0132_0231_3012_3021_3102
Counting sequence:
1, 1, 2, 6, 19, 55, 149, 391, 1010, 2589, 6613, 16864, 42974, 109474, 278840, 710187, 1808752, 4606600, 11732213, 29879857, 76098611, 193809381, 493597324, 1257102739, 3201612287, 8153924746, 20766564632, 52888666416, 134697822334, 343050875845, 873688240574, 2225125179466, 5666989475495, 14432792371179, 36757699397473, 93615180641779, 238420853052374, 607214586144169, 1546465205982665, 3938565851162052, 10030811494451122, 25546654046047150, 65062685437707668, 165702836415913807, 422015012315582636, 1074795546484786956, 2737308941701070569, 6971428442202510949, 17754961372590879647, 45218660129085341041, 115163710072964072912, 293301041647604366751, 746984453497258147699, 1902433658715084735310, 4845152812575031985324, 12339723737362406853916, 31427033946014930428730, 80038944441967299696969, 203844646567311936676858, 519155271522606103479694, 1322194134054491443333499, 3367388186198900926823839, 8576125777974899400461165, 21841833876203191171079967, 55627181716580182669445242, 141672323087338455910431925, 360813661767460285661389373, 918926828338839209902656232, 2340339641532477161377134086, 5960419773171253818318314106, 15180106016213824007916418864, 38660971447131378995528286259, 98462468683647835817291305832, 250766014830640874638027317136, 638655469792060964088874226717, 1626539423098410638633067076761, 4142500330819523115993035697739, 10550195554529517834708012699325, 26869430862976969424042128173524, 68431557611302979798785304744491, 174282741640112446856320750362215, 443866471754504842935468933642834, 1130447242760425112526043922392768, 2879043698915467514843877528790984, 7332401112345864985149267913617974, 18674293166367622597668457278420109, 47560031143996577695330059999249590, 121126756566706642973477845190537682, 308487837443777486239954207658745487, 785662462598258193148716320507278675, 2000939519207000515510864693863840953, 5096029338456036710410399915893706507, 12978660658717332129480380846158533086, 33054290175097701484882026302074614081, 84183270347368771809654833366201468209, 214399491528552577233672073880636084044, 546036543579571627761881007429548250842, 1390655849035064604567088922105934054406, 3541747733178253414129730925522052444172, 9020187858971143060588431780579587194071, 22972779300155604139873683408787160887204
Generating function in Maple syntax:
-(x^5+3*x^4+x^3+2*x^2-3*x+1)/(x^3+x^2+2*x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{x^{5}+3 x^{4}+x^{3}+2 x^{2}-3 x +1}{\left(x^{3}+x^{2}+2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**5 - 3*x**4 - x**3 - 2*x**2 + 3*x - 1)/((x - 1)**2*(x**3 + x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3+x^2+2*x-1)*(x-1)^2*F(x)+x^5+3*x^4+x^3+2*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+x^{2}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{5}+3 x^{4}+x^{3}+2 x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/31320*(-183*2^(2/3)*(29^(1/2)-841/183)*(1+I*3^(1/2))*(43+9*29^(1/2))^(1/3)+5800-6*(29^(1/2)-116/3)*(I*3^(1/2)-1)*2^(1/3)*(43+9*29^(1/2))^(2/3))*(-3/200*(I*3^(1/2)-1)*(29^(1/2)-43/9)*2^(1/3)*(43+9*29^(1/2))^(2/3)-1/12*I*3^(1/2)*(172+36*29^(1/2))^(1/3)-1/12*(172+36*29^(1/2))^(1/3)-1/3)^(-n)+1/31320*(183*2^(2/3)*(29^(1/2)-841/183)*(I*3^(1/2)-1)*(43+9*29^(1/2))^(1/3)+5800+6*(29^(1/2)-116/3)*(1+I*3^(1/2))*2^(1/3)*(43+9*29^(1/2))^(2/3))*(3/200*(29^(1/2)-43/9)*2^(1/3)*(1+I*3^(1/2))*(43+9*29^(1/2))^(2/3)+1/12*I*3^(1/2)*(172+36*29^(1/2))^(1/3)-1/12*(172+36*29^(1/2))^(1/3)-1/3)^(-n)+1/31320*((366*2^(2/3)*29^(1/2)-1682*2^(2/3))*(43+9*29^(1/2))^(1/3)+5800+(-12*29^(1/2)*2^(1/3)+464*2^(1/3))*(43+9*29^(1/2))^(2/3))*(1/300*(43-9*29^(1/2))*2^(1/3)*(43+9*29^(1/2))^(2/3)+1/6*(172+36*29^(1/2))^(1/3)-1/3)^(-n)-5/3*n+13/9)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-183 \,2^{\frac{2}{3}} \left(\sqrt{29}-\frac{841}{183}\right) \left(1+\mathrm{I} \sqrt{3}\right) \left(43+9 \sqrt{29}\right)^{\frac{1}{3}}+5800-6 \left(\sqrt{29}-\frac{116}{3}\right) \left(\mathrm{I} \sqrt{3}-1\right) 2^{\frac{1}{3}} \left(43+9 \sqrt{29}\right)^{\frac{2}{3}}\right) \left(-\frac{3 \left(\mathrm{I} \sqrt{3}-1\right) \left(\sqrt{29}-\frac{43}{9}\right) 2^{\frac{1}{3}} \left(43+9 \sqrt{29}\right)^{\frac{2}{3}}}{200}-\frac{\mathrm{I} \sqrt{3}\, \left(172+36 \sqrt{29}\right)^{\frac{1}{3}}}{12}-\frac{\left(172+36 \sqrt{29}\right)^{\frac{1}{3}}}{12}-\frac{1}{3}\right)^{-n}}{31320}\\+\\\frac{\left(183 \,2^{\frac{2}{3}} \left(\sqrt{29}-\frac{841}{183}\right) \left(\mathrm{I} \sqrt{3}-1\right) \left(43+9 \sqrt{29}\right)^{\frac{1}{3}}+5800+6 \left(\sqrt{29}-\frac{116}{3}\right) \left(1+\mathrm{I} \sqrt{3}\right) 2^{\frac{1}{3}} \left(43+9 \sqrt{29}\right)^{\frac{2}{3}}\right) \left(\frac{3 \left(\sqrt{29}-\frac{43}{9}\right) 2^{\frac{1}{3}} \left(1+\mathrm{I} \sqrt{3}\right) \left(43+9 \sqrt{29}\right)^{\frac{2}{3}}}{200}+\frac{\mathrm{I} \sqrt{3}\, \left(172+36 \sqrt{29}\right)^{\frac{1}{3}}}{12}-\frac{\left(172+36 \sqrt{29}\right)^{\frac{1}{3}}}{12}-\frac{1}{3}\right)^{-n}}{31320}\\+\\\frac{\left(\left(366 \,2^{\frac{2}{3}} \sqrt{29}-1682 \,2^{\frac{2}{3}}\right) \left(43+9 \sqrt{29}\right)^{\frac{1}{3}}+5800+\left(-12 \sqrt{29}\, 2^{\frac{1}{3}}+464 \,2^{\frac{1}{3}}\right) \left(43+9 \sqrt{29}\right)^{\frac{2}{3}}\right) \left(\frac{\left(43-9 \sqrt{29}\right) 2^{\frac{1}{3}} \left(43+9 \sqrt{29}\right)^{\frac{2}{3}}}{300}+\frac{\left(172+36 \sqrt{29}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{31320}\\-\frac{5 n}{3}+\frac{13}{9} & \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) = 55
a(n+3) = a(n)+a(n+1)+2*a(n+2)+5*n-1, 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) = 19
a \! \left(5\right) = 55
a \! \left(n +3\right) = a \! \left(n \right)+a \! \left(n +1\right)+2 a \! \left(n +2\right)+5 n -1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19295/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[26,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[19,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[4,x]
F[18,x] = F[14,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[11,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[11,x]*F[4,x]
F[25,x] = F[19,x]
F[26,x] = F[27,x]+F[33,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[17,x]+F[30,x]
F[30,x] = F[27,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[27,x]*F[4,x]
F[33,x] = F[34,x]+F[35,x]+F[86,x]
F[34,x] = 0
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[55,x]
F[37,x] = F[38,x]+F[7,x]
F[38,x] = F[24,x]+F[34,x]+F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[23,x]+F[4,x]
F[42,x] = F[43,x]+F[49,x]
F[43,x] = F[34,x]+F[44,x]+F[45,x]
F[44,x] = x^2
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[4,x]
F[48,x] = F[43,x]
F[49,x] = 2*F[34,x]+F[50,x]+F[51,x]
F[50,x] = F[23,x]*F[4,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[23,x]
F[54,x] = F[49,x]
F[55,x] = F[33,x]+F[56,x]
F[56,x] = 2*F[34,x]+F[57,x]+F[71,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[34,x]+F[59,x]+F[69,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[64,x]
F[61,x] = F[11,x]+F[62,x]
F[62,x] = F[24,x]+F[34,x]+F[63,x]
F[63,x] = F[4,x]*F[41,x]
F[64,x] = F[58,x]+F[65,x]
F[65,x] = 2*F[34,x]+F[57,x]+F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[31,x]+F[68,x]
F[68,x] = F[57,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[27,x]+F[58,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[67,x]+F[73,x]
F[73,x] = F[74,x]+F[80,x]
F[74,x] = 2*F[34,x]+F[75,x]+F[76,x]
F[75,x] = F[31,x]*F[4,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[78,x]+F[79,x]
F[78,x] = F[31,x]
F[79,x] = F[74,x]
F[80,x] = 3*F[34,x]+F[81,x]+F[82,x]
F[81,x] = F[4,x]*F[68,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[84,x]+F[85,x]
F[84,x] = F[68,x]
F[85,x] = F[80,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = F[70,x]+F[88,x]
F[88,x] = F[89,x]+F[97,x]
F[89,x] = F[34,x]+F[90,x]+F[94,x]
F[90,x] = F[4,x]*F[91,x]
F[91,x] = F[92,x]+F[93,x]
F[92,x] = F[4,x]
F[93,x] = F[31,x]
F[94,x] = F[4,x]*F[95,x]
F[95,x] = F[30,x]+F[96,x]
F[96,x] = F[89,x]
F[97,x] = 2*F[34,x]+F[102,x]+F[98,x]
F[98,x] = F[4,x]*F[99,x]
F[99,x] = F[100,x]+F[101,x]
F[100,x] = F[23,x]
F[101,x] = F[68,x]
F[102,x] = F[103,x]*F[4,x]
F[103,x] = F[104,x]+F[105,x]
F[104,x] = F[58,x]+F[68,x]
F[105,x] = F[97,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{26}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{19}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{33}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{17}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{86}\! \left(x \right)
F_{34}\! \left(x \right) = 0
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{55}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{7}\! \left(x \right)
F_{38}\! \left(x \right) = F_{24}\! \left(x \right)+F_{34}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{23}\! \left(x \right)+F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{49}\! \left(x \right)
F_{43}\! \left(x \right) = F_{34}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = x^{2}
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right)
F_{48}\! \left(x \right) = F_{43}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \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_{23}\! \left(x \right)
F_{54}\! \left(x \right) = F_{49}\! \left(x \right)
F_{55}\! \left(x \right) = F_{33}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{57}\! \left(x \right)+F_{71}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{34}\! \left(x \right)+F_{59}\! \left(x \right)+F_{69}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{64}\! \left(x \right)
F_{61}\! \left(x \right) = F_{11}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{24}\! \left(x \right)+F_{34}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{64}\! \left(x \right) = F_{58}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{57}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{31}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{57}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{27}\! \left(x \right)+F_{58}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{67}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{80}\! \left(x \right)
F_{74}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{75}\! \left(x \right)+F_{76}\! \left(x \right)
F_{75}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{79}\! \left(x \right)
F_{78}\! \left(x \right) = F_{31}\! \left(x \right)
F_{79}\! \left(x \right) = F_{74}\! \left(x \right)
F_{80}\! \left(x \right) = 3 F_{34}\! \left(x \right)+F_{81}\! \left(x \right)+F_{82}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{85}\! \left(x \right)
F_{84}\! \left(x \right) = F_{68}\! \left(x \right)
F_{85}\! \left(x \right) = F_{80}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{70}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{97}\! \left(x \right)
F_{89}\! \left(x \right) = F_{34}\! \left(x \right)+F_{90}\! \left(x \right)+F_{94}\! \left(x \right)
F_{90}\! \left(x \right) = F_{4}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)+F_{93}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right)
F_{93}\! \left(x \right) = F_{31}\! \left(x \right)
F_{94}\! \left(x \right) = F_{4}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{30}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{89}\! \left(x \right)
F_{97}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{102}\! \left(x \right)+F_{98}\! \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_{101}\! \left(x \right)
F_{100}\! \left(x \right) = F_{23}\! \left(x \right)
F_{101}\! \left(x \right) = F_{68}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right) F_{4}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{105}\! \left(x \right)
F_{104}\! \left(x \right) = F_{58}\! \left(x \right)+F_{68}\! \left(x \right)
F_{105}\! \left(x \right) = F_{97}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_26(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_19(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_4(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_11(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_11(x)*F_4(x))
Eq(F_25(x), F_19(x))
Eq(F_26(x), F_27(x) + F_33(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_17(x) + F_30(x))
Eq(F_30(x), F_27(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_27(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_35(x) + F_86(x))
Eq(F_34(x), 0)
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_55(x))
Eq(F_37(x), F_38(x) + F_7(x))
Eq(F_38(x), F_24(x) + F_34(x) + F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_23(x) + F_4(x))
Eq(F_42(x), F_43(x) + F_49(x))
Eq(F_43(x), F_34(x) + F_44(x) + F_45(x))
Eq(F_44(x), x**2)
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_4(x))
Eq(F_48(x), F_43(x))
Eq(F_49(x), 2*F_34(x) + F_50(x) + F_51(x))
Eq(F_50(x), F_23(x)*F_4(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_23(x))
Eq(F_54(x), F_49(x))
Eq(F_55(x), F_33(x) + F_56(x))
Eq(F_56(x), 2*F_34(x) + F_57(x) + F_71(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_34(x) + F_59(x) + F_69(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_64(x))
Eq(F_61(x), F_11(x) + F_62(x))
Eq(F_62(x), F_24(x) + F_34(x) + F_63(x))
Eq(F_63(x), F_4(x)*F_41(x))
Eq(F_64(x), F_58(x) + F_65(x))
Eq(F_65(x), 2*F_34(x) + F_57(x) + F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_31(x) + F_68(x))
Eq(F_68(x), F_57(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_27(x) + F_58(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_67(x) + F_73(x))
Eq(F_73(x), F_74(x) + F_80(x))
Eq(F_74(x), 2*F_34(x) + F_75(x) + F_76(x))
Eq(F_75(x), F_31(x)*F_4(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_78(x) + F_79(x))
Eq(F_78(x), F_31(x))
Eq(F_79(x), F_74(x))
Eq(F_80(x), 3*F_34(x) + F_81(x) + F_82(x))
Eq(F_81(x), F_4(x)*F_68(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_84(x) + F_85(x))
Eq(F_84(x), F_68(x))
Eq(F_85(x), F_80(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), F_70(x) + F_88(x))
Eq(F_88(x), F_89(x) + F_97(x))
Eq(F_89(x), F_34(x) + F_90(x) + F_94(x))
Eq(F_90(x), F_4(x)*F_91(x))
Eq(F_91(x), F_92(x) + F_93(x))
Eq(F_92(x), F_4(x))
Eq(F_93(x), F_31(x))
Eq(F_94(x), F_4(x)*F_95(x))
Eq(F_95(x), F_30(x) + F_96(x))
Eq(F_96(x), F_89(x))
Eq(F_97(x), F_102(x) + 2*F_34(x) + F_98(x))
Eq(F_98(x), F_4(x)*F_99(x))
Eq(F_99(x), F_100(x) + F_101(x))
Eq(F_100(x), F_23(x))
Eq(F_101(x), F_68(x))
Eq(F_102(x), F_103(x)*F_4(x))
Eq(F_103(x), F_104(x) + F_105(x))
Eq(F_104(x), F_58(x) + F_68(x))
Eq(F_105(x), F_97(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 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, 1], [0, 1], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 1], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 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, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 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, 2], [1, 2], [1, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[1, 2], [1, 0], [1, 0], [1, 2]]}, {"patt": [3, 0, 1, 2], "pos": [[1, 2], [1, 0], [1, 2], [1, 2]]}, {"patt": [3, 0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [3, 0, 2, 1], "pos": [[1, 2], [1, 0], [1, 2], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[1, 2], [1, 0], [1, 2], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 0], [1, 0], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}, {"patt": [3, 1, 0, 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": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 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, 1], [0, 1], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 1], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 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": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 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, 1], [0, 1], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 1], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 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