0132_0321_1302_2310
Counting sequence:
1, 1, 2, 6, 20, 63, 184, 505, 1322, 3339, 8204, 19725, 46606, 108559, 249872, 569361, 1286162, 2883603, 6422548, 14221333, 31326230, 68681751, 149946392, 326107161, 706740250, 1526726683, 3288334364, 7063207965, 15133048862, 32346472479, 68987912224, 146834194465, 311921999906, 661424963619, 1400159338532, 2959232466981, 6244882448422, 13159779794983, 27693949124648, 58205396795433, 122183229636650, 256186209271851, 536561674354732, 1122601371959341, 2346357813674030, 4899423813369903, 10221060091805744, 21304137299787825, 44367493204017202, 92323792361095219, 191965934116667444, 398850041998999605, 828099381482750006, 1718123257841844279, 3562347305250062392, 7381399789260243001, 15285217135295463482, 31633283782650363963, 65428295386438565948, 135252104009190735933, 279439349679084535870, 577037213055726911551, 1190967914258872926272, 2456875726317190905921, 5065937091242485612610, 10440857145719606214723, 21508903589945337184324, 44290632520976633430085, 91163809212272604086342, 187566493741478720831559, 385758312069414143393864, 793062421216921043075145, 1629806732400386304376906, 3348157836354578456510539, 6875765599058203431141452, 14115153417297369543737421, 28966996005922403740811342, 59426259820431615369150543, 121874834189899803675066448, 249869855341598667547082833, 512131200334247284134903890, 1049347611425498123644960851, 2149470107274810672627581012, 4401698909216864825105186901, 9011333059407445868259835990, 18443372304040782689317421143, 37737828385090264317627990104, 77197167137311760580037574745, 157876040635113653183228936282, 322792865243662906679946641499, 659822040939107686521233211484, 1348426187791600464433871061085, 2755035557429613801788000960606, 5627675418591338729691418722399, 11493035324725470472163469295712, 23466391384693668490987798790241, 47903327760187075117496510972002, 97767552542602192590433234714723, 199496513210917602060543666946148, 406995070835775902218035272876133, 830152686824461729305153511620710
Generating function in Maple syntax:
(3*x^5-12*x^4+23*x^3-19*x^2+7*x-1)/(x-1)^2/(2*x-1)^3
Generating function in latex syntax:
\frac{3 x^{5}-12 x^{4}+23 x^{3}-19 x^{2}+7 x -1}{\left(x -1\right)^{2} \left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(3*x**5 - 12*x**4 + 23*x**3 - 19*x**2 + 7*x - 1)/((x - 1)**2*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
(x-1)^2*(2*x-1)^3*F(x)-3*x^5+12*x^4-23*x^3+19*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} \left(2 x -1\right)^{3} F \! \left(x \right)-3 x^{5}+12 x^{4}-23 x^{3}+19 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/16*(n^2+5*n-22)*2^n+n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(n^{2}+5 n -22\right) 2^{n}}{16}+n +2 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 63
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)-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) = 20
a \! \left(5\right) = 63
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-n +1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15455/
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[20,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[10,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[16,x]
F[19,x] = F[12,x]*F[13,x]
F[20,x] = F[21,x]+F[52,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[32,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[9,x]
F[28,x] = F[12,x]+F[29,x]
F[29,x] = F[15,x]+F[30,x]+F[31,x]
F[30,x] = F[10,x]*F[12,x]
F[31,x] = F[12,x]*F[28,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[1,x]+F[33,x]
F[36,x] = F[15,x]+F[37,x]+F[39,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[25,x]+F[36,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[46,x]
F[41,x] = F[33,x]+F[42,x]
F[42,x] = F[15,x]+F[43,x]+F[45,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[10,x]+F[42,x]
F[45,x] = F[12,x]*F[41,x]
F[46,x] = F[47,x]+F[49,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[33,x]
F[49,x] = 2*F[15,x]+F[50,x]+F[51,x]
F[50,x] = F[12,x]*F[42,x]
F[51,x] = F[12,x]*F[46,x]
F[52,x] = F[15,x]+F[53,x]+F[88,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[64,x]
F[55,x] = F[56,x]+F[6,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[17,x]+F[59,x]
F[59,x] = F[60,x]+F[61,x]
F[60,x] = F[30,x]
F[61,x] = 2*F[15,x]+F[62,x]+F[63,x]
F[62,x] = F[12,x]*F[18,x]
F[63,x] = F[12,x]*F[59,x]
F[64,x] = F[65,x]+F[78,x]
F[65,x] = F[15,x]+F[66,x]+F[68,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[6,x]+F[65,x]
F[68,x] = F[12,x]*F[69,x]
F[69,x] = F[41,x]+F[70,x]
F[70,x] = F[42,x]+F[71,x]
F[71,x] = F[15,x]+F[72,x]+F[75,x]+F[77,x]
F[72,x] = F[12,x]*F[73,x]
F[73,x] = F[18,x]+F[74,x]
F[74,x] = 2*F[15,x]+F[72,x]+F[75,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[42,x]+F[74,x]
F[77,x] = F[12,x]*F[70,x]
F[78,x] = 2*F[15,x]+F[79,x]+F[81,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[56,x]+F[78,x]
F[81,x] = F[12,x]*F[82,x]
F[82,x] = F[76,x]+F[83,x]
F[83,x] = F[84,x]+F[85,x]
F[84,x] = F[50,x]
F[85,x] = 3*F[15,x]+F[86,x]+F[87,x]
F[86,x] = F[12,x]*F[74,x]
F[87,x] = F[12,x]*F[83,x]
F[88,x] = F[12,x]*F[89,x]
F[89,x] = F[101,x]+F[90,x]
F[90,x] = F[21,x]+F[91,x]
F[91,x] = F[100,x]+F[15,x]+F[92,x]
F[92,x] = F[12,x]*F[93,x]
F[93,x] = F[94,x]+F[96,x]
F[94,x] = F[10,x]+F[95,x]
F[95,x] = F[57,x]
F[96,x] = F[42,x]+F[97,x]
F[97,x] = 2*F[15,x]+F[81,x]+F[98,x]
F[98,x] = F[12,x]*F[99,x]
F[99,x] = F[95,x]+F[97,x]
F[100,x] = F[12,x]*F[90,x]
F[101,x] = F[102,x]+F[103,x]
F[102,x] = F[15,x]+F[45,x]+F[92,x]
F[103,x] = F[104,x]+F[134,x]+F[15,x]+F[75,x]
F[104,x] = F[105,x]*F[12,x]
F[105,x] = F[106,x]+F[118,x]
F[106,x] = F[107,x]+F[18,x]
F[107,x] = F[108,x]
F[108,x] = F[109,x]*F[12,x]
F[109,x] = F[110,x]+F[113,x]
F[110,x] = F[111,x]+F[18,x]
F[111,x] = F[112,x]
F[112,x] = F[110,x]*F[12,x]
F[113,x] = F[114,x]+F[115,x]
F[114,x] = F[62,x]
F[115,x] = 3*F[15,x]+F[116,x]+F[117,x]
F[116,x] = F[111,x]*F[12,x]
F[117,x] = F[113,x]*F[12,x]
F[118,x] = F[119,x]+F[74,x]
F[119,x] = 3*F[15,x]+F[120,x]+F[122,x]
F[120,x] = F[12,x]*F[121,x]
F[121,x] = F[107,x]+F[119,x]
F[122,x] = F[12,x]*F[123,x]
F[123,x] = F[124,x]+F[129,x]
F[124,x] = F[125,x]+F[74,x]
F[125,x] = 3*F[15,x]+F[126,x]+F[128,x]
F[126,x] = F[12,x]*F[127,x]
F[127,x] = F[111,x]+F[125,x]
F[128,x] = F[12,x]*F[124,x]
F[129,x] = F[130,x]+F[131,x]
F[130,x] = F[86,x]
F[131,x] = 4*F[15,x]+F[132,x]+F[133,x]
F[132,x] = F[12,x]*F[125,x]
F[133,x] = F[12,x]*F[129,x]
F[134,x] = F[101,x]*F[12,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_{20}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \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_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{52}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{32}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \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_{9}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{25}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{46}\! \left(x \right)
F_{41}\! \left(x \right) = F_{33}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{49}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{52}\! \left(x \right) = F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{88}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{64}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{6}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{17}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = F_{30}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{78}\! \left(x \right)
F_{65}\! \left(x \right) = F_{15}\! \left(x \right)+F_{66}\! \left(x \right)+F_{68}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{6}\! \left(x \right)+F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{41}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{42}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{15}\! \left(x \right)+F_{72}\! \left(x \right)+F_{75}\! \left(x \right)+F_{77}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{18}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{72}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{42}\! \left(x \right)+F_{74}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{78}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{79}\! \left(x \right)+F_{81}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{56}\! \left(x \right)+F_{78}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{76}\! \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_{50}\! \left(x \right)
F_{85}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{86}\! \left(x \right)+F_{87}\! \left(x \right)
F_{86}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{83}\! \left(x \right)
F_{88}\! \left(x \right) = F_{12}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{101}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{21}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{100}\! \left(x \right)+F_{15}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{12}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)+F_{96}\! \left(x \right)
F_{94}\! \left(x \right) = F_{10}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{57}\! \left(x \right)
F_{96}\! \left(x \right) = F_{42}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{81}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{12}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{95}\! \left(x \right)+F_{97}\! \left(x \right)
F_{100}\! \left(x \right) = F_{12}\! \left(x \right) F_{90}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{103}\! \left(x \right)
F_{102}\! \left(x \right) = F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{92}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{134}\! \left(x \right)+F_{15}\! \left(x \right)+F_{75}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right) F_{12}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{118}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{18}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{12}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)+F_{113}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{18}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{110}\! \left(x \right) F_{12}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)+F_{115}\! \left(x \right)
F_{114}\! \left(x \right) = F_{62}\! \left(x \right)
F_{115}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{116}\! \left(x \right)+F_{117}\! \left(x \right)
F_{116}\! \left(x \right) = F_{111}\! \left(x \right) F_{12}\! \left(x \right)
F_{117}\! \left(x \right) = F_{113}\! \left(x \right) F_{12}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right)+F_{74}\! \left(x \right)
F_{119}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{120}\! \left(x \right)+F_{122}\! \left(x \right)
F_{120}\! \left(x \right) = F_{12}\! \left(x \right) F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = F_{107}\! \left(x \right)+F_{119}\! \left(x \right)
F_{122}\! \left(x \right) = F_{12}\! \left(x \right) F_{123}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)+F_{129}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)+F_{74}\! \left(x \right)
F_{125}\! \left(x \right) = 3 F_{15}\! \left(x \right)+F_{126}\! \left(x \right)+F_{128}\! \left(x \right)
F_{126}\! \left(x \right) = F_{12}\! \left(x \right) F_{127}\! \left(x \right)
F_{127}\! \left(x \right) = F_{111}\! \left(x \right)+F_{125}\! \left(x \right)
F_{128}\! \left(x \right) = F_{12}\! \left(x \right) F_{124}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)+F_{131}\! \left(x \right)
F_{130}\! \left(x \right) = F_{86}\! \left(x \right)
F_{131}\! \left(x \right) = 4 F_{15}\! \left(x \right)+F_{132}\! \left(x \right)+F_{133}\! \left(x \right)
F_{132}\! \left(x \right) = F_{12}\! \left(x \right) F_{125}\! \left(x \right)
F_{133}\! \left(x \right) = F_{12}\! \left(x \right) F_{129}\! \left(x \right)
F_{134}\! \left(x \right) = F_{101}\! \left(x \right) F_{12}\! \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_20(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_10(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_12(x)*F_13(x))
Eq(F_20(x), F_21(x) + F_52(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_32(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_9(x))
Eq(F_28(x), F_12(x) + F_29(x))
Eq(F_29(x), F_15(x) + F_30(x) + F_31(x))
Eq(F_30(x), F_10(x)*F_12(x))
Eq(F_31(x), F_12(x)*F_28(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_1(x) + F_33(x))
Eq(F_36(x), F_15(x) + F_37(x) + F_39(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_25(x) + F_36(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_46(x))
Eq(F_41(x), F_33(x) + F_42(x))
Eq(F_42(x), F_15(x) + F_43(x) + F_45(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_10(x) + F_42(x))
Eq(F_45(x), F_12(x)*F_41(x))
Eq(F_46(x), F_47(x) + F_49(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_33(x))
Eq(F_49(x), 2*F_15(x) + F_50(x) + F_51(x))
Eq(F_50(x), F_12(x)*F_42(x))
Eq(F_51(x), F_12(x)*F_46(x))
Eq(F_52(x), F_15(x) + F_53(x) + F_88(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_64(x))
Eq(F_55(x), F_56(x) + F_6(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_17(x) + F_59(x))
Eq(F_59(x), F_60(x) + F_61(x))
Eq(F_60(x), F_30(x))
Eq(F_61(x), 2*F_15(x) + F_62(x) + F_63(x))
Eq(F_62(x), F_12(x)*F_18(x))
Eq(F_63(x), F_12(x)*F_59(x))
Eq(F_64(x), F_65(x) + F_78(x))
Eq(F_65(x), F_15(x) + F_66(x) + F_68(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_6(x) + F_65(x))
Eq(F_68(x), F_12(x)*F_69(x))
Eq(F_69(x), F_41(x) + F_70(x))
Eq(F_70(x), F_42(x) + F_71(x))
Eq(F_71(x), F_15(x) + F_72(x) + F_75(x) + F_77(x))
Eq(F_72(x), F_12(x)*F_73(x))
Eq(F_73(x), F_18(x) + F_74(x))
Eq(F_74(x), 2*F_15(x) + F_72(x) + F_75(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_42(x) + F_74(x))
Eq(F_77(x), F_12(x)*F_70(x))
Eq(F_78(x), 2*F_15(x) + F_79(x) + F_81(x))
Eq(F_79(x), F_12(x)*F_80(x))
Eq(F_80(x), F_56(x) + F_78(x))
Eq(F_81(x), F_12(x)*F_82(x))
Eq(F_82(x), F_76(x) + F_83(x))
Eq(F_83(x), F_84(x) + F_85(x))
Eq(F_84(x), F_50(x))
Eq(F_85(x), 3*F_15(x) + F_86(x) + F_87(x))
Eq(F_86(x), F_12(x)*F_74(x))
Eq(F_87(x), F_12(x)*F_83(x))
Eq(F_88(x), F_12(x)*F_89(x))
Eq(F_89(x), F_101(x) + F_90(x))
Eq(F_90(x), F_21(x) + F_91(x))
Eq(F_91(x), F_100(x) + F_15(x) + F_92(x))
Eq(F_92(x), F_12(x)*F_93(x))
Eq(F_93(x), F_94(x) + F_96(x))
Eq(F_94(x), F_10(x) + F_95(x))
Eq(F_95(x), F_57(x))
Eq(F_96(x), F_42(x) + F_97(x))
Eq(F_97(x), 2*F_15(x) + F_81(x) + F_98(x))
Eq(F_98(x), F_12(x)*F_99(x))
Eq(F_99(x), F_95(x) + F_97(x))
Eq(F_100(x), F_12(x)*F_90(x))
Eq(F_101(x), F_102(x) + F_103(x))
Eq(F_102(x), F_15(x) + F_45(x) + F_92(x))
Eq(F_103(x), F_104(x) + F_134(x) + F_15(x) + F_75(x))
Eq(F_104(x), F_105(x)*F_12(x))
Eq(F_105(x), F_106(x) + F_118(x))
Eq(F_106(x), F_107(x) + F_18(x))
Eq(F_107(x), F_108(x))
Eq(F_108(x), F_109(x)*F_12(x))
Eq(F_109(x), F_110(x) + F_113(x))
Eq(F_110(x), F_111(x) + F_18(x))
Eq(F_111(x), F_112(x))
Eq(F_112(x), F_110(x)*F_12(x))
Eq(F_113(x), F_114(x) + F_115(x))
Eq(F_114(x), F_62(x))
Eq(F_115(x), F_116(x) + F_117(x) + 3*F_15(x))
Eq(F_116(x), F_111(x)*F_12(x))
Eq(F_117(x), F_113(x)*F_12(x))
Eq(F_118(x), F_119(x) + F_74(x))
Eq(F_119(x), F_120(x) + F_122(x) + 3*F_15(x))
Eq(F_120(x), F_12(x)*F_121(x))
Eq(F_121(x), F_107(x) + F_119(x))
Eq(F_122(x), F_12(x)*F_123(x))
Eq(F_123(x), F_124(x) + F_129(x))
Eq(F_124(x), F_125(x) + F_74(x))
Eq(F_125(x), F_126(x) + F_128(x) + 3*F_15(x))
Eq(F_126(x), F_12(x)*F_127(x))
Eq(F_127(x), F_111(x) + F_125(x))
Eq(F_128(x), F_12(x)*F_124(x))
Eq(F_129(x), F_130(x) + F_131(x))
Eq(F_130(x), F_86(x))
Eq(F_131(x), F_132(x) + F_133(x) + 4*F_15(x))
Eq(F_132(x), F_12(x)*F_125(x))
Eq(F_133(x), F_12(x)*F_129(x))
Eq(F_134(x), F_101(x)*F_12(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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 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": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 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], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 1, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "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": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 1, 0], "pos": [[0, 1], [0, 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": [2, 1, 0], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 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], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 1, 0], "pos": [[0, 0], [0, 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, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 0], [1, 0], [1, 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