0231_0312_2130
Counting sequence:
1, 1, 2, 6, 21, 75, 265, 929, 3249, 11362, 39746, 139060, 486549, 1702349, 5956172, 20839367, 72912441, 255104933, 892557394, 3122866871, 10926241348, 38228574601, 133753581322, 467975086412, 1637344431204, 5728716901208, 20043551442825, 70128086510956, 245363129966728, 858472953449577, 3003612693983412, 10508996444450101, 36768724040454751, 128645877340380916, 450104326124228902, 1574818475213996989, 5509951995419908613, 19278139969564632791, 67450075970725861751, 235993345604878635951, 825690088087596246288, 2888912481064502160946, 10107685006344489739868, 35364621412773117755645, 123733223471422658905975, 432916004159437490267006, 1514680224108601129275459, 5299541156396557241432355, 18541957583733054435079495, 64874331737566252111173616, 226981369113263934299322623, 794159115703047545798567110, 2778592373101477571275743577, 9721698615803061565730888040, 34014137838798239280773015624, 119008171168366883981960023728, 416384059827156414995129605048, 1456838497525196101014196964001, 5097165363997082743790199626190, 17833888102261773580896531928008, 62396948525638626267092538979221, 218313536733328530109979393700742, 763832229735253421644693808724765, 2672485105195306751279024142727701, 9350452048830500355345892567831354, 32715225745323283043715301384452426, 114463556411834489765010227889550573, 400483427760487459598944903058040487, 1401205597122325864601049270407257161, 4902517780538844841457143832572331369, 17152857965925811992770209424348577601, 60014170181527879831120511880059139922, 209976706490089050284416077292226713638, 734663449233124556454921229984155380524, 2570429799862525146005936994125461404237, 8993382429625572385280023347885844874497, 31465915750674745371647496195730251201044, 110092488757845772216156766620769661397875, 385189999774168103526230248163602324298285, 1347697173531740135229243536983681666786037, 4715303285677995708026131256182372044977410, 16497834611954878083321565215887994767123515, 57722383989618778912654757479566521446838816, 201958238266651807972560237643375540383839109, 706608549139362681853757139911888366310496134, 2472271723115347248524428574946042895508048424, 8649948377160447017152161880435967868398335972, 30264313678779937915795964533737154505473688648, 105888340890684253236617433509661915596187592041, 370480588312279354843838316789028852793881861696, 1296232098469756948274579694849143373865316050700, 4535238028954565543119821377930131953251308864465, 15867824908484615910889961411124211980372872916840, 55518115194576757399645035497398041102303536208425, 194245974639548243192944714111683586851215743044011, 679624993237410740893053008473208135278856834545796, 2377862049857430243316146995876405163859512132819594, 8319629184350803802201546905458555954243147097606445, 29108597687259286823267183343213476037764884918843249, 101844738574707752274129074803443886681023276804206111, 356332891291788350475184529506277612190243738105629695
Generating function in Maple syntax:
-(2*x^4-7*x^3+8*x^2-5*x+1)/(x^5-6*x^4+13*x^3-12*x^2+6*x-1)
Generating function in latex syntax:
-\frac{2 x^{4}-7 x^{3}+8 x^{2}-5 x +1}{x^{5}-6 x^{4}+13 x^{3}-12 x^{2}+6 x -1}
Generating function in sympy syntax:
(-2*x**4 + 7*x**3 - 8*x**2 + 5*x - 1)/(x**5 - 6*x**4 + 13*x**3 - 12*x**2 + 6*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^5-6*x^4+13*x^3-12*x^2+6*x-1)*F(x)+2*x^4-7*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}-6 x^{4}+13 x^{3}-12 x^{2}+6 x -1\right) F \! \left(x \right)+2 x^{4}-7 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
-249/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n+3)-249/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n+3)-249/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n+3)-249/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n+3)-249/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n+3)+662/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n+2)+662/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n+2)+662/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n+2)+662/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n+2)+662/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n+2)-107/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n+1)-107/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n+1)-107/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n+1)-107/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n+1)-107/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n+1)+48/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n-1)+48/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n-1)+48/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n-1)+48/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n-1)+48/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n-1)+261/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 1)^(-n)+261/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 2)^(-n)+261/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 3)^(-n)+261/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 4)^(-n)+261/3089*RootOf(_Z^5-6*_Z^4+13*_Z^3-12*_Z^2+6*_Z-1,index = 5)^(-n)
Explicit closed form in latex syntax:
-\frac{249 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{3089}-\frac{249 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{3089}-\frac{249 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{3089}-\frac{249 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{3089}-\frac{249 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{3089}+\frac{662 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{3089}+\frac{662 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{3089}+\frac{662 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{3089}+\frac{662 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{3089}+\frac{662 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{3089}-\frac{107 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{3089}-\frac{107 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{3089}-\frac{107 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{3089}-\frac{107 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{3089}-\frac{107 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{3089}+\frac{48 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{3089}+\frac{48 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{3089}+\frac{48 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{3089}+\frac{48 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{3089}+\frac{48 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{3089}+\frac{261 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{3089}+\frac{261 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{3089}+\frac{261 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{3089}+\frac{261 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{3089}+\frac{261 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-6 \textit{\_Z}^{4}+13 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{3089}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(n+5) = a(n)-6*a(n+1)+13*a(n+2)-12*a(n+3)+6*a(n+4), n >= 5
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) = 21
a \! \left(n +5\right) = a \! \left(n \right)-6 a \! \left(n +1\right)+13 a \! \left(n +2\right)-12 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14752/
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[13,x]*F[4,x]
F[4,x] = F[17,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[13,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = F[1,x]+F[10,x]
F[13,x] = x
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = F[14,x]+F[6,x]
F[17,x] = F[18,x]+F[34,x]
F[18,x] = F[19,x]
F[19,x] = F[13,x]*F[20,x]
F[20,x] = F[21,x]+F[26,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[13,x]*F[24,x]
F[24,x] = F[21,x]+F[25,x]
F[25,x] = F[10,x]
F[26,x] = F[18,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[13,x]*F[29,x]
F[29,x] = F[26,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[13,x]*F[33,x]
F[33,x] = F[18,x]+F[31,x]
F[34,x] = F[35,x]+F[36,x]+F[70,x]
F[35,x] = 0
F[36,x] = F[13,x]*F[37,x]
F[37,x] = F[38,x]+F[52,x]
F[38,x] = F[39,x]+F[6,x]
F[39,x] = F[35,x]+F[40,x]+F[43,x]
F[40,x] = F[13,x]*F[41,x]
F[41,x] = F[38,x]+F[42,x]
F[42,x] = F[14,x]
F[43,x] = F[13,x]*F[44,x]
F[44,x] = F[45,x]+F[9,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[47,x]
F[47,x] = F[13,x]*F[48,x]
F[48,x] = F[10,x]+F[46,x]
F[49,x] = F[50,x]
F[50,x] = F[13,x]*F[51,x]
F[51,x] = F[14,x]+F[49,x]
F[52,x] = F[34,x]+F[53,x]
F[53,x] = 2*F[35,x]+F[54,x]+F[60,x]
F[54,x] = F[13,x]*F[55,x]
F[55,x] = F[52,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[13,x]*F[59,x]
F[59,x] = F[34,x]+F[57,x]
F[60,x] = F[13,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[31,x]+F[57,x]
F[63,x] = F[64,x]+F[67,x]
F[64,x] = F[65,x]
F[65,x] = F[13,x]*F[66,x]
F[66,x] = F[31,x]+F[64,x]
F[67,x] = F[68,x]
F[68,x] = F[13,x]*F[69,x]
F[69,x] = F[57,x]+F[67,x]
F[70,x] = F[13,x]*F[71,x]
F[71,x] = F[17,x]+F[72,x]
F[72,x] = F[73,x]+F[89,x]
F[73,x] = F[35,x]+F[74,x]+F[87,x]
F[74,x] = F[13,x]*F[75,x]
F[75,x] = F[76,x]+F[80,x]
F[76,x] = F[10,x]+F[77,x]
F[77,x] = F[35,x]+F[47,x]+F[78,x]
F[78,x] = F[13,x]*F[79,x]
F[79,x] = F[76,x]
F[80,x] = F[31,x]+F[81,x]
F[81,x] = 2*F[35,x]+F[82,x]+F[84,x]
F[82,x] = F[13,x]*F[83,x]
F[83,x] = F[80,x]
F[84,x] = F[13,x]*F[85,x]
F[85,x] = F[31,x]+F[86,x]
F[86,x] = F[84,x]
F[87,x] = F[13,x]*F[88,x]
F[88,x] = F[18,x]+F[73,x]
F[89,x] = 2*F[35,x]+F[103,x]+F[90,x]
F[90,x] = F[13,x]*F[91,x]
F[91,x] = F[92,x]+F[96,x]
F[92,x] = F[14,x]+F[93,x]
F[93,x] = 2*F[35,x]+F[50,x]+F[94,x]
F[94,x] = F[13,x]*F[95,x]
F[95,x] = F[92,x]
F[96,x] = F[57,x]+F[97,x]
F[97,x] = 3*F[35,x]+F[100,x]+F[98,x]
F[98,x] = F[13,x]*F[99,x]
F[99,x] = F[96,x]
F[100,x] = F[101,x]*F[13,x]
F[101,x] = F[102,x]+F[57,x]
F[102,x] = F[100,x]
F[103,x] = F[104,x]*F[13,x]
F[104,x] = F[34,x]+F[89,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{17}\! \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_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right)
F_{26}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{70}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{13}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{52}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{6}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{13}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{38}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{14}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{9}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{13}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{13}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{14}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{34}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{54}\! \left(x \right)+F_{60}\! \left(x \right)
F_{54}\! \left(x \right) = F_{13}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{52}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{13}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{34}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{13}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{31}\! \left(x \right)+F_{57}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{67}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{13}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{31}\! \left(x \right)+F_{64}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{13}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{57}\! \left(x \right)+F_{67}\! \left(x \right)
F_{70}\! \left(x \right) = F_{13}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{17}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{89}\! \left(x \right)
F_{73}\! \left(x \right) = F_{35}\! \left(x \right)+F_{74}\! \left(x \right)+F_{87}\! \left(x \right)
F_{74}\! \left(x \right) = F_{13}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{80}\! \left(x \right)
F_{76}\! \left(x \right) = F_{10}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{35}\! \left(x \right)+F_{47}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{13}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{76}\! \left(x \right)
F_{80}\! \left(x \right) = F_{31}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{82}\! \left(x \right)+F_{84}\! \left(x \right)
F_{82}\! \left(x \right) = F_{13}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{80}\! \left(x \right)
F_{84}\! \left(x \right) = F_{13}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{31}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{84}\! \left(x \right)
F_{87}\! \left(x \right) = F_{13}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{18}\! \left(x \right)+F_{73}\! \left(x \right)
F_{89}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{103}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{13}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)+F_{96}\! \left(x \right)
F_{92}\! \left(x \right) = F_{14}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{50}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{13}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{92}\! \left(x \right)
F_{96}\! \left(x \right) = F_{57}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = 3 F_{35}\! \left(x \right)+F_{100}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{13}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{96}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{13}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{57}\! \left(x \right)
F_{102}\! \left(x \right) = F_{100}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{13}\! \left(x \right)
F_{104}\! \left(x \right) = F_{34}\! \left(x \right)+F_{89}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_17(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_13(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), F_1(x) + F_10(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), F_14(x) + F_6(x))
Eq(F_17(x), F_18(x) + F_34(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_13(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_26(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_13(x)*F_24(x))
Eq(F_24(x), F_21(x) + F_25(x))
Eq(F_25(x), F_10(x))
Eq(F_26(x), F_18(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_13(x)*F_29(x))
Eq(F_29(x), F_26(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_13(x)*F_33(x))
Eq(F_33(x), F_18(x) + F_31(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_70(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_13(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_52(x))
Eq(F_38(x), F_39(x) + F_6(x))
Eq(F_39(x), F_35(x) + F_40(x) + F_43(x))
Eq(F_40(x), F_13(x)*F_41(x))
Eq(F_41(x), F_38(x) + F_42(x))
Eq(F_42(x), F_14(x))
Eq(F_43(x), F_13(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_9(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_13(x)*F_48(x))
Eq(F_48(x), F_10(x) + F_46(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_13(x)*F_51(x))
Eq(F_51(x), F_14(x) + F_49(x))
Eq(F_52(x), F_34(x) + F_53(x))
Eq(F_53(x), 2*F_35(x) + F_54(x) + F_60(x))
Eq(F_54(x), F_13(x)*F_55(x))
Eq(F_55(x), F_52(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_13(x)*F_59(x))
Eq(F_59(x), F_34(x) + F_57(x))
Eq(F_60(x), F_13(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_31(x) + F_57(x))
Eq(F_63(x), F_64(x) + F_67(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_13(x)*F_66(x))
Eq(F_66(x), F_31(x) + F_64(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_13(x)*F_69(x))
Eq(F_69(x), F_57(x) + F_67(x))
Eq(F_70(x), F_13(x)*F_71(x))
Eq(F_71(x), F_17(x) + F_72(x))
Eq(F_72(x), F_73(x) + F_89(x))
Eq(F_73(x), F_35(x) + F_74(x) + F_87(x))
Eq(F_74(x), F_13(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_80(x))
Eq(F_76(x), F_10(x) + F_77(x))
Eq(F_77(x), F_35(x) + F_47(x) + F_78(x))
Eq(F_78(x), F_13(x)*F_79(x))
Eq(F_79(x), F_76(x))
Eq(F_80(x), F_31(x) + F_81(x))
Eq(F_81(x), 2*F_35(x) + F_82(x) + F_84(x))
Eq(F_82(x), F_13(x)*F_83(x))
Eq(F_83(x), F_80(x))
Eq(F_84(x), F_13(x)*F_85(x))
Eq(F_85(x), F_31(x) + F_86(x))
Eq(F_86(x), F_84(x))
Eq(F_87(x), F_13(x)*F_88(x))
Eq(F_88(x), F_18(x) + F_73(x))
Eq(F_89(x), F_103(x) + 2*F_35(x) + F_90(x))
Eq(F_90(x), F_13(x)*F_91(x))
Eq(F_91(x), F_92(x) + F_96(x))
Eq(F_92(x), F_14(x) + F_93(x))
Eq(F_93(x), 2*F_35(x) + F_50(x) + F_94(x))
Eq(F_94(x), F_13(x)*F_95(x))
Eq(F_95(x), F_92(x))
Eq(F_96(x), F_57(x) + F_97(x))
Eq(F_97(x), F_100(x) + 3*F_35(x) + F_98(x))
Eq(F_98(x), F_13(x)*F_99(x))
Eq(F_99(x), F_96(x))
Eq(F_100(x), F_101(x)*F_13(x))
Eq(F_101(x), F_102(x) + F_57(x))
Eq(F_102(x), F_100(x))
Eq(F_103(x), F_104(x)*F_13(x))
Eq(F_104(x), F_34(x) + F_89(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, 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, 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": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"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": []}, "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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 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