0132_1023_1320
Counting sequence:
1, 1, 2, 6, 21, 72, 233, 719, 2146, 6260, 17968, 50967, 143278, 399960, 1110203, 3067479, 8442903, 23163006, 63371999, 172967309, 471115792, 1280844662, 3476636122, 9423007521, 25506378316, 68958653982, 186231636833, 502435772049, 1354261840761, 3647103861120, 9813907809653, 26387986672907, 70902355932718, 190380679033640, 510868144637428, 1370039225878299, 3672050302657498, 9836663086593876, 26336719386518519, 70479134632789179, 188518513524291147, 504023617762422642, 1346974840547491067, 3598218514351661561, 9608205535579672636, 25646643711138096842, 68431914531270476158, 182529373785230170629, 486696742841631449176, 1297301990976648577530, 3456891698161777049213, 9208678544659342378197, 24523376102344906478733, 65288837372742221723220, 173771059643579218525553, 462380710475497156447319, 1230012225634322380853818, 3271222999391503590971996, 8697716595983689137593464, 23120538978228052948165599, 61445676579859278074946502, 163263206268337002213971472, 433702310415284355910905587, 1151872109789528678375510751, 3058640796552295605890631231, 8120202210270274361210212902, 21553694812127715573994234583, 57199917156683465637389980709, 151771432324191320482097535016, 402631471584366228766270400990, 1067948881459323371112479964898, 2832175776880444314562560647337, 7509634359903357663229110118948, 19908934425815531152751266732950, 52772734365449418455972924100953, 139863757870309868452651659971289, 370626441354404138607659478070113, 981984783233714069937503861456616, 2601427657185521127334529072785709, 6890628217444209471672713778110819, 18249347332954605004977640486206982, 48325754764266249658811811207459152, 127954049567629821915507916879007932, 338746450773164534569963475581013091, 896689340635612801100940702828474226, 2373323627925898860323085434858486412, 6280883675585443776728238018481815983, 16620131739267946625823154847336761539, 43974322430884479034070625085692396531, 116336463878535190341746415950082308250, 307741143290672367690017445951218388083, 813971559924503215218591132112031554577, 2152721244186545574629727962067615915988, 5692750701808351054703698651151450033330, 15052658741040525105474753142793280911878, 39798050631517896127773867933133443965709, 105212840604268216118206259480270298329008, 278121688423232110458216289457196523231986, 735124528940276404382811685424532205296629, 1942887593979726194623026172962058155026589, 5134473035469490428208148803314280985550837
Generating function in Maple syntax:
(x^10-4*x^9-6*x^8+68*x^7-186*x^6+291*x^5-283*x^4+170*x^3-61*x^2+12*x-1)/(2*x-1)^2/(x^2-3*x+1)^2/(x-1)^3
Generating function in latex syntax:
\frac{x^{10}-4 x^{9}-6 x^{8}+68 x^{7}-186 x^{6}+291 x^{5}-283 x^{4}+170 x^{3}-61 x^{2}+12 x -1}{\left(2 x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**10 - 4*x**9 - 6*x**8 + 68*x**7 - 186*x**6 + 291*x**5 - 283*x**4 + 170*x**3 - 61*x**2 + 12*x - 1)/((x - 1)**3*(2*x - 1)**2*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x^2-3*x+1)^2*(x-1)^3*F(x)+x^10-4*x^9-6*x^8+68*x^7-186*x^6+291*x^5-283*x^4+170*x^3-61*x^2+12*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+x^{10}-4 x^{9}-6 x^{8}+68 x^{7}-186 x^{6}+291 x^{5}-283 x^{4}+170 x^{3}-61 x^{2}+12 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/200*((-20*n+4)*5^(1/2)+60*n+100)*(3/2-1/2*5^(1/2))^(-n)+1/200*((20*n-4)*5^(1/2)+60*n+100)*(3/2+1/2*5^(1/2))^(-n)-1/2*n^2-1/8*2^n*n+3/2*n-2^(-1+n)-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(\left(-20 n +4\right) \sqrt{5}+60 n +100\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{200}+\\\frac{\left(\left(20 n -4\right) \sqrt{5}+60 n +100\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{200}-\frac{n^{2}}{2}-\frac{2^{n} n}{8}+\frac{3 n}{2}-2^{-1+n}-\\1 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 72
a(6) = 233
a(7) = 719
a(8) = 2146
a(9) = 6260
a(10) = 17968
a(n+6) = -1/2*n^2-4*a(n)+28*a(n+1)-69*a(n+2)+74*a(n+3)-39*a(n+4)+10*a(n+5)+3/2*n+3, n >= 11
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(5\right) = 72
a \! \left(6\right) = 233
a \! \left(7\right) = 719
a \! \left(8\right) = 2146
a \! \left(9\right) = 6260
a \! \left(10\right) = 17968
a \! \left(n +6\right) = -\frac{n^{2}}{2}-4 a \! \left(n \right)+28 a \! \left(n +1\right)-69 a \! \left(n +2\right)+74 a \! \left(n +3\right)-39 a \! \left(n +4\right)+10 a \! \left(n +5\right)+\frac{3 n}{2}+3, \quad n \geq 11
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14706/
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[27,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[7,x]
F[14,x] = F[15,x]+F[16,x]+F[26,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]+F[4,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,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[4,x]*F[7,x]
F[24,x] = F[25,x]
F[25,x] = F[21,x]*F[4,x]
F[26,x] = F[13,x]*F[4,x]
F[27,x] = F[2,x]+F[28,x]
F[28,x] = F[116,x]+F[15,x]+F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[45,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[1,x]+F[32,x]
F[35,x] = F[15,x]+F[16,x]+F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[7,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[18,x]+F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[39,x]
F[43,x] = F[44,x]
F[44,x] = F[4,x]*F[41,x]
F[45,x] = F[46,x]+F[66,x]
F[46,x] = F[15,x]+F[29,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[2,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[32,x]+F[38,x]
F[53,x] = F[49,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[58,x]+F[64,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[4,x]+F[42,x]
F[62,x] = F[58,x]+F[63,x]
F[63,x] = F[55,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[57,x]
F[66,x] = F[114,x]+F[15,x]+F[67,x]+F[80,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[69,x]+F[76,x]
F[69,x] = F[70,x]+F[73,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[32,x]
F[73,x] = F[74,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[38,x]
F[76,x] = F[77,x]
F[77,x] = 2*F[15,x]+F[67,x]+F[78,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[49,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[104,x]+F[82,x]
F[82,x] = F[102,x]+F[83,x]
F[83,x] = F[101,x]+F[15,x]+F[84,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[86,x]+F[87,x]
F[86,x] = F[22,x]+F[4,x]
F[87,x] = F[83,x]+F[88,x]
F[88,x] = F[15,x]+F[80,x]+F[89,x]+F[99,x]
F[89,x] = F[4,x]*F[90,x]
F[90,x] = F[91,x]+F[96,x]
F[91,x] = F[92,x]+F[94,x]
F[92,x] = F[93,x]
F[93,x] = F[32,x]*F[4,x]
F[94,x] = F[95,x]
F[95,x] = F[38,x]*F[4,x]
F[96,x] = F[97,x]
F[97,x] = 2*F[15,x]+F[89,x]+F[98,x]
F[98,x] = F[4,x]*F[49,x]
F[99,x] = F[100,x]*F[4,x]
F[100,x] = F[50,x]
F[101,x] = F[2,x]*F[4,x]
F[102,x] = F[103,x]
F[103,x] = F[4,x]*F[82,x]
F[104,x] = F[105,x]+F[112,x]
F[105,x] = 2*F[15,x]+F[106,x]+F[99,x]
F[106,x] = F[107,x]*F[4,x]
F[107,x] = F[108,x]+F[110,x]
F[108,x] = F[109,x]+F[92,x]
F[109,x] = F[95,x]
F[110,x] = F[111,x]
F[111,x] = 2*F[15,x]+F[106,x]+F[98,x]
F[112,x] = F[113,x]
F[113,x] = F[104,x]*F[4,x]
F[114,x] = F[115,x]*F[4,x]
F[115,x] = F[100,x]
F[116,x] = F[117,x]*F[4,x]
F[117,x] = F[118,x]+F[121,x]
F[118,x] = F[119,x]+F[2,x]
F[119,x] = F[120,x]+F[15,x]+F[84,x]
F[120,x] = F[118,x]*F[4,x]
F[121,x] = F[100,x]+F[122,x]
F[122,x] = 2*F[15,x]+F[106,x]+F[123,x]
F[123,x] = F[121,x]*F[4,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_{27}\! \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_{7}\! \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_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \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_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{2}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{116}\! \left(x \right)+F_{15}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{45}\! \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_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{1}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{7}\! \left(x \right)
F_{38}\! \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_{18}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{39}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{66}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{32}\! \left(x \right)+F_{38}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \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_{64}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \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_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right)+F_{42}\! \left(x \right)
F_{62}\! \left(x \right) = F_{58}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{55}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{66}\! \left(x \right) = F_{114}\! \left(x \right)+F_{15}\! \left(x \right)+F_{67}\! \left(x \right)+F_{80}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{76}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{73}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{32}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{38}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{67}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{49}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{104}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{102}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{101}\! \left(x \right)+F_{15}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)+F_{87}\! \left(x \right)
F_{86}\! \left(x \right) = F_{22}\! \left(x \right)+F_{4}\! \left(x \right)
F_{87}\! \left(x \right) = F_{83}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{15}\! \left(x \right)+F_{80}\! \left(x \right)+F_{89}\! \left(x \right)+F_{99}\! \left(x \right)
F_{89}\! \left(x \right) = F_{4}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)+F_{96}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)+F_{94}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{89}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right) F_{4}\! \left(x \right)
F_{100}\! \left(x \right) = F_{50}\! \left(x \right)
F_{101}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)
F_{103}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{112}\! \left(x \right)
F_{105}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{106}\! \left(x \right)+F_{99}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right) F_{4}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)+F_{110}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{92}\! \left(x \right)
F_{109}\! \left(x \right) = F_{95}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)
F_{111}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{106}\! \left(x \right)+F_{98}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)
F_{113}\! \left(x \right) = F_{104}\! \left(x \right) F_{4}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right) F_{4}\! \left(x \right)
F_{115}\! \left(x \right) = F_{100}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right) F_{4}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)+F_{121}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right)+F_{2}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)+F_{15}\! \left(x \right)+F_{84}\! \left(x \right)
F_{120}\! \left(x \right) = F_{118}\! \left(x \right) F_{4}\! \left(x \right)
F_{121}\! \left(x \right) = F_{100}\! \left(x \right)+F_{122}\! \left(x \right)
F_{122}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{106}\! \left(x \right)+F_{123}\! \left(x \right)
F_{123}\! \left(x \right) = F_{121}\! \left(x \right) F_{4}\! \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_27(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_7(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_26(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x) + F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(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_4(x)*F_7(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_21(x)*F_4(x))
Eq(F_26(x), F_13(x)*F_4(x))
Eq(F_27(x), F_2(x) + F_28(x))
Eq(F_28(x), F_116(x) + F_15(x) + F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_45(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_1(x) + F_32(x))
Eq(F_35(x), F_15(x) + F_16(x) + F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_7(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_18(x) + F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_39(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_4(x)*F_41(x))
Eq(F_45(x), F_46(x) + F_66(x))
Eq(F_46(x), F_15(x) + F_29(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_2(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_32(x) + F_38(x))
Eq(F_53(x), F_49(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_58(x) + F_64(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_4(x) + F_42(x))
Eq(F_62(x), F_58(x) + F_63(x))
Eq(F_63(x), F_55(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_57(x))
Eq(F_66(x), F_114(x) + F_15(x) + F_67(x) + F_80(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_76(x))
Eq(F_69(x), F_70(x) + F_73(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_32(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_38(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), 2*F_15(x) + F_67(x) + F_78(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_49(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_104(x) + F_82(x))
Eq(F_82(x), F_102(x) + F_83(x))
Eq(F_83(x), F_101(x) + F_15(x) + F_84(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_86(x) + F_87(x))
Eq(F_86(x), F_22(x) + F_4(x))
Eq(F_87(x), F_83(x) + F_88(x))
Eq(F_88(x), F_15(x) + F_80(x) + F_89(x) + F_99(x))
Eq(F_89(x), F_4(x)*F_90(x))
Eq(F_90(x), F_91(x) + F_96(x))
Eq(F_91(x), F_92(x) + F_94(x))
Eq(F_92(x), F_93(x))
Eq(F_93(x), F_32(x)*F_4(x))
Eq(F_94(x), F_95(x))
Eq(F_95(x), F_38(x)*F_4(x))
Eq(F_96(x), F_97(x))
Eq(F_97(x), 2*F_15(x) + F_89(x) + F_98(x))
Eq(F_98(x), F_4(x)*F_49(x))
Eq(F_99(x), F_100(x)*F_4(x))
Eq(F_100(x), F_50(x))
Eq(F_101(x), F_2(x)*F_4(x))
Eq(F_102(x), F_103(x))
Eq(F_103(x), F_4(x)*F_82(x))
Eq(F_104(x), F_105(x) + F_112(x))
Eq(F_105(x), F_106(x) + 2*F_15(x) + F_99(x))
Eq(F_106(x), F_107(x)*F_4(x))
Eq(F_107(x), F_108(x) + F_110(x))
Eq(F_108(x), F_109(x) + F_92(x))
Eq(F_109(x), F_95(x))
Eq(F_110(x), F_111(x))
Eq(F_111(x), F_106(x) + 2*F_15(x) + F_98(x))
Eq(F_112(x), F_113(x))
Eq(F_113(x), F_104(x)*F_4(x))
Eq(F_114(x), F_115(x)*F_4(x))
Eq(F_115(x), F_100(x))
Eq(F_116(x), F_117(x)*F_4(x))
Eq(F_117(x), F_118(x) + F_121(x))
Eq(F_118(x), F_119(x) + F_2(x))
Eq(F_119(x), F_120(x) + F_15(x) + F_84(x))
Eq(F_120(x), F_118(x)*F_4(x))
Eq(F_121(x), F_100(x) + F_122(x))
Eq(F_122(x), F_106(x) + F_123(x) + 2*F_15(x))
Eq(F_123(x), F_121(x)*F_4(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, 2, 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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 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, 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, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 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, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"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], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}], "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, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}], "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, 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], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 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, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"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], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [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, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 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, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"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], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [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", 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