0132_0231_0321_2130
Counting sequence:
1, 1, 2, 6, 20, 64, 196, 587, 1742, 5151, 15209, 44880, 132405, 390587, 1152171, 3398684, 10025422, 29572893, 87233781, 257321151, 759042755, 2239014885, 6604618212, 19482220443, 57468410846, 169519601386, 500046805301, 1475031827780, 4351030483254, 12834615436440, 37859388490811, 111677151839495, 329423869220832, 971730419561372, 2866398268389866, 8455265851137010, 24941237720451894, 73571351862867797, 217019855854684812, 640162463277486769, 1888343247559359363, 5570211352828677054, 16430939955055046299, 48467781688306068341, 142969657743930760009, 421730112734835228558, 1244014225073481032964, 3669577640898069429355, 10824474343758928511275, 31929899373929044198061, 94186418818301190002293, 277829923167867299321447, 819539241176292641359574, 2417466629114675893961377, 7131012902438833718369312, 21034972893657641922389068, 62048700611042967856603635, 183030482947743128998004882, 539901035125334736614402936, 1592593337649837692113014110, 4697811958330198241131855765, 13857547105149703276920570605, 40876819565102198482879498814, 120577932377150737855118266082, 355679280605313069078584925184, 1049178304502806217065584562308, 3094853073156322607030473847992, 9129158984052854029240193652271, 26929079276489231270957047978922, 79435062084711670570893254288339, 234316555111891996680434932346557, 691184050954136144221797662517436, 2038846090346622370607173635569681, 6014162760820909225011915434960383, 17740502279647663853603925252768631, 52330712295392749481967762515424672, 154364482255096447817883921719935706, 455342423916936875679464497141389721, 1343163401253892296038621663979321521, 3962046643817718275408312669996109723, 11687195759751018096699273593771386343, 34474744243577939297600244189897681401, 101693170465506947333790198730514187960, 299973245523151368159449326273784075623, 884857337201560483098737969149001208594, 2610141133866579362460924923502513414446, 7699361752764022674593305597504886862401, 22711481241671303329666287277182713342064, 66994044020027300224575963520069844953482, 197618195237849476240957696840560673618516, 582931686843538591379932489516006567824015, 1719524617241167057899960773984810398340467, 5072232263284034994578410264876827497967964, 14962007449464238044612637735275527428363608, 44134742909600774782456859714033224348780054, 130188114013157084877212157935676459777229302, 384027274499333919518280695827522696500919946, 1132798863224045424867032092974209092512739225, 3341515954028716501244185202679741739030185064, 9856762072703528979186242368369022215795313989, 29075353789872052266851044245852226488097813527
Generating function in Maple syntax:
-(2*x^5-6*x^4+8*x^3-9*x^2+5*x-1)/(x^4-3*x^3+4*x^2-4*x+1)/(x-1)^2
Generating function in latex syntax:
-\frac{2 x^{5}-6 x^{4}+8 x^{3}-9 x^{2}+5 x -1}{\left(x^{4}-3 x^{3}+4 x^{2}-4 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**5 + 6*x**4 - 8*x**3 + 9*x**2 - 5*x + 1)/((x - 1)**2*(x**4 - 3*x**3 + 4*x**2 - 4*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^4-3*x^3+4*x^2-4*x+1)*(x-1)^2*F(x)+2*x^5-6*x^4+8*x^3-9*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}-3 x^{3}+4 x^{2}-4 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+2 x^{5}-6 x^{4}+8 x^{3}-9 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
1+5/331*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^4-3*_Z^3+4*_Z^2-4*_Z+1))+61/331*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^4-3*_Z^3+4*_Z^2-4*_Z+1))-112/331*Sum(_alpha^(-n),_alpha = RootOf(_Z^4-3*_Z^3+4*_Z^2-4*_Z+1))+65/331*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^4-3*_Z^3+4*_Z^2-4*_Z+1))-n
Explicit closed form in latex syntax:
1+\frac{5 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{331}+\frac{61 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{331}-\frac{112 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{331}+\frac{65 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{331}-n
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(n+4) = -a(n)+3*a(n+1)-4*a(n+2)+4*a(n+3)+n+2, 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 +4\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)+n +2, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15315/
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[18,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[12,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[17,x]
F[15,x] = 0
F[16,x] = F[10,x]*F[12,x]
F[17,x] = F[12,x]*F[13,x]
F[18,x] = F[19,x]+F[41,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]+F[29,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[9,x]
F[26,x] = F[12,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[26,x]
F[29,x] = F[19,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[19,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[39,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[19,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[36,x]
F[41,x] = F[15,x]+F[42,x]+F[81,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[56,x]
F[44,x] = F[45,x]+F[6,x]
F[45,x] = F[15,x]+F[46,x]+F[54,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[14,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[12,x]*F[51,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[13,x]
F[56,x] = F[41,x]+F[57,x]
F[57,x] = 2*F[15,x]+F[58,x]+F[79,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[76,x]
F[60,x] = F[61,x]+F[74,x]
F[61,x] = F[15,x]+F[62,x]+F[72,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[64,x]+F[66,x]
F[64,x] = F[10,x]+F[65,x]
F[65,x] = F[15,x]+F[17,x]+F[46,x]
F[66,x] = F[61,x]+F[67,x]
F[67,x] = 2*F[15,x]+F[58,x]+F[68,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[37,x]+F[70,x]
F[70,x] = 2*F[15,x]+F[68,x]+F[71,x]
F[71,x] = F[12,x]*F[61,x]
F[72,x] = F[12,x]*F[73,x]
F[73,x] = F[19,x]+F[61,x]
F[74,x] = F[75,x]
F[75,x] = F[12,x]*F[60,x]
F[76,x] = F[70,x]+F[77,x]
F[77,x] = F[78,x]
F[78,x] = F[12,x]*F[76,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[69,x]
F[81,x] = F[12,x]*F[82,x]
F[82,x] = F[73,x]+F[83,x]
F[83,x] = F[111,x]+F[84,x]
F[84,x] = F[15,x]+F[38,x]+F[85,x]
F[85,x] = F[12,x]*F[86,x]
F[86,x] = F[87,x]+F[99,x]
F[87,x] = F[12,x]+F[88,x]
F[88,x] = F[15,x]+F[89,x]+F[96,x]
F[89,x] = F[12,x]*F[90,x]
F[90,x] = F[91,x]+F[94,x]
F[91,x] = F[12,x]+F[92,x]
F[92,x] = F[93,x]
F[93,x] = F[12,x]*F[91,x]
F[94,x] = F[95,x]+F[97,x]
F[95,x] = F[96,x]
F[96,x] = x^2
F[97,x] = F[98,x]
F[98,x] = F[12,x]*F[94,x]
F[99,x] = F[100,x]+F[37,x]
F[100,x] = 2*F[15,x]+F[101,x]+F[108,x]
F[101,x] = F[102,x]*F[12,x]
F[102,x] = F[103,x]+F[106,x]
F[103,x] = F[104,x]+F[37,x]
F[104,x] = F[105,x]
F[105,x] = F[103,x]*F[12,x]
F[106,x] = F[107,x]+F[109,x]
F[107,x] = F[108,x]
F[108,x] = F[12,x]*F[37,x]
F[109,x] = F[110,x]
F[110,x] = F[106,x]*F[12,x]
F[111,x] = F[112,x]+F[142,x]+F[15,x]+F[71,x]
F[112,x] = F[113,x]*F[12,x]
F[113,x] = F[114,x]+F[128,x]
F[114,x] = F[115,x]+F[14,x]
F[115,x] = F[116,x]+F[124,x]+F[127,x]+F[15,x]
F[116,x] = F[117,x]*F[12,x]
F[117,x] = F[118,x]+F[122,x]
F[118,x] = F[119,x]+F[120,x]
F[119,x] = F[16,x]
F[120,x] = F[121,x]
F[121,x] = F[118,x]*F[12,x]
F[122,x] = F[123,x]+F[125,x]
F[123,x] = F[124,x]
F[124,x] = F[12,x]*F[14,x]
F[125,x] = F[126,x]
F[126,x] = F[12,x]*F[122,x]
F[127,x] = 0
F[128,x] = F[129,x]+F[70,x]
F[129,x] = 2*F[15,x]+F[130,x]+F[138,x]+F[141,x]
F[130,x] = F[12,x]*F[131,x]
F[131,x] = F[132,x]+F[136,x]
F[132,x] = F[133,x]+F[134,x]
F[133,x] = F[71,x]
F[134,x] = F[135,x]
F[135,x] = F[12,x]*F[132,x]
F[136,x] = F[137,x]+F[139,x]
F[137,x] = F[138,x]
F[138,x] = F[12,x]*F[70,x]
F[139,x] = F[140,x]
F[140,x] = F[12,x]*F[136,x]
F[141,x] = 0
F[142,x] = F[12,x]*F[83,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_{18}\! \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_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{41}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{29}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{9}\! \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_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{19}\! \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_{19}\! \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_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{81}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{56}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{6}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{54}\! \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_{14}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13}\! \left(x \right)
F_{56}\! \left(x \right) = F_{41}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{79}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{76}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{74}\! \left(x \right)
F_{61}\! \left(x \right) = F_{15}\! \left(x \right)+F_{62}\! \left(x \right)+F_{72}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = F_{10}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)+F_{46}\! \left(x \right)
F_{66}\! \left(x \right) = F_{61}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{37}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{19}\! \left(x \right)+F_{61}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{76}\! \left(x \right) = F_{70}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \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_{69}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{73}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{111}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{12}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)+F_{99}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{15}\! \left(x \right)+F_{89}\! \left(x \right)+F_{96}\! \left(x \right)
F_{89}\! \left(x \right) = F_{12}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)+F_{94}\! \left(x \right)
F_{91}\! \left(x \right) = F_{12}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{12}\! \left(x \right) F_{91}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)+F_{97}\! \left(x \right)
F_{95}\! \left(x \right) = F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = x^{2}
F_{97}\! \left(x \right) = F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{12}\! \left(x \right) F_{94}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{37}\! \left(x \right)
F_{100}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{101}\! \left(x \right)+F_{108}\! \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_{106}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{37}\! \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_{109}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)
F_{108}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)
F_{110}\! \left(x \right) = F_{106}\! \left(x \right) F_{12}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)+F_{142}\! \left(x \right)+F_{15}\! \left(x \right)+F_{71}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{12}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)+F_{128}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{14}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{124}\! \left(x \right)+F_{127}\! \left(x \right)+F_{15}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right) F_{12}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)+F_{122}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right)+F_{120}\! \left(x \right)
F_{119}\! \left(x \right) = F_{16}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = F_{118}\! \left(x \right) F_{12}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)+F_{125}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)
F_{124}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)
F_{126}\! \left(x \right) = F_{12}\! \left(x \right) F_{122}\! \left(x \right)
F_{127}\! \left(x \right) = 0
F_{128}\! \left(x \right) = F_{129}\! \left(x \right)+F_{70}\! \left(x \right)
F_{129}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{130}\! \left(x \right)+F_{138}\! \left(x \right)+F_{141}\! \left(x \right)
F_{130}\! \left(x \right) = F_{12}\! \left(x \right) F_{131}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)+F_{136}\! \left(x \right)
F_{132}\! \left(x \right) = F_{133}\! \left(x \right)+F_{134}\! \left(x \right)
F_{133}\! \left(x \right) = F_{71}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)
F_{135}\! \left(x \right) = F_{12}\! \left(x \right) F_{132}\! \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_{70}\! \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) = 0
F_{142}\! \left(x \right) = F_{12}\! \left(x \right) F_{83}\! \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_18(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_12(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_17(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_10(x)*F_12(x))
Eq(F_17(x), F_12(x)*F_13(x))
Eq(F_18(x), F_19(x) + F_41(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_29(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_9(x))
Eq(F_26(x), F_12(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_26(x))
Eq(F_29(x), F_19(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_19(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) + F_39(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_19(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_36(x))
Eq(F_41(x), F_15(x) + F_42(x) + F_81(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_56(x))
Eq(F_44(x), F_45(x) + F_6(x))
Eq(F_45(x), F_15(x) + F_46(x) + F_54(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_14(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_12(x)*F_51(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_13(x))
Eq(F_56(x), F_41(x) + F_57(x))
Eq(F_57(x), 2*F_15(x) + F_58(x) + F_79(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_76(x))
Eq(F_60(x), F_61(x) + F_74(x))
Eq(F_61(x), F_15(x) + F_62(x) + F_72(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_66(x))
Eq(F_64(x), F_10(x) + F_65(x))
Eq(F_65(x), F_15(x) + F_17(x) + F_46(x))
Eq(F_66(x), F_61(x) + F_67(x))
Eq(F_67(x), 2*F_15(x) + F_58(x) + F_68(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_37(x) + F_70(x))
Eq(F_70(x), 2*F_15(x) + F_68(x) + F_71(x))
Eq(F_71(x), F_12(x)*F_61(x))
Eq(F_72(x), F_12(x)*F_73(x))
Eq(F_73(x), F_19(x) + F_61(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_12(x)*F_60(x))
Eq(F_76(x), F_70(x) + F_77(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_12(x)*F_76(x))
Eq(F_79(x), F_12(x)*F_80(x))
Eq(F_80(x), F_69(x))
Eq(F_81(x), F_12(x)*F_82(x))
Eq(F_82(x), F_73(x) + F_83(x))
Eq(F_83(x), F_111(x) + F_84(x))
Eq(F_84(x), F_15(x) + F_38(x) + F_85(x))
Eq(F_85(x), F_12(x)*F_86(x))
Eq(F_86(x), F_87(x) + F_99(x))
Eq(F_87(x), F_12(x) + F_88(x))
Eq(F_88(x), F_15(x) + F_89(x) + F_96(x))
Eq(F_89(x), F_12(x)*F_90(x))
Eq(F_90(x), F_91(x) + F_94(x))
Eq(F_91(x), F_12(x) + F_92(x))
Eq(F_92(x), F_93(x))
Eq(F_93(x), F_12(x)*F_91(x))
Eq(F_94(x), F_95(x) + F_97(x))
Eq(F_95(x), F_96(x))
Eq(F_96(x), x**2)
Eq(F_97(x), F_98(x))
Eq(F_98(x), F_12(x)*F_94(x))
Eq(F_99(x), F_100(x) + F_37(x))
Eq(F_100(x), F_101(x) + F_108(x) + 2*F_15(x))
Eq(F_101(x), F_102(x)*F_12(x))
Eq(F_102(x), F_103(x) + F_106(x))
Eq(F_103(x), F_104(x) + F_37(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_109(x))
Eq(F_107(x), F_108(x))
Eq(F_108(x), F_12(x)*F_37(x))
Eq(F_109(x), F_110(x))
Eq(F_110(x), F_106(x)*F_12(x))
Eq(F_111(x), F_112(x) + F_142(x) + F_15(x) + F_71(x))
Eq(F_112(x), F_113(x)*F_12(x))
Eq(F_113(x), F_114(x) + F_128(x))
Eq(F_114(x), F_115(x) + F_14(x))
Eq(F_115(x), F_116(x) + F_124(x) + F_127(x) + F_15(x))
Eq(F_116(x), F_117(x)*F_12(x))
Eq(F_117(x), F_118(x) + F_122(x))
Eq(F_118(x), F_119(x) + F_120(x))
Eq(F_119(x), F_16(x))
Eq(F_120(x), F_121(x))
Eq(F_121(x), F_118(x)*F_12(x))
Eq(F_122(x), F_123(x) + F_125(x))
Eq(F_123(x), F_124(x))
Eq(F_124(x), F_12(x)*F_14(x))
Eq(F_125(x), F_126(x))
Eq(F_126(x), F_12(x)*F_122(x))
Eq(F_127(x), 0)
Eq(F_128(x), F_129(x) + F_70(x))
Eq(F_129(x), F_130(x) + F_138(x) + F_141(x) + 2*F_15(x))
Eq(F_130(x), F_12(x)*F_131(x))
Eq(F_131(x), F_132(x) + F_136(x))
Eq(F_132(x), F_133(x) + F_134(x))
Eq(F_133(x), F_71(x))
Eq(F_134(x), F_135(x))
Eq(F_135(x), F_12(x)*F_132(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_70(x))
Eq(F_139(x), F_140(x))
Eq(F_140(x), F_12(x)*F_136(x))
Eq(F_141(x), 0)
Eq(F_142(x), F_12(x)*F_83(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": [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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "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": [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": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "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", 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