0132_0231_0321_1023_3012
Counting sequence:
1, 1, 2, 6, 19, 53, 143, 384, 1022, 2706, 7141, 18799, 49407, 129700, 340204, 891852, 2337081, 6122571, 16036481, 41997630, 109976196, 287967352, 753992799, 1974134165, 5168636149, 13532190794, 35428702318, 92755325210, 242839864959, 635769036449, 1664476011867, 4357675125060, 11408579023482, 29868116498942, 78195870812977, 204719680493347, 535963510113611, 1403171474187024, 3673552060786904, 9617486820299216, 25178912284915253, 65919257179716023, 172578872396432333, 451817384181854482, 1182873324608873616, 3096802671418981892, 8107534840054303595, 21225802125384118457, 55569872044918688401, 145483814945239004470, 380881574512126208922, 997160911757155200558, 2610601166582549912651, 6834642598701048519469, 17893326649220375725991, 46845337385193623240712, 122642685573004372640662, 321082719456396697988234, 840605473021640347857725, 2200733700023200054070103, 5761595627810667352678391, 15084053184811639877309724, 39490563929204473399406404, 103387638607547546852735572, 270672351902166992684126673, 708629417115008244376952515, 1855215899472387145681191385, 4857018281356466236603716582, 12715838944696908826478821884, 33290498552917999954353168048, 87155656714395041054387059703, 228176471590888710200483671005, 597373758059414366268066409677, 1563944802589457202334200051778, 4094460649712824918177696355718, 10719437146556131320093538575554, 28063850789968653301171216034935, 73472115223373894287826218363481, 192352494880197293293676494113907, 503585369417299399288046724534860, 1318403613371850647701082303939922, 3451625470698527964372031327349174, 9036472798724239822797304903177737, 23657792925475123245089626371767691, 61936905977702843651480441566843395, 162152925007636559766812601898134344, 424521869045212633186496878041233200, 1111412682128012003126687317063328728, 2909716177338842989122574775469561869, 7617735849888553038041595510417602799, 19943491372326882475065789244014064805, 52212738267092216423948917913248464698, 136694723428949991257438246176658266568, 357871432019758170195879825477507963644, 936919572630325278675165662489198063363, 2452887285871219062482752880765127231361, 6421742284983334477618707135675338703273, 16812339569078789095217082833138417394926, 44015276422253041498375005545261638075442, 115233489697680351383939726446914905014358, 301685192670788042052662144928150738260995
Generating function in Maple syntax:
(x-1)*(3*x^5+5*x^4+x^3-x^2-2*x+1)/(x^2-3*x+1)/(x^3+x^2+x-1)
Generating function in latex syntax:
\frac{\left(x -1\right) \left(3 x^{5}+5 x^{4}+x^{3}-x^{2}-2 x +1\right)}{\left(x^{2}-3 x +1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(x - 1)*(3*x**5 + 5*x**4 + x**3 - x**2 - 2*x + 1)/((x**2 - 3*x + 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)*(x^3+x^2+x-1)*F(x)+(x-1)*(3*x^5+5*x^4+x^3-x^2-2*x+1) = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}-3 x +1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+\left(x -1\right) \left(3 x^{5}+5 x^{4}+x^{3}-x^{2}-2 x +1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/1320*(((195*I-25*11^(1/2))*3^(1/2)-75*I*11^(1/2)+195)*(17+3*11^(1/2)*3^(1/2))^(1/3)-480+((-210*I-40*11^(1/2))*3^(1/2)+120*I*11^(1/2)+210)*(17+3*11^(1/2)*3^(1/2))^(2/3))*(1/24*((17*I+3*11^(1/2))*3^(1/2)-9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/1320*(((210*I-40*11^(1/2))*3^(1/2)-120*I*11^(1/2)+210)*(17+3*11^(1/2)*3^(1/2))^(2/3)-480+((-195*I-25*11^(1/2))*3^(1/2)+75*I*11^(1/2)+195)*(17+3*11^(1/2)*3^(1/2))^(1/3))*(1/24*((-17*I+3*11^(1/2))*3^(1/2)+9*I*11^(1/2)-17)*(17+3*11^(1/2)*3^(1/2))^(2/3)+1/6*I*3^(1/2)*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/6*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3)^(-n)+1/1320*((80*11^(1/2)*3^(1/2)-420)*(17+3*11^(1/2)*3^(1/2))^(2/3)+50*(17+3*11^(1/2)*3^(1/2))^(1/3)*11^(1/2)*3^(1/2)-390*(17+3*11^(1/2)*3^(1/2))^(1/3)-480)*(1/3*(17+3*11^(1/2)*3^(1/2))^(1/3)-1/3+17/12*(17+3*11^(1/2)*3^(1/2))^(2/3)-1/4*(17+3*11^(1/2)*3^(1/2))^(2/3)*11^(1/2)*3^(1/2))^(-n)+1/1320*(2028*5^(1/2)-3900)*(3/2-1/2*5^(1/2))^(-n)+1/1320*(-2028*5^(1/2)-3900)*(3/2+1/2*5^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(\left(195 \,\mathrm{I}-25 \sqrt{11}\right) \sqrt{3}-75 \,\mathrm{I} \sqrt{11}+195\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-480+\left(\left(-210 \,\mathrm{I}-40 \sqrt{11}\right) \sqrt{3}+120 \,\mathrm{I} \sqrt{11}+210\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{1320}\\+\\\frac{\left(\left(\left(210 \,\mathrm{I}-40 \sqrt{11}\right) \sqrt{3}-120 \,\mathrm{I} \sqrt{11}+210\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-480+\left(\left(-195 \,\mathrm{I}-25 \sqrt{11}\right) \sqrt{3}+75 \,\mathrm{I} \sqrt{11}+195\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{1320}\\+\\\frac{\left(\left(80 \sqrt{11}\, \sqrt{3}-420\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+50 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-390 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-480\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{1320}\\+\frac{\left(2028 \sqrt{5}-3900\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{1320}+\frac{\left(-2028 \sqrt{5}-3900\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{1320} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(6) = 143
a(n+5) = a(n)-2*a(n+1)-a(n+2)-3*a(n+3)+4*a(n+4), 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) = 19
a \! \left(5\right) = 53
a \! \left(6\right) = 143
a \! \left(n +5\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)-3 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18645/
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[22,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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[1,x]+F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]+F[4,x]
F[20,x] = F[21,x]
F[21,x] = x^2
F[22,x] = F[23,x]+F[62,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[16,x]*F[4,x]
F[29,x] = F[23,x]+F[30,x]
F[30,x] = F[31,x]+F[32,x]+F[60,x]
F[31,x] = 0
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[38,x]
F[34,x] = F[27,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[4,x]
F[38,x] = F[30,x]+F[39,x]
F[39,x] = 2*F[31,x]+F[40,x]+F[44,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[35,x]
F[43,x] = F[39,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[31,x]+F[47,x]+F[59,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[4,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = x^2
F[52,x] = F[46,x]+F[53,x]
F[53,x] = 2*F[31,x]+F[54,x]+F[58,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[50,x]
F[57,x] = F[53,x]
F[58,x] = F[4,x]*F[46,x]
F[59,x] = F[23,x]*F[4,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[23,x]+F[46,x]
F[62,x] = F[31,x]+F[63,x]+F[98,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[70,x]
F[65,x] = F[27,x]+F[66,x]
F[66,x] = F[31,x]+F[67,x]+F[68,x]
F[67,x] = F[19,x]*F[4,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[20,x]+F[4,x]
F[70,x] = F[71,x]+F[72,x]
F[71,x] = F[31,x]+F[60,x]+F[63,x]
F[72,x] = F[31,x]+F[73,x]+F[78,x]+F[94,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[75,x]+F[76,x]
F[75,x] = F[35,x]
F[76,x] = F[77,x]
F[77,x] = 2*F[31,x]+F[44,x]+F[73,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[80,x]+F[92,x]
F[80,x] = F[31,x]+F[59,x]+F[81,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[83,x]+F[85,x]
F[83,x] = F[4,x]+F[84,x]
F[84,x] = F[31,x]+F[51,x]+F[67,x]
F[85,x] = F[80,x]+F[86,x]
F[86,x] = F[31,x]+F[58,x]+F[78,x]+F[87,x]
F[87,x] = F[4,x]*F[88,x]
F[88,x] = F[89,x]+F[90,x]
F[89,x] = F[50,x]
F[90,x] = F[91,x]
F[91,x] = 2*F[31,x]+F[58,x]+F[87,x]
F[92,x] = F[93,x]
F[93,x] = F[4,x]*F[80,x]
F[94,x] = F[4,x]*F[95,x]
F[95,x] = F[46,x]+F[96,x]
F[96,x] = F[97,x]
F[97,x] = F[4,x]*F[46,x]
F[98,x] = F[4,x]*F[99,x]
F[99,x] = F[100,x]+F[103,x]
F[100,x] = F[101,x]+F[23,x]
F[101,x] = F[102,x]+F[31,x]+F[81,x]
F[102,x] = F[100,x]*F[4,x]
F[103,x] = F[104,x]+F[106,x]
F[104,x] = F[105,x]+F[31,x]+F[47,x]
F[105,x] = F[4,x]*F[61,x]
F[106,x] = F[107,x]
F[107,x] = F[4,x]*F[79,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_{22}\! \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_{17}\! \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_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = x^{2}
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{62}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{60}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{38}\! \left(x \right)
F_{34}\! \left(x \right) = F_{27}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{30}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \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_{35}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{31}\! \left(x \right)+F_{47}\! \left(x \right)+F_{59}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = x^{2}
F_{52}\! \left(x \right) = F_{46}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{54}\! \left(x \right)+F_{58}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{50}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{59}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{23}\! \left(x \right)+F_{46}\! \left(x \right)
F_{62}\! \left(x \right) = F_{31}\! \left(x \right)+F_{63}\! \left(x \right)+F_{98}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{70}\! \left(x \right)
F_{65}\! \left(x \right) = F_{27}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{31}\! \left(x \right)+F_{67}\! \left(x \right)+F_{68}\! \left(x \right)
F_{67}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{20}\! \left(x \right)+F_{4}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{72}\! \left(x \right)
F_{71}\! \left(x \right) = F_{31}\! \left(x \right)+F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{72}\! \left(x \right) = F_{31}\! \left(x \right)+F_{73}\! \left(x \right)+F_{78}\! \left(x \right)+F_{94}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{76}\! \left(x \right)
F_{75}\! \left(x \right) = F_{35}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{44}\! \left(x \right)+F_{73}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{92}\! \left(x \right)
F_{80}\! \left(x \right) = F_{31}\! \left(x \right)+F_{59}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{85}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{31}\! \left(x \right)+F_{51}\! \left(x \right)+F_{67}\! \left(x \right)
F_{85}\! \left(x \right) = F_{80}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{31}\! \left(x \right)+F_{58}\! \left(x \right)+F_{78}\! \left(x \right)+F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{90}\! \left(x \right)
F_{89}\! \left(x \right) = F_{50}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{58}\! \left(x \right)+F_{87}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{94}\! \left(x \right) = F_{4}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{46}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{98}\! \left(x \right) = F_{4}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{103}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{23}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{31}\! \left(x \right)+F_{81}\! \left(x \right)
F_{102}\! \left(x \right) = F_{100}\! \left(x \right) F_{4}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{106}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{31}\! \left(x \right)+F_{47}\! \left(x \right)
F_{105}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)
F_{107}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \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_22(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_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_1(x) + F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x) + F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), x**2)
Eq(F_22(x), F_23(x) + F_62(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_16(x)*F_4(x))
Eq(F_29(x), F_23(x) + F_30(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_60(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_38(x))
Eq(F_34(x), F_27(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_4(x))
Eq(F_38(x), F_30(x) + F_39(x))
Eq(F_39(x), 2*F_31(x) + F_40(x) + F_44(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_35(x))
Eq(F_43(x), F_39(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_31(x) + F_47(x) + F_59(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_4(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), x**2)
Eq(F_52(x), F_46(x) + F_53(x))
Eq(F_53(x), 2*F_31(x) + F_54(x) + F_58(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_50(x))
Eq(F_57(x), F_53(x))
Eq(F_58(x), F_4(x)*F_46(x))
Eq(F_59(x), F_23(x)*F_4(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_23(x) + F_46(x))
Eq(F_62(x), F_31(x) + F_63(x) + F_98(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_70(x))
Eq(F_65(x), F_27(x) + F_66(x))
Eq(F_66(x), F_31(x) + F_67(x) + F_68(x))
Eq(F_67(x), F_19(x)*F_4(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_20(x) + F_4(x))
Eq(F_70(x), F_71(x) + F_72(x))
Eq(F_71(x), F_31(x) + F_60(x) + F_63(x))
Eq(F_72(x), F_31(x) + F_73(x) + F_78(x) + F_94(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_75(x) + F_76(x))
Eq(F_75(x), F_35(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), 2*F_31(x) + F_44(x) + F_73(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_80(x) + F_92(x))
Eq(F_80(x), F_31(x) + F_59(x) + F_81(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_85(x))
Eq(F_83(x), F_4(x) + F_84(x))
Eq(F_84(x), F_31(x) + F_51(x) + F_67(x))
Eq(F_85(x), F_80(x) + F_86(x))
Eq(F_86(x), F_31(x) + F_58(x) + F_78(x) + F_87(x))
Eq(F_87(x), F_4(x)*F_88(x))
Eq(F_88(x), F_89(x) + F_90(x))
Eq(F_89(x), F_50(x))
Eq(F_90(x), F_91(x))
Eq(F_91(x), 2*F_31(x) + F_58(x) + F_87(x))
Eq(F_92(x), F_93(x))
Eq(F_93(x), F_4(x)*F_80(x))
Eq(F_94(x), F_4(x)*F_95(x))
Eq(F_95(x), F_46(x) + F_96(x))
Eq(F_96(x), F_97(x))
Eq(F_97(x), F_4(x)*F_46(x))
Eq(F_98(x), F_4(x)*F_99(x))
Eq(F_99(x), F_100(x) + F_103(x))
Eq(F_100(x), F_101(x) + F_23(x))
Eq(F_101(x), F_102(x) + F_31(x) + F_81(x))
Eq(F_102(x), F_100(x)*F_4(x))
Eq(F_103(x), F_104(x) + F_106(x))
Eq(F_104(x), F_105(x) + F_31(x) + F_47(x))
Eq(F_105(x), F_4(x)*F_61(x))
Eq(F_106(x), F_107(x))
Eq(F_107(x), F_4(x)*F_79(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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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": 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, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [3, 0, 1, 2], "pos": [[1, 2], [1, 0], [1, 0], [1, 2]]}, {"patt": [3, 0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": 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