,x]*F[4,x]
F[36,x] = F[101,x]+F[37,x]+F[38,x]
F[37,x] = 0
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[63,x]
F[40,x] = F[41,x]+F[7,x]
F[41,x] = F[37,x]+F[42,x]+F[45,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[11,x]+F[44,x]
F[44,x] = F[42,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[31,x]+F[44,x]
F[48,x] = F[49,x]+F[56,x]
F[49,x] = F[37,x]+F[50,x]+F[52,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[4,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[31,x]
F[55,x] = F[49,x]
F[56,x] = 2*F[37,x]+F[57,x]+F[59,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[23,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[44,x]
F[62,x] = F[56,x]
F[63,x] = F[36,x]+F[64,x]
F[64,x] = 2*F[37,x]+F[65,x]+F[80,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[77,x]
F[67,x] = F[37,x]+F[68,x]+F[78,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[73,x]
F[70,x] = F[11,x]+F[71,x]
F[71,x] = F[37,x]+F[42,x]+F[72,x]
F[72,x] = F[4,x]*F[47,x]
F[73,x] = F[67,x]+F[74,x]
F[74,x] = 2*F[37,x]+F[65,x]+F[75,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[34,x]+F[77,x]
F[77,x] = F[65,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[27,x]+F[67,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[76,x]+F[82,x]
F[82,x] = F[83,x]+F[92,x]
F[83,x] = 2*F[37,x]+F[84,x]+F[88,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[86,x]
F[86,x] = F[87,x]
F[87,x] = F[27,x]*F[4,x]
F[88,x] = F[4,x]*F[89,x]
F[89,x] = F[90,x]+F[91,x]
F[90,x] = F[34,x]
F[91,x] = F[83,x]
F[92,x] = 3*F[37,x]+F[93,x]+F[97,x]
F[93,x] = F[4,x]*F[94,x]
F[94,x] = F[95,x]
F[95,x] = F[96,x]
F[96,x] = F[4,x]*F[67,x]
F[97,x] = F[4,x]*F[98,x]
F[98,x] = F[100,x]+F[99,x]
F[99,x] = F[77,x]
F[100,x] = F[92,x]
F[101,x] = F[102,x]*F[4,x]
F[102,x] = F[103,x]+F[79,x]
F[103,x] = F[104,x]+F[119,x]
F[104,x] = F[105,x]+F[115,x]+F[37,x]
F[105,x] = F[106,x]*F[4,x]
F[106,x] = F[107,x]+F[111,x]
F[107,x] = F[108,x]+F[31,x]
F[108,x] = F[109,x]
F[109,x] = F[110,x]*F[4,x]
F[110,x] = F[108,x]+F[31,x]
F[111,x] = F[112,x]+F[34,x]
F[112,x] = F[113,x]
F[113,x] = F[114,x]*F[4,x]
F[114,x] = F[112,x]+F[34,x]
F[115,x] = F[116,x]*F[4,x]
F[116,x] = F[117,x]+F[118,x]
F[117,x] = F[27,x]+F[86,x]
F[118,x] = F[104,x]
F[119,x] = 2*F[37,x]+F[120,x]+F[132,x]
F[120,x] = F[121,x]*F[4,x]
F[121,x] = F[122,x]+F[127,x]
F[122,x] = F[123,x]+F[44,x]
F[123,x] = F[124,x]
F[124,x] = F[125,x]*F[4,x]
F[125,x] = F[126,x]+F[44,x]
F[126,x] = F[124,x]
F[127,x] = F[128,x]+F[77,x]
F[128,x] = F[129,x]
F[129,x] = F[130,x]*F[4,x]
F[130,x] = F[131,x]+F[77,x]
F[131,x] = F[129,x]
F[132,x] = F[133,x]*F[4,x]
F[133,x] = F[134,x]+F[135,x]
F[134,x] = F[67,x]+F[95,x]
F[135,x] = F[119,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_{26}\! \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_{19}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{19}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{36}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{101}\! \left(x \right)+F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = 0
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{63}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{7}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)+F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{11}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{31}\! \left(x \right)+F_{44}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{56}\! \left(x \right)
F_{49}\! \left(x \right) = F_{37}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{31}\! \left(x \right)
F_{55}\! \left(x \right) = F_{49}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{37}\! \left(x \right)+F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{23}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{44}\! \left(x \right)
F_{62}\! \left(x \right) = F_{56}\! \left(x \right)
F_{63}\! \left(x \right) = F_{36}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{37}\! \left(x \right)+F_{65}\! \left(x \right)+F_{80}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{77}\! \left(x \right)
F_{67}\! \left(x \right) = F_{37}\! \left(x \right)+F_{68}\! \left(x \right)+F_{78}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{73}\! \left(x \right)
F_{70}\! \left(x \right) = F_{11}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{37}\! \left(x \right)+F_{42}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{73}\! \left(x \right) = F_{67}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = 2 F_{37}\! \left(x \right)+F_{65}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{34}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{65}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{27}\! \left(x \right)+F_{67}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{76}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{92}\! \left(x \right)
F_{83}\! \left(x \right) = 2 F_{37}\! \left(x \right)+F_{84}\! \left(x \right)+F_{88}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{90}\! \left(x \right)+F_{91}\! \left(x \right)
F_{90}\! \left(x \right) = F_{34}\! \left(x \right)
F_{91}\! \left(x \right) = F_{83}\! \left(x \right)
F_{92}\! \left(x \right) = 3 F_{37}\! \left(x \right)+F_{93}\! \left(x \right)+F_{97}\! \left(x \right)
F_{93}\! \left(x \right) = F_{4}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{97}\! \left(x \right) = F_{4}\! \left(x \right) F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{100}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{77}\! \left(x \right)
F_{100}\! \left(x \right) = F_{92}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{4}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{79}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{119}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{115}\! \left(x \right)+F_{37}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{4}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{111}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)+F_{31}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right) F_{4}\! \left(x \right)
F_{110}\! \left(x \right) = F_{108}\! \left(x \right)+F_{31}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)+F_{34}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right) F_{4}\! \left(x \right)
F_{114}\! \left(x \right) = F_{112}\! \left(x \right)+F_{34}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right) F_{4}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)+F_{118}\! \left(x \right)
F_{117}\! \left(x \right) = F_{27}\! \left(x \right)+F_{86}\! \left(x \right)
F_{118}\! \left(x \right) = F_{104}\! \left(x \right)
F_{119}\! \left(x \right) = 2 F_{37}\! \left(x \right)+F_{120}\! \left(x \right)+F_{132}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right) F_{4}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)+F_{127}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)+F_{44}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right) F_{4}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)+F_{44}\! \left(x \right)
F_{126}\! \left(x \right) = F_{124}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)+F_{77}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right) F_{4}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right)+F_{77}\! \left(x \right)
F_{131}\! \left(x \right) = F_{129}\! \left(x \right)
F_{132}\! \left(x \right) = F_{133}\! \left(x \right) F_{4}\! \left(x \right)
F_{133}\! \left(x \right) = F_{134}\! \left(x \right)+F_{135}\! \left(x \right)
F_{134}\! \left(x \right) = F_{67}\! \left(x \right)+F_{95}\! \left(x \right)
F_{135}\! \left(x \right) = F_{119}\! \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_26(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_19(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_4(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_11(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_11(x)*F_4(x))
Eq(F_25(x), F_19(x))
Eq(F_26(x), F_27(x) + F_36(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_1(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_30(x)*F_4(x))
Eq(F_33(x), F_27(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_33(x)*F_4(x))
Eq(F_36(x), F_101(x) + F_37(x) + F_38(x))
Eq(F_37(x), 0)
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_63(x))
Eq(F_40(x), F_41(x) + F_7(x))
Eq(F_41(x), F_37(x) + F_42(x) + F_45(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_11(x) + F_44(x))
Eq(F_44(x), F_42(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_31(x) + F_44(x))
Eq(F_48(x), F_49(x) + F_56(x))
Eq(F_49(x), F_37(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_4(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_31(x))
Eq(F_55(x), F_49(x))
Eq(F_56(x), 2*F_37(x) + F_57(x) + F_59(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_23(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_44(x))
Eq(F_62(x), F_56(x))
Eq(F_63(x), F_36(x) + F_64(x))
Eq(F_64(x), 2*F_37(x) + F_65(x) + F_80(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_77(x))
Eq(F_67(x), F_37(x) + F_68(x) + F_78(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_73(x))
Eq(F_70(x), F_11(x) + F_71(x))
Eq(F_71(x), F_37(x) + F_42(x) + F_72(x))
Eq(F_72(x), F_4(x)*F_47(x))
Eq(F_73(x), F_67(x) + F_74(x))
Eq(F_74(x), 2*F_37(x) + F_65(x) + F_75(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_34(x) + F_77(x))
Eq(F_77(x), F_65(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_27(x) + F_67(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_76(x) + F_82(x))
Eq(F_82(x), F_83(x) + F_92(x))
Eq(F_83(x), 2*F_37(x) + F_84(x) + F_88(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_86(x))
Eq(F_86(x), F_87(x))
Eq(F_87(x), F_27(x)*F_4(x))
Eq(F_88(x), F_4(x)*F_89(x))
Eq(F_89(x), F_90(x) + F_91(x))
Eq(F_90(x), F_34(x))
Eq(F_91(x), F_83(x))
Eq(F_92(x), 3*F_37(x) + F_93(x) + F_97(x))
Eq(F_93(x), F_4(x)*F_94(x))
Eq(F_94(x), F_95(x))
Eq(F_95(x), F_96(x))
Eq(F_96(x), F_4(x)*F_67(x))
Eq(F_97(x), F_4(x)*F_98(x))
Eq(F_98(x), F_100(x) + F_99(x))
Eq(F_99(x), F_77(x))
Eq(F_100(x), F_92(x))
Eq(F_101(x), F_102(x)*F_4(x))
Eq(F_102(x), F_103(x) + F_79(x))
Eq(F_103(x), F_104(x) + F_119(x))
Eq(F_104(x), F_105(x) + F_115(x) + F_37(x))
Eq(F_105(x), F_106(x)*F_4(x))
Eq(F_106(x), F_107(x) + F_111(x))
Eq(F_107(x), F_108(x) + F_31(x))
Eq(F_108(x), F_109(x))
Eq(F_109(x), F_110(x)*F_4(x))
Eq(F_110(x), F_108(x) + F_31(x))
Eq(F_111(x), F_112(x) + F_34(x))
Eq(F_112(x), F_113(x))
Eq(F_113(x), F_114(x)*F_4(x))
Eq(F_114(x), F_112(x) + F_34(x))
Eq(F_115(x), F_116(x)*F_4(x))
Eq(F_116(x), F_117(x) + F_118(x))
Eq(F_117(x), F_27(x) + F_86(x))
Eq(F_118(x), F_104(x))
Eq(F_119(x), F_120(x) + F_132(x) + 2*F_37(x))
Eq(F_120(x), F_121(x)*F_4(x))
Eq(F_121(x), F_122(x) + F_127(x))
Eq(F_122(x), F_123(x) + F_44(x))
Eq(F_123(x), F_124(x))
Eq(F_124(x), F_125(x)*F_4(x))
Eq(F_125(x), F_126(x) + F_44(x))
Eq(F_126(x), F_124(x))
Eq(F_127(x), F_128(x) + F_77(x))
Eq(F_128(x), F_129(x))
Eq(F_129(x), F_130(x)*F_4(x))
Eq(F_130(x), F_131(x) + F_77(x))
Eq(F_131(x), F_129(x))
Eq(F_132(x), F_133(x)*F_4(x))
Eq(F_133(x), F_134(x) + F_135(x))
Eq(F_134(x), F_67(x) + F_95(x))
Eq(F_135(x), F_119(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": [3, 0, 1, 2], "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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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, 1, 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