0132_0213_1320_2310
Counting sequence:
1, 1, 2, 6, 20, 64, 187, 506, 1298, 3214, 7771, 18472, 43334, 100552, 231107, 526670, 1191042, 2674810, 5969235, 13245228, 29237798, 64236132, 140522507, 306200386, 664816690, 1438669574, 3103812587, 6677364496, 14327780518, 30668795104, 65498302451, 139586496022, 296889681794, 630286527602, 1335734916387, 2826088579540, 5970004652742, 12592844236860, 26525718159995, 55800215265738, 117235427485298, 246015726907582, 515670953640187, 1079720418711416, 2258396883711110, 4719103906699384, 9851624185185187, 20547673300219486, 42819380832481538, 89157198873506794, 185492009652764531, 385620718094078972, 801077783719047654, 1662954162407119252, 3449757314566409067, 7151716208265005010, 14816842774049640242, 30678520661648581110, 63482740347415330955, 131288936337105578848, 271368899146837549798, 560608081615080351632, 1157533190625275443667, 2388853357545388087846, 4927586510689665206530, 10159544298595536864994, 20937054523660342481987, 43128487644332933165476, 88802625970837783060358, 182770340282314239090284, 376018431198495501922011, 773287511569904405631322, 1589666617295994322060914, 3266697014525077078691502, 6710482772157765850901147, 13779865397013624735931080, 28286975232389174832537542, 58047328807433678969387560, 119079193232040973711338115, 244203015562155093293576558, 500646405047908306978279298, 1026075789397916512035112282, 2102322000309840134817452947, 4306193769467309120306985676, 8817904928269105200310614950, 18051680338485642836716574404, 36944773047423067579024886987, 75591713648863533036032122466, 154626448031989529961623292978, 316216308784959323969549817574, 646514185516789848566072622891, 1321500991937143443454820337072, 2700566195701057069692444961574, 5517498755094939885225402380352, 11270206117654102022681632904627, 23015781210393790070924523805750, 46992203891273035235170762048642, 95925497764145546741383344656210, 195772789572747178193647108357475, 399468395396920790146648604970868, 814940879621722976487193080841798
Generating function in Maple syntax:
(2*x^9-14*x^8-x^7+60*x^6-126*x^5+145*x^4-102*x^3+43*x^2-10*x+1)/(2*x-1)^3/(-1+x)^5
Generating function in latex syntax:
\frac{2 x^{9}-14 x^{8}-x^{7}+60 x^{6}-126 x^{5}+145 x^{4}-102 x^{3}+43 x^{2}-10 x +1}{\left(2 x -1\right)^{3} \left(-1+x \right)^{5}}
Generating function in sympy syntax:
(2*x**9 - 14*x**8 - x**7 + 60*x**6 - 126*x**5 + 145*x**4 - 102*x**3 + 43*x**2 - 10*x + 1)/((x - 1)**5*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^3*(-1+x)^5*F(x)+2*x^9-14*x^8-x^7+60*x^6-126*x^5+145*x^4-102*x^3+43*x^2-10*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{3} \left(-1+x \right)^{5} F \! \left(x \right)+2 x^{9}-14 x^{8}-x^{7}+60 x^{6}-126 x^{5}+145 x^{4}-102 x^{3}+43 x^{2}-10 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/48*(3*n^2+9*n-42)*2^n+1/12*n^4-2/3*n^3+23/12*n^2-4/3*n+2)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(3 n^{2}+9 n -42\right) 2^{n}}{48}+\frac{n^{4}}{12}-\frac{2 n^{3}}{3}+\frac{23 n^{2}}{12}-\frac{4 n}{3}+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) = 64
a(6) = 187
a(7) = 506
a(8) = 1298
a(9) = 3214
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)-1/12*(n-2)*(n^3-18*n^2+77*n-36), n >= 10
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) = 64
a \! \left(6\right) = 187
a \! \left(7\right) = 506
a \! \left(8\right) = 1298
a \! \left(9\right) = 3214
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-\frac{\left(n -2\right) \left(n^{3}-18 n^{2}+77 n -36\right)}{12}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15292/
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[25,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[14,x]+F[23,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[17,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[13,x]
F[25,x] = F[26,x]+F[35,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[9,x]
F[29,x] = F[18,x]+F[30,x]
F[30,x] = F[31,x]+F[32,x]+F[34,x]
F[31,x] = 0
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[10,x]+F[30,x]
F[34,x] = F[12,x]*F[29,x]
F[35,x] = F[121,x]+F[31,x]+F[36,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[63,x]
F[38,x] = F[39,x]+F[6,x]
F[39,x] = F[31,x]+F[40,x]+F[43,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[10,x]+F[42,x]
F[42,x] = F[40,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[48,x]
F[45,x] = F[10,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[12,x]*F[45,x]
F[48,x] = F[49,x]+F[61,x]
F[49,x] = F[31,x]+F[40,x]+F[50,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[12,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[12,x]*F[52,x]
F[55,x] = F[56,x]+F[59,x]
F[56,x] = F[57,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[12,x]+F[56,x]
F[59,x] = F[60,x]
F[60,x] = F[12,x]*F[55,x]
F[61,x] = F[62,x]
F[62,x] = F[12,x]*F[48,x]
F[63,x] = F[64,x]+F[86,x]
F[64,x] = F[31,x]+F[65,x]+F[67,x]
F[65,x] = F[12,x]*F[66,x]
F[66,x] = F[6,x]+F[64,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[29,x]+F[69,x]
F[69,x] = F[70,x]+F[79,x]
F[70,x] = F[31,x]+F[32,x]+F[71,x]
F[71,x] = F[12,x]*F[72,x]
F[72,x] = F[29,x]+F[73,x]
F[73,x] = F[74,x]+F[77,x]
F[74,x] = F[75,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[18,x]+F[74,x]
F[77,x] = F[78,x]
F[78,x] = F[12,x]*F[73,x]
F[79,x] = 2*F[31,x]+F[80,x]+F[85,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[46,x]+F[82,x]
F[82,x] = 2*F[31,x]+F[80,x]+F[83,x]
F[83,x] = F[12,x]*F[84,x]
F[84,x] = F[30,x]+F[82,x]
F[85,x] = F[12,x]*F[69,x]
F[86,x] = F[31,x]+F[87,x]+F[89,x]+F[94,x]
F[87,x] = F[12,x]*F[88,x]
F[88,x] = F[39,x]+F[86,x]
F[89,x] = F[12,x]*F[90,x]
F[90,x] = F[30,x]+F[91,x]
F[91,x] = 2*F[31,x]+F[89,x]+F[92,x]
F[92,x] = F[12,x]*F[93,x]
F[93,x] = F[42,x]+F[91,x]
F[94,x] = F[12,x]*F[95,x]
F[95,x] = F[84,x]+F[96,x]
F[96,x] = F[111,x]+F[97,x]
F[97,x] = F[31,x]+F[89,x]+F[92,x]+F[98,x]
F[98,x] = F[12,x]*F[99,x]
F[99,x] = F[100,x]+F[105,x]
F[100,x] = F[101,x]+F[103,x]
F[101,x] = F[102,x]
F[102,x] = F[12,x]*F[18,x]
F[103,x] = F[104,x]
F[104,x] = F[100,x]*F[12,x]
F[105,x] = F[106,x]+F[109,x]
F[106,x] = F[107,x]
F[107,x] = F[108,x]*F[12,x]
F[108,x] = F[101,x]+F[106,x]
F[109,x] = F[110,x]
F[110,x] = F[105,x]*F[12,x]
F[111,x] = 3*F[31,x]+F[112,x]+F[120,x]
F[112,x] = F[113,x]*F[12,x]
F[113,x] = F[114,x]+F[117,x]
F[114,x] = F[115,x]
F[115,x] = F[116,x]*F[12,x]
F[116,x] = F[114,x]+F[42,x]
F[117,x] = 3*F[31,x]+F[112,x]+F[118,x]
F[118,x] = F[119,x]*F[12,x]
F[119,x] = F[117,x]+F[91,x]
F[120,x] = F[12,x]*F[96,x]
F[121,x] = F[12,x]*F[122,x]
F[122,x] = F[123,x]+F[134,x]
F[123,x] = F[124,x]+F[26,x]
F[124,x] = F[125,x]+F[133,x]+F[31,x]
F[125,x] = F[12,x]*F[126,x]
F[126,x] = F[127,x]+F[129,x]
F[127,x] = F[10,x]+F[128,x]
F[128,x] = F[31,x]+F[40,x]+F[47,x]
F[129,x] = F[130,x]+F[30,x]
F[130,x] = F[131,x]+F[31,x]+F[83,x]+F[89,x]
F[131,x] = F[12,x]*F[132,x]
F[132,x] = F[128,x]+F[130,x]
F[133,x] = F[12,x]*F[123,x]
F[134,x] = F[135,x]+F[138,x]
F[135,x] = F[136,x]+F[31,x]+F[71,x]
F[136,x] = F[12,x]*F[137,x]
F[137,x] = F[41,x]+F[90,x]
F[138,x] = 2*F[31,x]+F[139,x]+F[148,x]
F[139,x] = F[12,x]*F[140,x]
F[140,x] = F[141,x]+F[143,x]
F[141,x] = F[142,x]+F[46,x]
F[142,x] = F[115,x]
F[143,x] = F[144,x]+F[82,x]
F[144,x] = 2*F[31,x]+F[118,x]+F[145,x]+F[147,x]
F[145,x] = F[12,x]*F[146,x]
F[146,x] = F[142,x]+F[144,x]
F[147,x] = 0
F[148,x] = F[12,x]*F[134,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_{25}\! \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_{14}\! \left(x \right)+F_{23}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{35}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{9}\! \left(x \right)
F_{29}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = 0
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_{30}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{35}\! \left(x \right) = F_{121}\! \left(x \right)+F_{31}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{63}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{6}\! \left(x \right)
F_{39}\! \left(x \right) = F_{31}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{48}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{61}\! \left(x \right)
F_{49}\! \left(x \right) = F_{31}\! \left(x \right)+F_{40}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{59}\! \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_{12}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{86}\! \left(x \right)
F_{64}\! \left(x \right) = F_{31}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{6}\! \left(x \right)+F_{64}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{29}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{79}\! \left(x \right)
F_{70}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{29}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{77}\! \left(x \right)
F_{74}\! \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_{18}\! \left(x \right)+F_{74}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{73}\! \left(x \right)
F_{79}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{80}\! \left(x \right)+F_{85}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{46}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{80}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{30}\! \left(x \right)+F_{82}\! \left(x \right)
F_{85}\! \left(x \right) = F_{12}\! \left(x \right) F_{69}\! \left(x \right)
F_{86}\! \left(x \right) = F_{31}\! \left(x \right)+F_{87}\! \left(x \right)+F_{89}\! \left(x \right)+F_{94}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{39}\! \left(x \right)+F_{86}\! \left(x \right)
F_{89}\! \left(x \right) = F_{12}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{30}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{89}\! \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_{42}\! \left(x \right)+F_{91}\! \left(x \right)
F_{94}\! \left(x \right) = F_{12}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{84}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{111}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{31}\! \left(x \right)+F_{89}\! \left(x \right)+F_{92}\! \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_{100}\! \left(x \right)+F_{105}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{103}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)
F_{102}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)
F_{104}\! \left(x \right) = F_{100}\! \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_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{12}\! \left(x \right)
F_{108}\! \left(x \right) = F_{101}\! \left(x \right)+F_{106}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)
F_{110}\! \left(x \right) = F_{105}\! \left(x \right) F_{12}\! \left(x \right)
F_{111}\! \left(x \right) = 3 F_{31}\! \left(x \right)+F_{112}\! \left(x \right)+F_{120}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{12}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)+F_{117}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right) F_{12}\! \left(x \right)
F_{116}\! \left(x \right) = F_{114}\! \left(x \right)+F_{42}\! \left(x \right)
F_{117}\! \left(x \right) = 3 F_{31}\! \left(x \right)+F_{112}\! \left(x \right)+F_{118}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right) F_{12}\! \left(x \right)
F_{119}\! \left(x \right) = F_{117}\! \left(x \right)+F_{91}\! \left(x \right)
F_{120}\! \left(x \right) = F_{12}\! \left(x \right) F_{96}\! \left(x \right)
F_{121}\! \left(x \right) = F_{12}\! \left(x \right) F_{122}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)+F_{134}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)+F_{26}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)+F_{133}\! \left(x \right)+F_{31}\! \left(x \right)
F_{125}\! \left(x \right) = F_{12}\! \left(x \right) F_{126}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)+F_{129}\! \left(x \right)
F_{127}\! \left(x \right) = F_{10}\! \left(x \right)+F_{128}\! \left(x \right)
F_{128}\! \left(x \right) = F_{31}\! \left(x \right)+F_{40}\! \left(x \right)+F_{47}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)+F_{30}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right)+F_{31}\! \left(x \right)+F_{83}\! \left(x \right)+F_{89}\! \left(x \right)
F_{131}\! \left(x \right) = F_{12}\! \left(x \right) F_{132}\! \left(x \right)
F_{132}\! \left(x \right) = F_{128}\! \left(x \right)+F_{130}\! \left(x \right)
F_{133}\! \left(x \right) = F_{12}\! \left(x \right) F_{123}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)+F_{138}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right)+F_{31}\! \left(x \right)+F_{71}\! \left(x \right)
F_{136}\! \left(x \right) = F_{12}\! \left(x \right) F_{137}\! \left(x \right)
F_{137}\! \left(x \right) = F_{41}\! \left(x \right)+F_{90}\! \left(x \right)
F_{138}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{139}\! \left(x \right)+F_{148}\! \left(x \right)
F_{139}\! \left(x \right) = F_{12}\! \left(x \right) F_{140}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right)+F_{143}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right)+F_{46}\! \left(x \right)
F_{142}\! \left(x \right) = F_{115}\! \left(x \right)
F_{143}\! \left(x \right) = F_{144}\! \left(x \right)+F_{82}\! \left(x \right)
F_{144}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{118}\! \left(x \right)+F_{145}\! \left(x \right)+F_{147}\! \left(x \right)
F_{145}\! \left(x \right) = F_{12}\! \left(x \right) F_{146}\! \left(x \right)
F_{146}\! \left(x \right) = F_{142}\! \left(x \right)+F_{144}\! \left(x \right)
F_{147}\! \left(x \right) = 0
F_{148}\! \left(x \right) = F_{12}\! \left(x \right) F_{134}\! \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_25(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_14(x) + F_23(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_17(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_13(x))
Eq(F_25(x), F_26(x) + F_35(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_9(x))
Eq(F_29(x), F_18(x) + F_30(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_34(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_10(x) + F_30(x))
Eq(F_34(x), F_12(x)*F_29(x))
Eq(F_35(x), F_121(x) + F_31(x) + F_36(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_63(x))
Eq(F_38(x), F_39(x) + F_6(x))
Eq(F_39(x), F_31(x) + F_40(x) + F_43(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_10(x) + F_42(x))
Eq(F_42(x), F_40(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_48(x))
Eq(F_45(x), F_10(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_12(x)*F_45(x))
Eq(F_48(x), F_49(x) + F_61(x))
Eq(F_49(x), F_31(x) + F_40(x) + F_50(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_12(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_12(x)*F_52(x))
Eq(F_55(x), F_56(x) + F_59(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_12(x) + F_56(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_12(x)*F_55(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_12(x)*F_48(x))
Eq(F_63(x), F_64(x) + F_86(x))
Eq(F_64(x), F_31(x) + F_65(x) + F_67(x))
Eq(F_65(x), F_12(x)*F_66(x))
Eq(F_66(x), F_6(x) + F_64(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_29(x) + F_69(x))
Eq(F_69(x), F_70(x) + F_79(x))
Eq(F_70(x), F_31(x) + F_32(x) + F_71(x))
Eq(F_71(x), F_12(x)*F_72(x))
Eq(F_72(x), F_29(x) + F_73(x))
Eq(F_73(x), F_74(x) + F_77(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_18(x) + F_74(x))
Eq(F_77(x), F_78(x))
Eq(F_78(x), F_12(x)*F_73(x))
Eq(F_79(x), 2*F_31(x) + F_80(x) + F_85(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_46(x) + F_82(x))
Eq(F_82(x), 2*F_31(x) + F_80(x) + F_83(x))
Eq(F_83(x), F_12(x)*F_84(x))
Eq(F_84(x), F_30(x) + F_82(x))
Eq(F_85(x), F_12(x)*F_69(x))
Eq(F_86(x), F_31(x) + F_87(x) + F_89(x) + F_94(x))
Eq(F_87(x), F_12(x)*F_88(x))
Eq(F_88(x), F_39(x) + F_86(x))
Eq(F_89(x), F_12(x)*F_90(x))
Eq(F_90(x), F_30(x) + F_91(x))
Eq(F_91(x), 2*F_31(x) + F_89(x) + F_92(x))
Eq(F_92(x), F_12(x)*F_93(x))
Eq(F_93(x), F_42(x) + F_91(x))
Eq(F_94(x), F_12(x)*F_95(x))
Eq(F_95(x), F_84(x) + F_96(x))
Eq(F_96(x), F_111(x) + F_97(x))
Eq(F_97(x), F_31(x) + F_89(x) + F_92(x) + F_98(x))
Eq(F_98(x), F_12(x)*F_99(x))
Eq(F_99(x), F_100(x) + F_105(x))
Eq(F_100(x), F_101(x) + F_103(x))
Eq(F_101(x), F_102(x))
Eq(F_102(x), F_12(x)*F_18(x))
Eq(F_103(x), F_104(x))
Eq(F_104(x), F_100(x)*F_12(x))
Eq(F_105(x), F_106(x) + F_109(x))
Eq(F_106(x), F_107(x))
Eq(F_107(x), F_108(x)*F_12(x))
Eq(F_108(x), F_101(x) + F_106(x))
Eq(F_109(x), F_110(x))
Eq(F_110(x), F_105(x)*F_12(x))
Eq(F_111(x), F_112(x) + F_120(x) + 3*F_31(x))
Eq(F_112(x), F_113(x)*F_12(x))
Eq(F_113(x), F_114(x) + F_117(x))
Eq(F_114(x), F_115(x))
Eq(F_115(x), F_116(x)*F_12(x))
Eq(F_116(x), F_114(x) + F_42(x))
Eq(F_117(x), F_112(x) + F_118(x) + 3*F_31(x))
Eq(F_118(x), F_119(x)*F_12(x))
Eq(F_119(x), F_117(x) + F_91(x))
Eq(F_120(x), F_12(x)*F_96(x))
Eq(F_121(x), F_12(x)*F_122(x))
Eq(F_122(x), F_123(x) + F_134(x))
Eq(F_123(x), F_124(x) + F_26(x))
Eq(F_124(x), F_125(x) + F_133(x) + F_31(x))
Eq(F_125(x), F_12(x)*F_126(x))
Eq(F_126(x), F_127(x) + F_129(x))
Eq(F_127(x), F_10(x) + F_128(x))
Eq(F_128(x), F_31(x) + F_40(x) + F_47(x))
Eq(F_129(x), F_130(x) + F_30(x))
Eq(F_130(x), F_131(x) + F_31(x) + F_83(x) + F_89(x))
Eq(F_131(x), F_12(x)*F_132(x))
Eq(F_132(x), F_128(x) + F_130(x))
Eq(F_133(x), F_12(x)*F_123(x))
Eq(F_134(x), F_135(x) + F_138(x))
Eq(F_135(x), F_136(x) + F_31(x) + F_71(x))
Eq(F_136(x), F_12(x)*F_137(x))
Eq(F_137(x), F_41(x) + F_90(x))
Eq(F_138(x), F_139(x) + F_148(x) + 2*F_31(x))
Eq(F_139(x), F_12(x)*F_140(x))
Eq(F_140(x), F_141(x) + F_143(x))
Eq(F_141(x), F_142(x) + F_46(x))
Eq(F_142(x), F_115(x))
Eq(F_143(x), F_144(x) + F_82(x))
Eq(F_144(x), F_118(x) + F_145(x) + F_147(x) + 2*F_31(x))
Eq(F_145(x), F_12(x)*F_146(x))
Eq(F_146(x), F_142(x) + F_144(x))
Eq(F_147(x), 0)
Eq(F_148(x), F_12(x)*F_134(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, 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]]}, {"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, 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]]}, {"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, 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]]}, {"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], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 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