0132_1023_1302
Counting sequence:
1, 1, 2, 6, 21, 74, 257, 883, 3015, 10258, 34826, 118075, 399978, 1354163, 4582981, 15506825, 52460374, 177458522, 600253133, 2030270534, 6866914597, 23225323035, 78551944115, 265673909334, 898542853066, 3038976432991, 10278151013062, 34761786525051, 117567914206641, 397626480643509, 1344812168063782, 4548286987930038, 15382751512147265, 52025965294262518, 175956875543603281, 595103245147300455, 2012696928416606843, 6807136221480689218, 23022394660292645766, 77863970220057183575, 263343493117633873994, 890653211732279857767, 3012275460461966318233, 10187807463623639359209, 34456151899535339751922, 116534043985189923666310, 394129426940914835531645, 1332983906101903302832394, 4508280712389579684281881, 15247442138429117607680731, 51568326504448345279115647, 174409076238378122764002946, 589868392779820759349740586, 1994992051440866135016751219, 6747256394790002015493307082, 22819874807935685742746370843, 77179027411724232681403234365, 261026947883615399803370260753, 882818426278218801922599287686, 2985777445947291729157437332410, 10098188587095324254119297066133, 34153052123459479507337587894598, 115508931060956762086164717446301, 390662395460612816324869426119075, 1321258069183085645252542108620139, 4468622794684255882492720338039462, 15113315215941713754104544136341258, 51114696252303693164998230644516759, 172874854764445059046877955202110022, 584679507089520206070414190550501635, 1977442737269321091153916382759581225, 6687902913929998681192981536624174429, 22619135585144052265521497248469987446, 76500107911776356352389070794533141846, 258730776358982835184395201377361809257, 875052551723485560553916395878325940566, 2959512505831036342117259569703452497481, 10009358015034995640864609692727415208079, 33852618522729478162840974399829249285907, 114492835516928000598294638906123771065298, 387225861890266661625688462969090085865190, 1309635379713260297181302961542646640684463, 4429313732879594594342001760686773493510666, 14980368160618289187696783891668005072188815, 50665056431162160317842652041502228464207473, 171354129327813511692187146364302670354977553, 579536266333524426713033256365145367453439586, 1960047798750572293020906443980847251211074950, 6629071546621737456969365663066165345263158309, 22420162201269931277346930889382923631928945482, 75827160650787820513667035660528797276078398833, 256454803526564353956892815644212583283072141059, 867354991105886865241535802296142748778992134647, 2933478610075506418005446975792403210915909329010, 9921308857401834824238040339296016549174258582730, 33554827741330111958619393696302283822062783736939, 113485678243988404559680332081644583342075038761386, 383819558418854815581921266475354018337517722246211, 1298114931367107957824277455436420298990689464198517, 4390350460461194036636010958467796814486167842579161, 14848590598501314619213194430923928557087028643542326
Generating function in Maple syntax:
(x^2-3*x+1)^2/(x^5+4*x^4-14*x^3+16*x^2-7*x+1)
Generating function in latex syntax:
\frac{\left(x^{2}-3 x +1\right)^{2}}{x^{5}+4 x^{4}-14 x^{3}+16 x^{2}-7 x +1}
Generating function in sympy syntax:
(x**2 - 3*x + 1)**2/(x**5 + 4*x**4 - 14*x**3 + 16*x**2 - 7*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^5+4*x^4-14*x^3+16*x^2-7*x+1)*F(x)-(x^2-3*x+1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}+4 x^{4}-14 x^{3}+16 x^{2}-7 x +1\right) F \! \left(x \right)-\left(x^{2}-3 x +1\right)^{2} = 0
Explicit closed form in Maple syntax:
-8390/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 1)^(-n+3)-8390/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 2)^(-n+3)-8390/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 3)^(-n+3)-8390/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 4)^(-n+3)-8390/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 5)^(-n+3)-36889/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 1)^(-n+2)-36889/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 2)^(-n+2)-36889/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 3)^(-n+2)-36889/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 4)^(-n+2)-36889/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 5)^(-n+2)+103870/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 1)^(-n+1)+103870/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 2)^(-n+1)+103870/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 3)^(-n+1)+103870/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 4)^(-n+1)+103870/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 5)^(-n+1)+17241/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 1)^(-n-1)+17241/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 2)^(-n-1)+17241/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 3)^(-n-1)+17241/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 4)^(-n-1)+17241/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 5)^(-n-1)-79776/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 1)^(-n)-79776/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 2)^(-n)-79776/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 3)^(-n)-79776/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 4)^(-n)-79776/32411*RootOf(_Z^5+4*_Z^4-14*_Z^3+16*_Z^2-7*_Z+1,index = 5)^(-n)
Explicit closed form in latex syntax:
-\frac{8390 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{32411}-\frac{8390 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{32411}-\frac{8390 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{32411}-\frac{8390 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{32411}-\frac{8390 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{32411}-\frac{36889 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{32411}-\frac{36889 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{32411}-\frac{36889 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{32411}-\frac{36889 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{32411}-\frac{36889 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{32411}+\frac{103870 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{32411}+\frac{103870 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{32411}+\frac{103870 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{32411}+\frac{103870 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{32411}+\frac{103870 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{32411}+\frac{17241 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{32411}+\frac{17241 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{32411}+\frac{17241 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{32411}+\frac{17241 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{32411}+\frac{17241 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{32411}-\frac{79776 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{32411}-\frac{79776 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{32411}-\frac{79776 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{32411}-\frac{79776 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{32411}-\frac{79776 \mathit{RootOf}\! \left(\textit{\_Z}^{5}+4 \textit{\_Z}^{4}-14 \textit{\_Z}^{3}+16 \textit{\_Z}^{2}-7 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{32411}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(n+5) = -a(n)-4*a(n+1)+14*a(n+2)-16*a(n+3)+7*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)-4 a \! \left(n +1\right)+14 a \! \left(n +2\right)-16 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14705/
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[27,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]+F[16,x]+F[26,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[12,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[18,x]
F[21,x] = F[22,x]+F[24,x]
F[22,x] = F[15,x]+F[16,x]+F[23,x]
F[23,x] = F[12,x]*F[6,x]
F[24,x] = F[25,x]
F[25,x] = F[12,x]*F[21,x]
F[26,x] = F[12,x]*F[13,x]
F[27,x] = F[2,x]+F[28,x]
F[28,x] = F[15,x]+F[29,x]+F[62,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[42,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[1,x]+F[32,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[18,x]+F[38,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[36,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[38,x]
F[42,x] = F[43,x]+F[46,x]
F[43,x] = F[15,x]+F[29,x]+F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[2,x]+F[43,x]
F[46,x] = F[47,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[49,x]+F[59,x]
F[49,x] = F[50,x]+F[57,x]
F[50,x] = F[15,x]+F[51,x]+F[56,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[12,x]+F[39,x]
F[54,x] = F[50,x]+F[55,x]
F[55,x] = F[47,x]
F[56,x] = F[12,x]*F[2,x]
F[57,x] = F[58,x]
F[58,x] = F[12,x]*F[49,x]
F[59,x] = F[55,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[12,x]*F[59,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[64,x]+F[67,x]
F[64,x] = F[2,x]+F[65,x]
F[65,x] = F[15,x]+F[51,x]+F[66,x]
F[66,x] = F[12,x]*F[64,x]
F[67,x] = F[28,x]+F[68,x]
F[68,x] = F[15,x]+F[69,x]+F[70,x]+F[77,x]
F[69,x] = 0
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[49,x]+F[72,x]
F[72,x] = F[73,x]+F[75,x]
F[73,x] = F[15,x]+F[69,x]+F[70,x]+F[74,x]
F[74,x] = F[12,x]*F[28,x]
F[75,x] = F[76,x]
F[76,x] = F[12,x]*F[72,x]
F[77,x] = F[12,x]*F[67,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_{27}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{6}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{27}\! \left(x \right) = F_{2}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{62}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{42}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{1}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{36}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{2}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{59}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{57}\! \left(x \right)
F_{50}\! \left(x \right) = F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right)+F_{39}\! \left(x \right)
F_{54}\! \left(x \right) = F_{50}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{47}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{2}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{59}\! \left(x \right) = F_{55}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \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_{67}\! \left(x \right)
F_{64}\! \left(x \right) = F_{2}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{67}\! \left(x \right) = F_{28}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{15}\! \left(x \right)+F_{69}\! \left(x \right)+F_{70}\! \left(x \right)+F_{77}\! \left(x \right)
F_{69}\! \left(x \right) = 0
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{49}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{75}\! \left(x \right)
F_{73}\! \left(x \right) = F_{15}\! \left(x \right)+F_{69}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \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_27(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_26(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_12(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_18(x))
Eq(F_21(x), F_22(x) + F_24(x))
Eq(F_22(x), F_15(x) + F_16(x) + F_23(x))
Eq(F_23(x), F_12(x)*F_6(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_12(x)*F_21(x))
Eq(F_26(x), F_12(x)*F_13(x))
Eq(F_27(x), F_2(x) + F_28(x))
Eq(F_28(x), F_15(x) + F_29(x) + F_62(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_42(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_1(x) + F_32(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_18(x) + F_38(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_36(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_38(x))
Eq(F_42(x), F_43(x) + F_46(x))
Eq(F_43(x), F_15(x) + F_29(x) + F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_2(x) + F_43(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_59(x))
Eq(F_49(x), F_50(x) + F_57(x))
Eq(F_50(x), F_15(x) + F_51(x) + F_56(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_12(x) + F_39(x))
Eq(F_54(x), F_50(x) + F_55(x))
Eq(F_55(x), F_47(x))
Eq(F_56(x), F_12(x)*F_2(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_12(x)*F_49(x))
Eq(F_59(x), F_55(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_12(x)*F_59(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_67(x))
Eq(F_64(x), F_2(x) + F_65(x))
Eq(F_65(x), F_15(x) + F_51(x) + F_66(x))
Eq(F_66(x), F_12(x)*F_64(x))
Eq(F_67(x), F_28(x) + F_68(x))
Eq(F_68(x), F_15(x) + F_69(x) + F_70(x) + F_77(x))
Eq(F_69(x), 0)
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_49(x) + F_72(x))
Eq(F_72(x), F_73(x) + F_75(x))
Eq(F_73(x), F_15(x) + F_69(x) + F_70(x) + F_74(x))
Eq(F_74(x), F_12(x)*F_28(x))
Eq(F_75(x), F_76(x))
Eq(F_76(x), F_12(x)*F_72(x))
Eq(F_77(x), F_12(x)*F_67(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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 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, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "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], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [1, 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": [0, 1, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "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, 0, 2, 3], "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], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "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], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 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, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "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], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 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], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [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], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "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], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}], [{"patt": [0], "pos": [[1, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, 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