0123_0132_0213_1320
Counting sequence:
1, 1, 2, 6, 20, 67, 215, 656, 1916, 5411, 14915, 40426, 108330, 288091, 762279, 2010198, 5289194, 13895829, 36469809, 95647954, 250728790, 657025397, 1721285157, 4508647276, 11808197100, 30922873317, 80974036825, 212026070096, 555157201156, 1453550557311, 3805702851115, 9963972060196, 26087037096640, 68298779709751, 178812571581079, 468145455668958, 1225636806828734, 3208790939299359, 8400787881848219, 21993676321301114, 57580448107487470, 150748081720853657, 394664623958616437, 1033247443095078246, 2705081009803950330, 7082002193001613417, 18540938778897928985, 48540840557143801536, 127081635709821656096, 332704172191340020361, 871031092077063549165, 2280389526424850866516, 5970138331901586940200, 15630027158581473120611, 40919946522273423579279, 107129819164820812172568, 280469524484901339144324, 734278781314576987040139, 1922366873507034612033579, 5032821947301023385078674, 13176099184581932597679250, 34495476038811559055298467, 90310329796578208383765967, 236435515080360878786403166, 618996218903358832936476210, 1620553148547390094584673117, 4242663240574104844160701945, 11107436600845421333443255226, 29079646617303007500340139486, 76131503361745062517858316061, 199314863689294721968815615565, 521813088148863571096707719636, 1366124401642743929829522074260, 3576560118550262482377221373421, 9363555957549829435333777832129, 24514107761182793436717227637528, 64178767340165679268094890189964, 168022194287648490098242011270919, 439887815579448264599191632601139, 1151641252564033221899785815015884, 3015035942339325190667614972351880, 7893466574907289853459832675431327, 20665363783289239253813848757365247, 54102624776773817477795938813857286, 141642510550658991998201130485709990, 370824906882456715634670784423880167, 970832210111218268301145604892005955, 2541671723480212314699651519186897666, 6654182960387447124459404373805830326, 17420877157798185952441793612495272161, 45608448513239224514632306095467525565, 119404468382383715145667690716731129806, 312604956634840376015715670530880621570, 818410401523994323063768808997976289681, 2142626247940856413460689184267442696129, 5609468342306002557824614730143160274088, 14685778778992006540922425561129938200904, 38447867994699727626593960776202757394449, 100657825205166597461891450708701126376149, 263525607620918907004706533240677095929244, 689918997657827808042771983098524585709360
Generating function in Maple syntax:
(3*x^10-4*x^8+18*x^7-22*x^6-13*x^5+56*x^4-61*x^3+33*x^2-9*x+1)/(2*x-1)/(x^2-3*x+1)/(x^2+x-1)/(x-1)^4
Generating function in latex syntax:
\frac{3 x^{10}-4 x^{8}+18 x^{7}-22 x^{6}-13 x^{5}+56 x^{4}-61 x^{3}+33 x^{2}-9 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(3*x**10 - 4*x**8 + 18*x**7 - 22*x**6 - 13*x**5 + 56*x**4 - 61*x**3 + 33*x**2 - 9*x + 1)/((x - 1)**4*(2*x - 1)*(x**2 - 3*x + 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(x^2-3*x+1)*(x^2+x-1)*(x-1)^4*F(x)+3*x^10-4*x^8+18*x^7-22*x^6-13*x^5+56*x^4-61*x^3+33*x^2-9*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+3 x^{10}-4 x^{8}+18 x^{7}-22 x^{6}-13 x^{5}+56 x^{4}-61 x^{3}+33 x^{2}-9 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/60*(432*5^(1/2)-900)*(3/2-1/2*5^(1/2))^(-n)+1/60*(-432*5^(1/2)-900)*(3/2+1/2*5^(1/2))^(-n)+1/60*(108*5^(1/2)-240)*(-1/2*5^(1/2)-1/2)^(-n)+1/60*(-108*5^(1/2)-240)*(1/2*5^(1/2)-1/2)^(-n)-1/3*n^3+9/2*n^2-67/6*n-3/4*2^n+30)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(432 \sqrt{5}-900\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{60}+\frac{\left(-432 \sqrt{5}-900\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{60}+\\\frac{\left(108 \sqrt{5}-240\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{60}+\frac{\left(-108 \sqrt{5}-240\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{60}-\frac{n^{3}}{3}+\\\frac{9 n^{2}}{2}-\frac{67 n}{6}-\frac{3 \,2^{n}}{4}+30 & \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) = 67
a(6) = 215
a(7) = 656
a(8) = 1916
a(9) = 5411
a(10) = 14915
a(n+5) = 1/3*n^3-11/2*n^2-2*a(n)+5*a(n+1)+4*a(n+2)-11*a(n+3)+6*a(n+4)+79/6*n-10, 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) = 20
a \! \left(5\right) = 67
a \! \left(6\right) = 215
a \! \left(7\right) = 656
a \! \left(8\right) = 1916
a \! \left(9\right) = 5411
a \! \left(10\right) = 14915
a \! \left(n +5\right) = \frac{n^{3}}{3}-\frac{11 n^{2}}{2}-2 a \! \left(n \right)+5 a \! \left(n +1\right)+4 a \! \left(n +2\right)-11 a \! \left(n +3\right)+6 a \! \left(n +4\right)+\frac{79 n}{6}-10, \quad n \geq 11
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14787/
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[14,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[4,x]*F[7,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[150,x]+F[16,x]+F[17,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[33,x]
F[19,x] = F[20,x]+F[7,x]
F[20,x] = F[16,x]+F[21,x]+F[25,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[24,x]
F[23,x] = F[4,x]
F[24,x] = F[12,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[11,x]+F[27,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[23,x]
F[31,x] = F[32,x]
F[32,x] = F[28,x]*F[4,x]
F[33,x] = F[15,x]+F[34,x]
F[34,x] = F[119,x]+F[147,x]+F[16,x]+F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[45,x]
F[37,x] = F[20,x]+F[38,x]
F[38,x] = F[16,x]+F[39,x]+F[40,x]+F[43,x]
F[39,x] = 0
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[24,x]+F[42,x]
F[42,x] = F[31,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[27,x]
F[45,x] = F[34,x]+F[46,x]
F[46,x] = F[116,x]+F[16,x]+F[47,x]+F[76,x]+F[77,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[53,x]
F[49,x] = F[38,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[42,x]
F[53,x] = F[46,x]+F[54,x]
F[54,x] = F[16,x]+F[55,x]+F[59,x]+F[60,x]+F[61,x]+F[75,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[50,x]
F[58,x] = F[54,x]
F[59,x] = 0
F[60,x] = 0
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[64,x]
F[64,x] = 3*F[16,x]+F[65,x]+F[69,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[68,x]
F[67,x] = F[31,x]
F[68,x] = F[64,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[73,x]+F[74,x]
F[73,x] = F[28,x]
F[74,x] = F[70,x]
F[75,x] = 0
F[76,x] = 0
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[63,x]+F[79,x]
F[79,x] = F[80,x]
F[80,x] = 2*F[16,x]+F[81,x]+F[92,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[83,x]+F[85,x]
F[83,x] = F[12,x]+F[84,x]
F[84,x] = F[32,x]
F[85,x] = F[80,x]+F[86,x]
F[86,x] = 2*F[16,x]+F[69,x]+F[87,x]+F[91,x]
F[87,x] = F[4,x]*F[88,x]
F[88,x] = F[89,x]+F[90,x]
F[89,x] = F[84,x]
F[90,x] = F[86,x]
F[91,x] = 0
F[92,x] = F[4,x]*F[93,x]
F[93,x] = F[94,x]
F[94,x] = F[4,x]*F[95,x]
F[95,x] = F[96,x]+F[97,x]
F[96,x] = F[28,x]+F[7,x]
F[97,x] = F[93,x]+F[98,x]
F[98,x] = 2*F[16,x]+F[103,x]+F[99,x]
F[99,x] = F[100,x]*F[4,x]
F[100,x] = F[101,x]+F[102,x]
F[101,x] = F[28,x]
F[102,x] = F[98,x]
F[103,x] = F[104,x]*F[4,x]
F[104,x] = F[105,x]
F[105,x] = F[106,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[4,x]
F[109,x] = F[29,x]
F[110,x] = F[105,x]+F[111,x]
F[111,x] = 2*F[16,x]+F[103,x]+F[112,x]
F[112,x] = F[113,x]*F[4,x]
F[113,x] = F[114,x]+F[115,x]
F[114,x] = F[109,x]
F[115,x] = F[111,x]
F[116,x] = F[117,x]*F[4,x]
F[117,x] = F[118,x]
F[118,x] = F[64,x]+F[70,x]
F[119,x] = F[120,x]*F[4,x]
F[120,x] = F[121,x]+F[79,x]
F[121,x] = F[122,x]
F[122,x] = F[123,x]+F[146,x]+F[16,x]
F[123,x] = F[124,x]*F[4,x]
F[124,x] = F[125,x]+F[127,x]
F[125,x] = F[126,x]+F[4,x]
F[126,x] = F[13,x]+F[16,x]+F[21,x]
F[127,x] = F[122,x]+F[128,x]
F[128,x] = F[119,x]+F[129,x]+F[16,x]+F[92,x]
F[129,x] = F[130,x]*F[4,x]
F[130,x] = F[131,x]+F[133,x]
F[131,x] = F[126,x]+F[132,x]
F[132,x] = F[16,x]+F[32,x]+F[39,x]+F[40,x]
F[133,x] = F[128,x]+F[134,x]
F[134,x] = F[135,x]+F[16,x]+F[69,x]+F[76,x]+F[77,x]
F[135,x] = F[136,x]*F[4,x]
F[136,x] = F[137,x]+F[139,x]
F[137,x] = F[132,x]+F[138,x]
F[138,x] = F[51,x]
F[139,x] = F[134,x]+F[140,x]
F[140,x] = F[141,x]+F[145,x]+F[16,x]+F[59,x]+F[60,x]+F[61,x]
F[141,x] = F[142,x]*F[4,x]
F[142,x] = F[143,x]+F[144,x]
F[143,x] = F[138,x]
F[144,x] = F[140,x]
F[145,x] = 0
F[146,x] = F[2,x]*F[4,x]
F[147,x] = F[148,x]*F[4,x]
F[148,x] = F[118,x]+F[149,x]
F[149,x] = F[80,x]+F[93,x]
F[150,x] = F[151,x]*F[4,x]
F[151,x] = F[149,x]+F[152,x]
F[152,x] = F[122,x]+F[2,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_{14}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{150}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{33}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)
F_{23}\! \left(x \right) = F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \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_{23}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{119}\! \left(x \right)+F_{147}\! \left(x \right)+F_{16}\! \left(x \right)+F_{35}\! \left(x \right)
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_{45}\! \left(x \right)
F_{37}\! \left(x \right) = F_{20}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{16}\! \left(x \right)+F_{39}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{39}\! \left(x \right) = 0
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{24}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{31}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{27}\! \left(x \right)
F_{45}\! \left(x \right) = F_{34}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{116}\! \left(x \right)+F_{16}\! \left(x \right)+F_{47}\! \left(x \right)+F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{53}\! \left(x \right)
F_{49}\! \left(x \right) = F_{38}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{42}\! \left(x \right)
F_{53}\! \left(x \right) = F_{46}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{16}\! \left(x \right)+F_{55}\! \left(x \right)+F_{59}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)+F_{75}\! \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_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{50}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \left(x \right)
F_{59}\! \left(x \right) = 0
F_{60}\! \left(x \right) = 0
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = 3 F_{16}\! \left(x \right)+F_{65}\! \left(x \right)+F_{69}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{68}\! \left(x \right)
F_{67}\! \left(x \right) = F_{31}\! \left(x \right)
F_{68}\! \left(x \right) = F_{64}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \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_{73}\! \left(x \right)+F_{74}\! \left(x \right)
F_{73}\! \left(x \right) = F_{28}\! \left(x \right)
F_{74}\! \left(x \right) = F_{70}\! \left(x \right)
F_{75}\! \left(x \right) = 0
F_{76}\! \left(x \right) = 0
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{63}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{81}\! \left(x \right)+F_{92}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{85}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{32}\! \left(x \right)
F_{85}\! \left(x \right) = F_{80}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{69}\! \left(x \right)+F_{87}\! \left(x \right)+F_{91}\! \left(x \right)
F_{87}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{90}\! \left(x \right)
F_{89}\! \left(x \right) = F_{84}\! \left(x \right)
F_{90}\! \left(x \right) = F_{86}\! \left(x \right)
F_{91}\! \left(x \right) = 0
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{4}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{96}\! \left(x \right)+F_{97}\! \left(x \right)
F_{96}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)
F_{97}\! \left(x \right) = F_{93}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{103}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right) F_{4}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{102}\! \left(x \right)
F_{101}\! \left(x \right) = F_{28}\! \left(x \right)
F_{102}\! \left(x \right) = F_{98}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{4}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \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_{4}\! \left(x \right)
F_{109}\! \left(x \right) = F_{29}\! \left(x \right)
F_{110}\! \left(x \right) = F_{105}\! \left(x \right)+F_{111}\! \left(x \right)
F_{111}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{103}\! \left(x \right)+F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{4}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)+F_{115}\! \left(x \right)
F_{114}\! \left(x \right) = F_{109}\! \left(x \right)
F_{115}\! \left(x \right) = F_{111}\! \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_{118}\! \left(x \right) = F_{64}\! \left(x \right)+F_{70}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right) F_{4}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)+F_{79}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)+F_{146}\! \left(x \right)+F_{16}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right) F_{4}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)+F_{127}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)+F_{4}\! \left(x \right)
F_{126}\! \left(x \right) = F_{13}\! \left(x \right)+F_{16}\! \left(x \right)+F_{21}\! \left(x \right)
F_{127}\! \left(x \right) = F_{122}\! \left(x \right)+F_{128}\! \left(x \right)
F_{128}\! \left(x \right) = F_{119}\! \left(x \right)+F_{129}\! \left(x \right)+F_{16}\! \left(x \right)+F_{92}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right) F_{4}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right)+F_{133}\! \left(x \right)
F_{131}\! \left(x \right) = F_{126}\! \left(x \right)+F_{132}\! \left(x \right)
F_{132}\! \left(x \right) = F_{16}\! \left(x \right)+F_{32}\! \left(x \right)+F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{133}\! \left(x \right) = F_{128}\! \left(x \right)+F_{134}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)+F_{16}\! \left(x \right)+F_{69}\! \left(x \right)+F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right) F_{4}\! \left(x \right)
F_{136}\! \left(x \right) = F_{137}\! \left(x \right)+F_{139}\! \left(x \right)
F_{137}\! \left(x \right) = F_{132}\! \left(x \right)+F_{138}\! \left(x \right)
F_{138}\! \left(x \right) = F_{51}\! \left(x \right)
F_{139}\! \left(x \right) = F_{134}\! \left(x \right)+F_{140}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right)+F_{145}\! \left(x \right)+F_{16}\! \left(x \right)+F_{59}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right) F_{4}\! \left(x \right)
F_{142}\! \left(x \right) = F_{143}\! \left(x \right)+F_{144}\! \left(x \right)
F_{143}\! \left(x \right) = F_{138}\! \left(x \right)
F_{144}\! \left(x \right) = F_{140}\! \left(x \right)
F_{145}\! \left(x \right) = 0
F_{146}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{147}\! \left(x \right) = F_{148}\! \left(x \right) F_{4}\! \left(x \right)
F_{148}\! \left(x \right) = F_{118}\! \left(x \right)+F_{149}\! \left(x \right)
F_{149}\! \left(x \right) = F_{80}\! \left(x \right)+F_{93}\! \left(x \right)
F_{150}\! \left(x \right) = F_{151}\! \left(x \right) F_{4}\! \left(x \right)
F_{151}\! \left(x \right) = F_{149}\! \left(x \right)+F_{152}\! \left(x \right)
F_{152}\! \left(x \right) = F_{122}\! \left(x \right)+F_{2}\! \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_14(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_4(x)*F_7(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_150(x) + F_16(x) + F_17(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_33(x))
Eq(F_19(x), F_20(x) + F_7(x))
Eq(F_20(x), F_16(x) + F_21(x) + F_25(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_24(x))
Eq(F_23(x), F_4(x))
Eq(F_24(x), F_12(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_11(x) + F_27(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_23(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_28(x)*F_4(x))
Eq(F_33(x), F_15(x) + F_34(x))
Eq(F_34(x), F_119(x) + F_147(x) + F_16(x) + F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_45(x))
Eq(F_37(x), F_20(x) + F_38(x))
Eq(F_38(x), F_16(x) + F_39(x) + F_40(x) + F_43(x))
Eq(F_39(x), 0)
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_24(x) + F_42(x))
Eq(F_42(x), F_31(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_27(x))
Eq(F_45(x), F_34(x) + F_46(x))
Eq(F_46(x), F_116(x) + F_16(x) + F_47(x) + F_76(x) + F_77(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_53(x))
Eq(F_49(x), F_38(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_42(x))
Eq(F_53(x), F_46(x) + F_54(x))
Eq(F_54(x), F_16(x) + F_55(x) + F_59(x) + F_60(x) + F_61(x) + F_75(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_50(x))
Eq(F_58(x), F_54(x))
Eq(F_59(x), 0)
Eq(F_60(x), 0)
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), 3*F_16(x) + F_65(x) + F_69(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_68(x))
Eq(F_67(x), F_31(x))
Eq(F_68(x), F_64(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_74(x))
Eq(F_73(x), F_28(x))
Eq(F_74(x), F_70(x))
Eq(F_75(x), 0)
Eq(F_76(x), 0)
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_63(x) + F_79(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), 2*F_16(x) + F_81(x) + F_92(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_85(x))
Eq(F_83(x), F_12(x) + F_84(x))
Eq(F_84(x), F_32(x))
Eq(F_85(x), F_80(x) + F_86(x))
Eq(F_86(x), 2*F_16(x) + F_69(x) + F_87(x) + F_91(x))
Eq(F_87(x), F_4(x)*F_88(x))
Eq(F_88(x), F_89(x) + F_90(x))
Eq(F_89(x), F_84(x))
Eq(F_90(x), F_86(x))
Eq(F_91(x), 0)
Eq(F_92(x), F_4(x)*F_93(x))
Eq(F_93(x), F_94(x))
Eq(F_94(x), F_4(x)*F_95(x))
Eq(F_95(x), F_96(x) + F_97(x))
Eq(F_96(x), F_28(x) + F_7(x))
Eq(F_97(x), F_93(x) + F_98(x))
Eq(F_98(x), F_103(x) + 2*F_16(x) + F_99(x))
Eq(F_99(x), F_100(x)*F_4(x))
Eq(F_100(x), F_101(x) + F_102(x))
Eq(F_101(x), F_28(x))
Eq(F_102(x), F_98(x))
Eq(F_103(x), F_104(x)*F_4(x))
Eq(F_104(x), F_105(x))
Eq(F_105(x), F_106(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_4(x))
Eq(F_109(x), F_29(x))
Eq(F_110(x), F_105(x) + F_111(x))
Eq(F_111(x), F_103(x) + F_112(x) + 2*F_16(x))
Eq(F_112(x), F_113(x)*F_4(x))
Eq(F_113(x), F_114(x) + F_115(x))
Eq(F_114(x), F_109(x))
Eq(F_115(x), F_111(x))
Eq(F_116(x), F_117(x)*F_4(x))
Eq(F_117(x), F_118(x))
Eq(F_118(x), F_64(x) + F_70(x))
Eq(F_119(x), F_120(x)*F_4(x))
Eq(F_120(x), F_121(x) + F_79(x))
Eq(F_121(x), F_122(x))
Eq(F_122(x), F_123(x) + F_146(x) + F_16(x))
Eq(F_123(x), F_124(x)*F_4(x))
Eq(F_124(x), F_125(x) + F_127(x))
Eq(F_125(x), F_126(x) + F_4(x))
Eq(F_126(x), F_13(x) + F_16(x) + F_21(x))
Eq(F_127(x), F_122(x) + F_128(x))
Eq(F_128(x), F_119(x) + F_129(x) + F_16(x) + F_92(x))
Eq(F_129(x), F_130(x)*F_4(x))
Eq(F_130(x), F_131(x) + F_133(x))
Eq(F_131(x), F_126(x) + F_132(x))
Eq(F_132(x), F_16(x) + F_32(x) + F_39(x) + F_40(x))
Eq(F_133(x), F_128(x) + F_134(x))
Eq(F_134(x), F_135(x) + F_16(x) + F_69(x) + F_76(x) + F_77(x))
Eq(F_135(x), F_136(x)*F_4(x))
Eq(F_136(x), F_137(x) + F_139(x))
Eq(F_137(x), F_132(x) + F_138(x))
Eq(F_138(x), F_51(x))
Eq(F_139(x), F_134(x) + F_140(x))
Eq(F_140(x), F_141(x) + F_145(x) + F_16(x) + F_59(x) + F_60(x) + F_61(x))
Eq(F_141(x), F_142(x)*F_4(x))
Eq(F_142(x), F_143(x) + F_144(x))
Eq(F_143(x), F_138(x))
Eq(F_144(x), F_140(x))
Eq(F_145(x), 0)
Eq(F_146(x), F_2(x)*F_4(x))
Eq(F_147(x), F_148(x)*F_4(x))
Eq(F_148(x), F_118(x) + F_149(x))
Eq(F_149(x), F_80(x) + F_93(x))
Eq(F_150(x), F_151(x)*F_4(x))
Eq(F_151(x), F_149(x) + F_152(x))
Eq(F_152(x), F_122(x) + F_2(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 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": 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