0132_0231_0312_0321_2310
Counting sequence:
1, 1, 2, 6, 19, 56, 149, 370, 878, 2021, 4555, 10112, 22196, 48295, 104345, 224138, 479098, 1019753, 2162519, 4570948, 9633584, 20250395, 42467077, 88866542, 185597654, 386924221, 805306019, 1673526920, 3472883308, 7197425231, 14898167345, 30802968082, 63619202546, 131264937425, 270582939055, 557272006028, 1146756267368, 2357937044803, 4844723109149, 9947144256758, 20409684589774, 41850161331365, 85761906965627, 175646982536272, 359540302281764, 735573278981111, 1504131906796489, 3074234511260570, 6280410417855338, 12824703626378041, 26177172834089735, 53409876830845652, 108930815987022496, 222083756624706155, 452611762550733365, 922112023704107518, 1878001044613495238, 3823556083637549453, 7782220156096215379, 15834656289834662168, 32209744534953785564, 65500352980476491935, 133162433782090823777, 270648323206457325602, 549943557697466005474, 1117180937964034717601, 2268949521066274846559, 4607074332408960513820, 9352499245370742666968, 18981699651847128610451, 38516801625905543771725, 78140407896233660642822, 158494425081312467482046, 321416068740315227354485, 651686574636011039487275, 1321082023582783248528608, 2677581795787088836162708, 5425999088817222350533703, 10993669172120534057481209, 22270680333213246827787178, 45108044644370851081220954, 91349457244630417013732105, 184965650401038263730041527, 374464772625631386865234532, 757996488898372492540768784, 1534126865090964422702133691, 3104521504770367720645456229, 6281578558717613191773286670, 12708228215788981884511318198, 25706598628285474770952122461, 51993481649985971545763213315, 105147532086801987099244359592, 212616201747264062213924581196, 429874678641848300458720882415, 869033907578336952979185200785, 1756636915745954610081857269298, 3550412032670470628410688269778, 7175100467698064073315323997553, 14498753740110373779618542906639, 29294613089649238825212875631788, 59183437398155460182377330895944
Generating function in Maple syntax:
(x^8-4*x^7+4*x^6+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{x^{8}-4 x^{7}+4 x^{6}+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:
(x**8 - 4*x**7 + 4*x**6 + 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)+x^8-4*x^7+4*x^6+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)+x^{8}-4 x^{7}+4 x^{6}+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 = 3,6,1/32*(15*n-6)*2^n-1/2*n^2-1/2*n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 6 & n =3 \\ \frac{\left(15 n -6\right) 2^{n}}{32}-\frac{n^{2}}{2}-\frac{n}{2}+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) = 19
a(5) = 56
a(6) = 149
a(7) = 370
a(8) = 878
a(n+2) = -1/2*n^2-4*a(n)+4*a(n+1)+3/2*n+3, n >= 9
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) = 56
a \! \left(6\right) = 149
a \! \left(7\right) = 370
a \! \left(8\right) = 878
a \! \left(n +2\right) = -\frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{3 n}{2}+3, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18582/
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[16,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[12,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[10,x]*F[12,x]
F[16,x] = F[17,x]+F[40,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[20,x]+F[5,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[1,x]+F[21,x]
F[24,x] = F[25,x]+F[26,x]+F[28,x]
F[25,x] = 0
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[24,x]+F[6,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[35,x]
F[30,x] = F[21,x]+F[31,x]
F[31,x] = F[25,x]+F[32,x]+F[34,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[10,x]+F[31,x]
F[34,x] = F[12,x]*F[30,x]
F[35,x] = F[36,x]+F[38,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[21,x]
F[38,x] = F[39,x]
F[39,x] = F[12,x]*F[31,x]
F[40,x] = F[25,x]+F[41,x]+F[83,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[43,x]+F[55,x]
F[43,x] = F[44,x]+F[6,x]
F[44,x] = F[25,x]+F[45,x]+F[53,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[10,x]+F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[12,x]*F[47,x]
F[50,x] = F[14,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[12,x]*F[48,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[13,x]
F[55,x] = F[56,x]+F[68,x]
F[56,x] = F[25,x]+F[57,x]+F[59,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[56,x]+F[6,x]
F[59,x] = F[12,x]*F[60,x]
F[60,x] = F[30,x]+F[61,x]
F[61,x] = F[62,x]+F[65,x]
F[62,x] = F[25,x]+F[37,x]+F[63,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[12,x]+F[62,x]
F[65,x] = 2*F[25,x]+F[39,x]+F[66,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[14,x]+F[65,x]
F[68,x] = F[25,x]+F[69,x]+F[71,x]+F[81,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[44,x]+F[68,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[73,x]+F[78,x]
F[73,x] = F[31,x]+F[74,x]
F[74,x] = 2*F[25,x]+F[75,x]+F[77,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[48,x]+F[74,x]
F[77,x] = F[12,x]*F[73,x]
F[78,x] = F[38,x]+F[79,x]
F[79,x] = F[80,x]
F[80,x] = F[12,x]*F[74,x]
F[81,x] = F[12,x]*F[82,x]
F[82,x] = F[61,x]
F[83,x] = F[12,x]*F[84,x]
F[84,x] = F[85,x]+F[98,x]
F[85,x] = F[17,x]+F[86,x]
F[86,x] = F[25,x]+F[87,x]+F[97,x]
F[87,x] = F[12,x]*F[88,x]
F[88,x] = F[89,x]+F[92,x]
F[89,x] = F[10,x]+F[90,x]
F[90,x] = F[25,x]+F[45,x]+F[91,x]
F[91,x] = F[12,x]*F[13,x]
F[92,x] = F[31,x]+F[93,x]
F[93,x] = F[25,x]+F[71,x]+F[94,x]+F[96,x]
F[94,x] = F[12,x]*F[95,x]
F[95,x] = F[90,x]+F[93,x]
F[96,x] = F[12,x]*F[61,x]
F[97,x] = F[12,x]*F[85,x]
F[98,x] = F[132,x]+F[99,x]
F[99,x] = F[100,x]+F[25,x]+F[37,x]
F[100,x] = F[101,x]*F[12,x]
F[101,x] = F[102,x]+F[115,x]
F[102,x] = F[103,x]+F[12,x]
F[103,x] = F[104,x]+F[114,x]+F[25,x]
F[104,x] = F[105,x]*F[12,x]
F[105,x] = F[106,x]+F[109,x]
F[106,x] = F[107,x]+F[12,x]
F[107,x] = F[108,x]
F[108,x] = F[106,x]*F[12,x]
F[109,x] = F[110,x]+F[112,x]
F[110,x] = F[111,x]
F[111,x] = x^2
F[112,x] = F[113,x]
F[113,x] = F[107,x]*F[12,x]
F[114,x] = x^2
F[115,x] = F[116,x]+F[62,x]
F[116,x] = F[117,x]+F[119,x]+F[131,x]+F[25,x]
F[117,x] = F[118,x]*F[12,x]
F[118,x] = F[103,x]+F[116,x]
F[119,x] = F[12,x]*F[120,x]
F[120,x] = F[121,x]+F[126,x]
F[121,x] = F[122,x]+F[62,x]
F[122,x] = 2*F[25,x]+F[123,x]+F[125,x]
F[123,x] = F[12,x]*F[124,x]
F[124,x] = F[107,x]+F[122,x]
F[125,x] = F[12,x]*F[121,x]
F[126,x] = F[127,x]+F[129,x]
F[127,x] = F[128,x]
F[128,x] = F[12,x]*F[62,x]
F[129,x] = F[130,x]
F[130,x] = F[12,x]*F[122,x]
F[131,x] = F[12,x]*F[36,x]
F[132,x] = 2*F[25,x]+F[133,x]+F[39,x]
F[133,x] = F[12,x]*F[134,x]
F[134,x] = F[135,x]+F[148,x]
F[135,x] = F[136,x]+F[14,x]
F[136,x] = 2*F[25,x]+F[137,x]+F[147,x]
F[137,x] = F[12,x]*F[138,x]
F[138,x] = F[139,x]+F[142,x]
F[139,x] = F[14,x]+F[140,x]
F[140,x] = F[141,x]
F[141,x] = F[12,x]*F[139,x]
F[142,x] = F[143,x]+F[145,x]
F[143,x] = F[144,x]
F[144,x] = F[12,x]*F[14,x]
F[145,x] = F[146,x]
F[146,x] = F[12,x]*F[140,x]
F[147,x] = F[12,x]*F[14,x]
F[148,x] = F[149,x]+F[65,x]
F[149,x] = 2*F[25,x]+F[150,x]+F[152,x]+F[164,x]
F[150,x] = F[12,x]*F[151,x]
F[151,x] = F[136,x]+F[149,x]
F[152,x] = F[12,x]*F[153,x]
F[153,x] = F[154,x]+F[159,x]
F[154,x] = F[155,x]+F[65,x]
F[155,x] = 3*F[25,x]+F[156,x]+F[158,x]
F[156,x] = F[12,x]*F[157,x]
F[157,x] = F[140,x]+F[155,x]
F[158,x] = F[12,x]*F[154,x]
F[159,x] = F[160,x]+F[162,x]
F[160,x] = F[161,x]
F[161,x] = F[12,x]*F[65,x]
F[162,x] = F[163,x]
F[163,x] = F[12,x]*F[155,x]
F[164,x] = F[12,x]*F[38,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_{16}\! \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_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{40}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{5}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \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_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{24}\! \left(x \right)+F_{6}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{40}\! \left(x \right) = F_{25}\! \left(x \right)+F_{41}\! \left(x \right)+F_{83}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{55}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{6}\! \left(x \right)
F_{44}\! \left(x \right) = F_{25}\! \left(x \right)+F_{45}\! \left(x \right)+F_{53}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = F_{14}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{13}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{68}\! \left(x \right)
F_{56}\! \left(x \right) = F_{25}\! \left(x \right)+F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{56}\! \left(x \right)+F_{6}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{30}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{65}\! \left(x \right)
F_{62}\! \left(x \right) = F_{25}\! \left(x \right)+F_{37}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right)+F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{39}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{14}\! \left(x \right)+F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{25}\! \left(x \right)+F_{69}\! \left(x \right)+F_{71}\! \left(x \right)+F_{81}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{44}\! \left(x \right)+F_{68}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{78}\! \left(x \right)
F_{73}\! \left(x \right) = F_{31}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{75}\! \left(x \right)+F_{77}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{48}\! \left(x \right)+F_{74}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{78}\! \left(x \right) = F_{38}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{81}\! \left(x \right) = F_{12}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{61}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)+F_{98}\! \left(x \right)
F_{85}\! \left(x \right) = F_{17}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{25}\! \left(x \right)+F_{87}\! \left(x \right)+F_{97}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{92}\! \left(x \right)
F_{89}\! \left(x \right) = F_{10}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{25}\! \left(x \right)+F_{45}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{92}\! \left(x \right) = F_{31}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{25}\! \left(x \right)+F_{71}\! \left(x \right)+F_{94}\! \left(x \right)+F_{96}\! \left(x \right)
F_{94}\! \left(x \right) = F_{12}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{90}\! \left(x \right)+F_{93}\! \left(x \right)
F_{96}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{97}\! \left(x \right) = F_{12}\! \left(x \right) F_{85}\! \left(x \right)
F_{98}\! \left(x \right) = F_{132}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{25}\! \left(x \right)+F_{37}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{12}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{115}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{12}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{114}\! \left(x \right)+F_{25}\! \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_{109}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{12}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)
F_{108}\! \left(x \right) = F_{106}\! \left(x \right) F_{12}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)+F_{112}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)
F_{111}\! \left(x \right) = x^{2}
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)
F_{113}\! \left(x \right) = F_{107}\! \left(x \right) F_{12}\! \left(x \right)
F_{114}\! \left(x \right) = x^{2}
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{62}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)+F_{119}\! \left(x \right)+F_{131}\! \left(x \right)+F_{25}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right) F_{12}\! \left(x \right)
F_{118}\! \left(x \right) = F_{103}\! \left(x \right)+F_{116}\! \left(x \right)
F_{119}\! \left(x \right) = F_{12}\! \left(x \right) F_{120}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)+F_{126}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)+F_{62}\! \left(x \right)
F_{122}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{123}\! \left(x \right)+F_{125}\! \left(x \right)
F_{123}\! \left(x \right) = F_{12}\! \left(x \right) F_{124}\! \left(x \right)
F_{124}\! \left(x \right) = F_{107}\! \left(x \right)+F_{122}\! \left(x \right)
F_{125}\! \left(x \right) = F_{12}\! \left(x \right) F_{121}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)+F_{129}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)
F_{128}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)
F_{130}\! \left(x \right) = F_{12}\! \left(x \right) F_{122}\! \left(x \right)
F_{131}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{132}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{133}\! \left(x \right)+F_{39}\! \left(x \right)
F_{133}\! \left(x \right) = F_{12}\! \left(x \right) F_{134}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)+F_{148}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right)+F_{14}\! \left(x \right)
F_{136}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{137}\! \left(x \right)+F_{147}\! \left(x \right)
F_{137}\! \left(x \right) = F_{12}\! \left(x \right) F_{138}\! \left(x \right)
F_{138}\! \left(x \right) = F_{139}\! \left(x \right)+F_{142}\! \left(x \right)
F_{139}\! \left(x \right) = F_{14}\! \left(x \right)+F_{140}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right)
F_{141}\! \left(x \right) = F_{12}\! \left(x \right) F_{139}\! \left(x \right)
F_{142}\! \left(x \right) = F_{143}\! \left(x \right)+F_{145}\! \left(x \right)
F_{143}\! \left(x \right) = F_{144}\! \left(x \right)
F_{144}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{145}\! \left(x \right) = F_{146}\! \left(x \right)
F_{146}\! \left(x \right) = F_{12}\! \left(x \right) F_{140}\! \left(x \right)
F_{147}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{148}\! \left(x \right) = F_{149}\! \left(x \right)+F_{65}\! \left(x \right)
F_{149}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{150}\! \left(x \right)+F_{152}\! \left(x \right)+F_{164}\! \left(x \right)
F_{150}\! \left(x \right) = F_{12}\! \left(x \right) F_{151}\! \left(x \right)
F_{151}\! \left(x \right) = F_{136}\! \left(x \right)+F_{149}\! \left(x \right)
F_{152}\! \left(x \right) = F_{12}\! \left(x \right) F_{153}\! \left(x \right)
F_{153}\! \left(x \right) = F_{154}\! \left(x \right)+F_{159}\! \left(x \right)
F_{154}\! \left(x \right) = F_{155}\! \left(x \right)+F_{65}\! \left(x \right)
F_{155}\! \left(x \right) = 3 F_{25}\! \left(x \right)+F_{156}\! \left(x \right)+F_{158}\! \left(x \right)
F_{156}\! \left(x \right) = F_{12}\! \left(x \right) F_{157}\! \left(x \right)
F_{157}\! \left(x \right) = F_{140}\! \left(x \right)+F_{155}\! \left(x \right)
F_{158}\! \left(x \right) = F_{12}\! \left(x \right) F_{154}\! \left(x \right)
F_{159}\! \left(x \right) = F_{160}\! \left(x \right)+F_{162}\! \left(x \right)
F_{160}\! \left(x \right) = F_{161}\! \left(x \right)
F_{161}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{162}\! \left(x \right) = F_{163}\! \left(x \right)
F_{163}\! \left(x \right) = F_{12}\! \left(x \right) F_{155}\! \left(x \right)
F_{164}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \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_16(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_12(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_10(x)*F_12(x))
Eq(F_16(x), F_17(x) + F_40(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_5(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_1(x) + F_21(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_28(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_24(x) + F_6(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_35(x))
Eq(F_30(x), F_21(x) + F_31(x))
Eq(F_31(x), F_25(x) + F_32(x) + F_34(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_10(x) + F_31(x))
Eq(F_34(x), F_12(x)*F_30(x))
Eq(F_35(x), F_36(x) + F_38(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_21(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_12(x)*F_31(x))
Eq(F_40(x), F_25(x) + F_41(x) + F_83(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_55(x))
Eq(F_43(x), F_44(x) + F_6(x))
Eq(F_44(x), F_25(x) + F_45(x) + F_53(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_10(x) + F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_12(x)*F_47(x))
Eq(F_50(x), F_14(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_12(x)*F_48(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_13(x))
Eq(F_55(x), F_56(x) + F_68(x))
Eq(F_56(x), F_25(x) + F_57(x) + F_59(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_56(x) + F_6(x))
Eq(F_59(x), F_12(x)*F_60(x))
Eq(F_60(x), F_30(x) + F_61(x))
Eq(F_61(x), F_62(x) + F_65(x))
Eq(F_62(x), F_25(x) + F_37(x) + F_63(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_12(x) + F_62(x))
Eq(F_65(x), 2*F_25(x) + F_39(x) + F_66(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_14(x) + F_65(x))
Eq(F_68(x), F_25(x) + F_69(x) + F_71(x) + F_81(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_44(x) + F_68(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_73(x) + F_78(x))
Eq(F_73(x), F_31(x) + F_74(x))
Eq(F_74(x), 2*F_25(x) + F_75(x) + F_77(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_48(x) + F_74(x))
Eq(F_77(x), F_12(x)*F_73(x))
Eq(F_78(x), F_38(x) + F_79(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_12(x)*F_74(x))
Eq(F_81(x), F_12(x)*F_82(x))
Eq(F_82(x), F_61(x))
Eq(F_83(x), F_12(x)*F_84(x))
Eq(F_84(x), F_85(x) + F_98(x))
Eq(F_85(x), F_17(x) + F_86(x))
Eq(F_86(x), F_25(x) + F_87(x) + F_97(x))
Eq(F_87(x), F_12(x)*F_88(x))
Eq(F_88(x), F_89(x) + F_92(x))
Eq(F_89(x), F_10(x) + F_90(x))
Eq(F_90(x), F_25(x) + F_45(x) + F_91(x))
Eq(F_91(x), F_12(x)*F_13(x))
Eq(F_92(x), F_31(x) + F_93(x))
Eq(F_93(x), F_25(x) + F_71(x) + F_94(x) + F_96(x))
Eq(F_94(x), F_12(x)*F_95(x))
Eq(F_95(x), F_90(x) + F_93(x))
Eq(F_96(x), F_12(x)*F_61(x))
Eq(F_97(x), F_12(x)*F_85(x))
Eq(F_98(x), F_132(x) + F_99(x))
Eq(F_99(x), F_100(x) + F_25(x) + F_37(x))
Eq(F_100(x), F_101(x)*F_12(x))
Eq(F_101(x), F_102(x) + F_115(x))
Eq(F_102(x), F_103(x) + F_12(x))
Eq(F_103(x), F_104(x) + F_114(x) + F_25(x))
Eq(F_104(x), F_105(x)*F_12(x))
Eq(F_105(x), F_106(x) + F_109(x))
Eq(F_106(x), F_107(x) + F_12(x))
Eq(F_107(x), F_108(x))
Eq(F_108(x), F_106(x)*F_12(x))
Eq(F_109(x), F_110(x) + F_112(x))
Eq(F_110(x), F_111(x))
Eq(F_111(x), x**2)
Eq(F_112(x), F_113(x))
Eq(F_113(x), F_107(x)*F_12(x))
Eq(F_114(x), x**2)
Eq(F_115(x), F_116(x) + F_62(x))
Eq(F_116(x), F_117(x) + F_119(x) + F_131(x) + F_25(x))
Eq(F_117(x), F_118(x)*F_12(x))
Eq(F_118(x), F_103(x) + F_116(x))
Eq(F_119(x), F_12(x)*F_120(x))
Eq(F_120(x), F_121(x) + F_126(x))
Eq(F_121(x), F_122(x) + F_62(x))
Eq(F_122(x), F_123(x) + F_125(x) + 2*F_25(x))
Eq(F_123(x), F_12(x)*F_124(x))
Eq(F_124(x), F_107(x) + F_122(x))
Eq(F_125(x), F_12(x)*F_121(x))
Eq(F_126(x), F_127(x) + F_129(x))
Eq(F_127(x), F_128(x))
Eq(F_128(x), F_12(x)*F_62(x))
Eq(F_129(x), F_130(x))
Eq(F_130(x), F_12(x)*F_122(x))
Eq(F_131(x), F_12(x)*F_36(x))
Eq(F_132(x), F_133(x) + 2*F_25(x) + F_39(x))
Eq(F_133(x), F_12(x)*F_134(x))
Eq(F_134(x), F_135(x) + F_148(x))
Eq(F_135(x), F_136(x) + F_14(x))
Eq(F_136(x), F_137(x) + F_147(x) + 2*F_25(x))
Eq(F_137(x), F_12(x)*F_138(x))
Eq(F_138(x), F_139(x) + F_142(x))
Eq(F_139(x), F_14(x) + F_140(x))
Eq(F_140(x), F_141(x))
Eq(F_141(x), F_12(x)*F_139(x))
Eq(F_142(x), F_143(x) + F_145(x))
Eq(F_143(x), F_144(x))
Eq(F_144(x), F_12(x)*F_14(x))
Eq(F_145(x), F_146(x))
Eq(F_146(x), F_12(x)*F_140(x))
Eq(F_147(x), F_12(x)*F_14(x))
Eq(F_148(x), F_149(x) + F_65(x))
Eq(F_149(x), F_150(x) + F_152(x) + F_164(x) + 2*F_25(x))
Eq(F_150(x), F_12(x)*F_151(x))
Eq(F_151(x), F_136(x) + F_149(x))
Eq(F_152(x), F_12(x)*F_153(x))
Eq(F_153(x), F_154(x) + F_159(x))
Eq(F_154(x), F_155(x) + F_65(x))
Eq(F_155(x), F_156(x) + F_158(x) + 3*F_25(x))
Eq(F_156(x), F_12(x)*F_157(x))
Eq(F_157(x), F_140(x) + F_155(x))
Eq(F_158(x), F_12(x)*F_154(x))
Eq(F_159(x), F_160(x) + F_162(x))
Eq(F_160(x), F_161(x))
Eq(F_161(x), F_12(x)*F_65(x))
Eq(F_162(x), F_163(x))
Eq(F_163(x), F_12(x)*F_155(x))
Eq(F_164(x), F_12(x)*F_38(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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": [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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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": [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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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": [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": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 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