0123_0231_1230_1302_3021
Counting sequence:
1, 1, 2, 6, 19, 53, 132, 308, 693, 1527, 3322, 7166, 15363, 32777, 69648, 147480, 311329, 655403, 1376310, 2883650, 6029391, 12583005, 26214508, 54526076, 113246349, 234881183, 486539442, 1006633158, 2080375003, 4294967537, 8858370312, 18253611296, 37580964153, 77309411667, 158913790318, 326417514890, 670014898599, 1374389535173, 2817498546660, 5772436046340, 11819749999141, 24189255811655, 49478023250538, 101155069756046, 206708186022579, 422212465066713, 862017116177152, 1759218604442408, 3588805953061713, 7318349394477947, 14918173765665702, 30399297484751826, 61924494876345343, 126100789566374957, 256705178760119388, 522417556774978700, 1062849512059438269, 2161727821137839343, 4395513236313605410, 8935141660703065430, 18158513697557841291, 36893488147419104705, 74939897799445054968, 152185638608103802416, 308982963234634991209, 627189298506124756643, 1272825341085959063262, 2582544170319337228058, 5238875316933512660823, 10625324586456701732757, 21545797078092756289492, 43681889966544218228756, 88544371553805847758933, 179449926349046518122647, 363622219180962681456858, 736689171327664653338910, 1492267808586807887530339, 3022314549036572936767913, 6120186961799060196952560, 12391489651049949040740920, 25085210757003555375155841, 50774884423814425337662155, 102758694667243479850027798, 207935240973716218049465186, 420706185225890952797752239, 851083777008698938993150973, 1721510367131231944781597772, 3481706360490132023153790108, 7040783973435600313488772333, 14236310451781873161339931967, 28782105913385091391404641682, 58183181846412872920258842086, 117604303732111126115416804923, 237684487542793012780631854737, 480320735242727546660860202728, 970544990799738135520913395520, 1960897022228042355440212774809, 3961408125713216879677197520883, 8002044413940698096947938988110, 16162545152909924869082965872810, 32642002955876907088540107542791
Generating function in Maple syntax:
-(4*x^7-4*x^5+6*x^4-14*x^3+14*x^2-6*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{4 x^{7}-4 x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-4*x**7 + 4*x**5 - 6*x**4 + 14*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^3*F(x)+4*x^7-4*x^5+6*x^4-14*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+4 x^{7}-4 x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/4*(n+3)*2^n+1/2*n^2-13/2*n+9)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(n +3\right) 2^{n}}{4}+\frac{n^{2}}{2}-\frac{13 n}{2}+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) = 53
a(6) = 132
a(7) = 308
a(n+2) = 1/2*n^2-4*a(n)+4*a(n+1)-17/2*n+22, n >= 8
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) = 53
a \! \left(6\right) = 132
a \! \left(7\right) = 308
a \! \left(n +2\right) = \frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{17 n}{2}+22, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17363/
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[29,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[24,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[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[4,x]
F[23,x] = F[19,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[11,x]
F[28,x] = F[24,x]
F[29,x] = F[30,x]+F[65,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[1,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[17,x]*F[4,x]
F[36,x] = F[30,x]+F[37,x]
F[37,x] = F[38,x]+F[39,x]+F[63,x]
F[38,x] = 0
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[34,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[22,x]*F[4,x]
F[44,x] = F[37,x]+F[45,x]
F[45,x] = 2*F[38,x]+F[46,x]+F[50,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[42,x]
F[49,x] = F[45,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[57,x]
F[55,x] = F[4,x]+F[56,x]
F[56,x] = F[43,x]
F[57,x] = F[52,x]+F[58,x]
F[58,x] = 2*F[38,x]+F[50,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[56,x]
F[62,x] = F[58,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[30,x]+F[52,x]
F[65,x] = F[157,x]+F[38,x]+F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[68,x]+F[91,x]
F[68,x] = F[69,x]+F[7,x]
F[69,x] = F[38,x]+F[70,x]+F[75,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[11,x]+F[72,x]
F[72,x] = F[73,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[4,x]+F[72,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[77,x]+F[80,x]
F[77,x] = F[34,x]+F[78,x]
F[78,x] = F[79,x]
F[79,x] = F[27,x]*F[4,x]
F[80,x] = F[81,x]+F[86,x]
F[81,x] = F[82,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[84,x]+F[85,x]
F[84,x] = F[34,x]
F[85,x] = F[81,x]
F[86,x] = F[87,x]
F[87,x] = F[4,x]*F[88,x]
F[88,x] = F[89,x]+F[90,x]
F[89,x] = F[78,x]
F[90,x] = F[86,x]
F[91,x] = F[65,x]+F[92,x]
F[92,x] = F[123,x]+F[137,x]+F[38,x]+F[93,x]
F[93,x] = F[4,x]*F[94,x]
F[94,x] = F[95,x]+F[99,x]
F[95,x] = F[69,x]+F[96,x]
F[96,x] = F[97,x]
F[97,x] = F[4,x]*F[98,x]
F[98,x] = F[72,x]
F[99,x] = F[100,x]+F[92,x]
F[100,x] = 2*F[38,x]+F[101,x]+F[105,x]+F[122,x]
F[101,x] = F[102,x]*F[4,x]
F[102,x] = F[103,x]+F[104,x]
F[103,x] = F[96,x]
F[104,x] = F[100,x]
F[105,x] = F[106,x]*F[4,x]
F[106,x] = F[107,x]
F[107,x] = F[108,x]
F[108,x] = F[109,x]*F[4,x]
F[109,x] = F[110,x]+F[114,x]
F[110,x] = F[111,x]+F[72,x]
F[111,x] = F[112,x]
F[112,x] = F[113,x]*F[4,x]
F[113,x] = F[72,x]
F[114,x] = F[107,x]+F[115,x]
F[115,x] = 3*F[38,x]+F[116,x]+F[120,x]
F[116,x] = F[117,x]*F[4,x]
F[117,x] = F[118,x]+F[119,x]
F[118,x] = F[111,x]
F[119,x] = F[115,x]
F[120,x] = F[121,x]*F[4,x]
F[121,x] = F[107,x]
F[122,x] = 0
F[123,x] = F[124,x]*F[4,x]
F[124,x] = F[107,x]+F[125,x]
F[125,x] = F[126,x]
F[126,x] = F[127,x]*F[4,x]
F[127,x] = F[128,x]+F[129,x]
F[128,x] = F[11,x]+F[78,x]
F[129,x] = F[125,x]+F[130,x]
F[130,x] = 2*F[38,x]+F[131,x]+F[135,x]
F[131,x] = F[132,x]*F[4,x]
F[132,x] = F[133,x]+F[134,x]
F[133,x] = F[78,x]
F[134,x] = F[130,x]
F[135,x] = F[136,x]*F[4,x]
F[136,x] = F[125,x]
F[137,x] = F[138,x]*F[4,x]
F[138,x] = F[139,x]+F[146,x]
F[139,x] = F[140,x]+F[143,x]
F[140,x] = F[141,x]
F[141,x] = F[142,x]*F[4,x]
F[142,x] = F[30,x]
F[143,x] = F[144,x]
F[144,x] = F[145,x]*F[4,x]
F[145,x] = F[125,x]
F[146,x] = F[147,x]+F[152,x]
F[147,x] = F[148,x]
F[148,x] = F[149,x]*F[4,x]
F[149,x] = F[150,x]+F[151,x]
F[150,x] = F[140,x]
F[151,x] = F[147,x]
F[152,x] = F[153,x]
F[153,x] = F[154,x]*F[4,x]
F[154,x] = F[155,x]+F[156,x]
F[155,x] = F[143,x]
F[156,x] = F[152,x]
F[157,x] = F[158,x]*F[4,x]
F[158,x] = F[159,x]+F[160,x]
F[159,x] = F[125,x]+F[30,x]
F[160,x] = F[161,x]+F[165,x]
F[161,x] = F[162,x]
F[162,x] = F[163,x]*F[4,x]
F[163,x] = F[142,x]+F[164,x]
F[164,x] = F[161,x]
F[165,x] = F[166,x]
F[166,x] = F[167,x]*F[4,x]
F[167,x] = F[145,x]+F[168,x]
F[168,x] = F[165,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_{29}\! \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_{24}\! \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_{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_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)
F_{28}\! \left(x \right) = F_{24}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{65}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{63}\! \left(x \right)
F_{38}\! \left(x \right) = 0
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_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{34}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{44}\! \left(x \right) = F_{37}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{38}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{42}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \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_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{57}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{43}\! \left(x \right)
F_{57}\! \left(x \right) = F_{52}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = 2 F_{38}\! \left(x \right)+F_{50}\! \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_{56}\! \left(x \right)
F_{62}\! \left(x \right) = F_{58}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{30}\! \left(x \right)+F_{52}\! \left(x \right)
F_{65}\! \left(x \right) = F_{157}\! \left(x \right)+F_{38}\! \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_{68}\! \left(x \right)+F_{91}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{7}\! \left(x \right)
F_{69}\! \left(x \right) = F_{38}\! \left(x \right)+F_{70}\! \left(x \right)+F_{75}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{11}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right)+F_{72}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{80}\! \left(x \right)
F_{77}\! \left(x \right) = F_{34}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{86}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \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_{34}\! \left(x \right)
F_{85}\! \left(x \right) = F_{81}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \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_{78}\! \left(x \right)
F_{90}\! \left(x \right) = F_{86}\! \left(x \right)
F_{91}\! \left(x \right) = F_{65}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{123}\! \left(x \right)+F_{137}\! \left(x \right)+F_{38}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{4}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)+F_{99}\! \left(x \right)
F_{95}\! \left(x \right) = F_{69}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{4}\! \left(x \right) F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{72}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{92}\! \left(x \right)
F_{100}\! \left(x \right) = 2 F_{38}\! \left(x \right)+F_{101}\! \left(x \right)+F_{105}\! \left(x \right)+F_{122}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{4}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{104}\! \left(x \right)
F_{103}\! \left(x \right) = F_{96}\! \left(x \right)
F_{104}\! \left(x \right) = F_{100}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{4}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{4}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)+F_{114}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{72}\! \left(x \right)
F_{111}\! \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_{72}\! \left(x \right)
F_{114}\! \left(x \right) = F_{107}\! \left(x \right)+F_{115}\! \left(x \right)
F_{115}\! \left(x \right) = 3 F_{38}\! \left(x \right)+F_{116}\! \left(x \right)+F_{120}\! \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_{119}\! \left(x \right)
F_{118}\! \left(x \right) = F_{111}\! \left(x \right)
F_{119}\! \left(x \right) = F_{115}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right) F_{4}\! \left(x \right)
F_{121}\! \left(x \right) = F_{107}\! \left(x \right)
F_{122}\! \left(x \right) = 0
F_{123}\! \left(x \right) = F_{124}\! \left(x \right) F_{4}\! \left(x \right)
F_{124}\! \left(x \right) = F_{107}\! \left(x \right)+F_{125}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right) F_{4}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)+F_{129}\! \left(x \right)
F_{128}\! \left(x \right) = F_{11}\! \left(x \right)+F_{78}\! \left(x \right)
F_{129}\! \left(x \right) = F_{125}\! \left(x \right)+F_{130}\! \left(x \right)
F_{130}\! \left(x \right) = 2 F_{38}\! \left(x \right)+F_{131}\! \left(x \right)+F_{135}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right) F_{4}\! \left(x \right)
F_{132}\! \left(x \right) = F_{133}\! \left(x \right)+F_{134}\! \left(x \right)
F_{133}\! \left(x \right) = F_{78}\! \left(x \right)
F_{134}\! \left(x \right) = F_{130}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right) F_{4}\! \left(x \right)
F_{136}\! \left(x \right) = F_{125}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right) F_{4}\! \left(x \right)
F_{138}\! \left(x \right) = F_{139}\! \left(x \right)+F_{146}\! \left(x \right)
F_{139}\! \left(x \right) = F_{140}\! \left(x \right)+F_{143}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right) F_{4}\! \left(x \right)
F_{142}\! \left(x \right) = F_{30}\! \left(x \right)
F_{143}\! \left(x \right) = F_{144}\! \left(x \right)
F_{144}\! \left(x \right) = F_{145}\! \left(x \right) F_{4}\! \left(x \right)
F_{145}\! \left(x \right) = F_{125}\! \left(x \right)
F_{146}\! \left(x \right) = F_{147}\! \left(x \right)+F_{152}\! \left(x \right)
F_{147}\! \left(x \right) = F_{148}\! \left(x \right)
F_{148}\! \left(x \right) = F_{149}\! \left(x \right) F_{4}\! \left(x \right)
F_{149}\! \left(x \right) = F_{150}\! \left(x \right)+F_{151}\! \left(x \right)
F_{150}\! \left(x \right) = F_{140}\! \left(x \right)
F_{151}\! \left(x \right) = F_{147}\! \left(x \right)
F_{152}\! \left(x \right) = F_{153}\! \left(x \right)
F_{153}\! \left(x \right) = F_{154}\! \left(x \right) F_{4}\! \left(x \right)
F_{154}\! \left(x \right) = F_{155}\! \left(x \right)+F_{156}\! \left(x \right)
F_{155}\! \left(x \right) = F_{143}\! \left(x \right)
F_{156}\! \left(x \right) = F_{152}\! \left(x \right)
F_{157}\! \left(x \right) = F_{158}\! \left(x \right) F_{4}\! \left(x \right)
F_{158}\! \left(x \right) = F_{159}\! \left(x \right)+F_{160}\! \left(x \right)
F_{159}\! \left(x \right) = F_{125}\! \left(x \right)+F_{30}\! \left(x \right)
F_{160}\! \left(x \right) = F_{161}\! \left(x \right)+F_{165}\! \left(x \right)
F_{161}\! \left(x \right) = F_{162}\! \left(x \right)
F_{162}\! \left(x \right) = F_{163}\! \left(x \right) F_{4}\! \left(x \right)
F_{163}\! \left(x \right) = F_{142}\! \left(x \right)+F_{164}\! \left(x \right)
F_{164}\! \left(x \right) = F_{161}\! \left(x \right)
F_{165}\! \left(x \right) = F_{166}\! \left(x \right)
F_{166}\! \left(x \right) = F_{167}\! \left(x \right) F_{4}\! \left(x \right)
F_{167}\! \left(x \right) = F_{145}\! \left(x \right)+F_{168}\! \left(x \right)
F_{168}\! \left(x \right) = F_{165}\! \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_29(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_24(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) + F_19(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_23(x))
Eq(F_22(x), F_4(x))
Eq(F_23(x), F_19(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_11(x))
Eq(F_28(x), F_24(x))
Eq(F_29(x), F_30(x) + F_65(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_1(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_17(x)*F_4(x))
Eq(F_36(x), F_30(x) + F_37(x))
Eq(F_37(x), F_38(x) + F_39(x) + F_63(x))
Eq(F_38(x), 0)
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_34(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_22(x)*F_4(x))
Eq(F_44(x), F_37(x) + F_45(x))
Eq(F_45(x), 2*F_38(x) + F_46(x) + F_50(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_42(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_57(x))
Eq(F_55(x), F_4(x) + F_56(x))
Eq(F_56(x), F_43(x))
Eq(F_57(x), F_52(x) + F_58(x))
Eq(F_58(x), 2*F_38(x) + F_50(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_56(x))
Eq(F_62(x), F_58(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_30(x) + F_52(x))
Eq(F_65(x), F_157(x) + F_38(x) + F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_68(x) + F_91(x))
Eq(F_68(x), F_69(x) + F_7(x))
Eq(F_69(x), F_38(x) + F_70(x) + F_75(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_11(x) + F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_4(x) + F_72(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_77(x) + F_80(x))
Eq(F_77(x), F_34(x) + F_78(x))
Eq(F_78(x), F_79(x))
Eq(F_79(x), F_27(x)*F_4(x))
Eq(F_80(x), F_81(x) + F_86(x))
Eq(F_81(x), F_82(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_34(x))
Eq(F_85(x), F_81(x))
Eq(F_86(x), F_87(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_78(x))
Eq(F_90(x), F_86(x))
Eq(F_91(x), F_65(x) + F_92(x))
Eq(F_92(x), F_123(x) + F_137(x) + F_38(x) + F_93(x))
Eq(F_93(x), F_4(x)*F_94(x))
Eq(F_94(x), F_95(x) + F_99(x))
Eq(F_95(x), F_69(x) + F_96(x))
Eq(F_96(x), F_97(x))
Eq(F_97(x), F_4(x)*F_98(x))
Eq(F_98(x), F_72(x))
Eq(F_99(x), F_100(x) + F_92(x))
Eq(F_100(x), F_101(x) + F_105(x) + F_122(x) + 2*F_38(x))
Eq(F_101(x), F_102(x)*F_4(x))
Eq(F_102(x), F_103(x) + F_104(x))
Eq(F_103(x), F_96(x))
Eq(F_104(x), F_100(x))
Eq(F_105(x), F_106(x)*F_4(x))
Eq(F_106(x), F_107(x))
Eq(F_107(x), F_108(x))
Eq(F_108(x), F_109(x)*F_4(x))
Eq(F_109(x), F_110(x) + F_114(x))
Eq(F_110(x), F_111(x) + F_72(x))
Eq(F_111(x), F_112(x))
Eq(F_112(x), F_113(x)*F_4(x))
Eq(F_113(x), F_72(x))
Eq(F_114(x), F_107(x) + F_115(x))
Eq(F_115(x), F_116(x) + F_120(x) + 3*F_38(x))
Eq(F_116(x), F_117(x)*F_4(x))
Eq(F_117(x), F_118(x) + F_119(x))
Eq(F_118(x), F_111(x))
Eq(F_119(x), F_115(x))
Eq(F_120(x), F_121(x)*F_4(x))
Eq(F_121(x), F_107(x))
Eq(F_122(x), 0)
Eq(F_123(x), F_124(x)*F_4(x))
Eq(F_124(x), F_107(x) + F_125(x))
Eq(F_125(x), F_126(x))
Eq(F_126(x), F_127(x)*F_4(x))
Eq(F_127(x), F_128(x) + F_129(x))
Eq(F_128(x), F_11(x) + F_78(x))
Eq(F_129(x), F_125(x) + F_130(x))
Eq(F_130(x), F_131(x) + F_135(x) + 2*F_38(x))
Eq(F_131(x), F_132(x)*F_4(x))
Eq(F_132(x), F_133(x) + F_134(x))
Eq(F_133(x), F_78(x))
Eq(F_134(x), F_130(x))
Eq(F_135(x), F_136(x)*F_4(x))
Eq(F_136(x), F_125(x))
Eq(F_137(x), F_138(x)*F_4(x))
Eq(F_138(x), F_139(x) + F_146(x))
Eq(F_139(x), F_140(x) + F_143(x))
Eq(F_140(x), F_141(x))
Eq(F_141(x), F_142(x)*F_4(x))
Eq(F_142(x), F_30(x))
Eq(F_143(x), F_144(x))
Eq(F_144(x), F_145(x)*F_4(x))
Eq(F_145(x), F_125(x))
Eq(F_146(x), F_147(x) + F_152(x))
Eq(F_147(x), F_148(x))
Eq(F_148(x), F_149(x)*F_4(x))
Eq(F_149(x), F_150(x) + F_151(x))
Eq(F_150(x), F_140(x))
Eq(F_151(x), F_147(x))
Eq(F_152(x), F_153(x))
Eq(F_153(x), F_154(x)*F_4(x))
Eq(F_154(x), F_155(x) + F_156(x))
Eq(F_155(x), F_143(x))
Eq(F_156(x), F_152(x))
Eq(F_157(x), F_158(x)*F_4(x))
Eq(F_158(x), F_159(x) + F_160(x))
Eq(F_159(x), F_125(x) + F_30(x))
Eq(F_160(x), F_161(x) + F_165(x))
Eq(F_161(x), F_162(x))
Eq(F_162(x), F_163(x)*F_4(x))
Eq(F_163(x), F_142(x) + F_164(x))
Eq(F_164(x), F_161(x))
Eq(F_165(x), F_166(x))
Eq(F_166(x), F_167(x)*F_4(x))
Eq(F_167(x), F_145(x) + F_168(x))
Eq(F_168(x), F_165(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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