0132_0231_1203_2013
Counting sequence:
1, 1, 2, 6, 20, 67, 225, 764, 2631, 9181, 32402, 115453, 414754, 1500578, 5463000, 19998453, 73568615, 271830530, 1008380162, 3754120497, 14021977177, 52529690405, 197327462648, 743126365765, 2805087665540, 10611194382674, 40220692493943, 152736381062327, 581017638928953, 2213815305898790, 8448014020056856, 32284120450660846, 123540067793142664, 473345042578691611, 1815801253643194490, 6973504929227843456, 26810215037281789599, 103179144697649710363, 397469530773190473339, 1532553079382495423390, 5914362102376772360311, 22843532625297379179662, 88300887626738995549262, 341584298417083434838611, 1322352806766310022547649, 5122711539109635358612847, 19858339400019166205476226, 77030824967343027035144167, 298987890629584348745996912, 1161180073855511916643194876, 4512244005883154816638431575, 17543783843211207828151667715, 68246884341788653029741616011, 265621316425815382207552152256, 1034321765763888806566070779381, 4029522384130756247097466917334, 15705420324752369780363805106302, 61240243857367220143229194139765, 238896190993620143521604602286995, 932310189396266393154890517092893, 3639860562835110578164747870997648, 14215975572423421826482018378201285, 55543206648340894203746785663784349, 217091382346030564134856290089676412, 848802787261612727068931282808616527, 3319854311290821886619571994823701965, 12988975830755371257702389035024740878, 50835871765248800459580642757124914301, 199022021874589148268512895459180741386, 779405849448000024080062123767692400258, 3053192540133949534081531947878132691757, 11963801984179885011500800007747839239821, 46892713206514854641604697849430125503263, 183848204775571657765383860391536916683596, 720988278504288038505499802853952024330325, 2828191467908831210595448599121943337959682, 11096814671006515055799909643741128621419426, 43550587460078214282315276817346250842426841, 170959486103416912525622265896946636571570209, 671263961050015455063058604959122208663204833, 2636282426329521014462362284467015397850403072, 10355871117753002337105526765752169701292717593, 40688824789919389099237449046754748857686185390, 159902499495093884153235865003599248903305031298, 628528277715688132513049634296251593509353653205, 2471051483490328634835038876860558906801234306315, 9716820497694777528720967782448192208752650709857, 38216424938244395923665436553851832671788427456992, 150334122822380527521999455013976675120397170435597, 591486606125650044176111226322012380893515614770240, 2327610778663725857663494606539575395009888070202988, 9161196966456400333532494714309682120234210766515991, 36063582718434453011841136157704903028363921497946351, 141990317041842383422894873819512970226821764754880763, 559139626406053174905104046694459631306586983684849022, 2202175964450067146727719583266378886057265539219324821, 8674661422210759275285229647496316565636928912610481658, 34175936566226947495383772999649259554544824043281272986, 134664878796480023463140997620888496287027576701031004055, 530704759496135371092831256294005823984756689866249162697, 2091775480868392732981927722145852803939523494224581680531
Generating function in Maple syntax:
-1/2*(-3*x^2+(x^2-x+1)*(1-4*x)^(1/2)+3*x-1)/(x^3-x+1)/x^2*(x-1)^2
Generating function in latex syntax:
-\frac{\left(-3 x^{2}+\left(x^{2}-x +1\right) \sqrt{1-4 x}+3 x -1\right) \left(x -1\right)^{2}}{2 \left(x^{3}-x +1\right) x^{2}}
Generating function in sympy syntax:
(x - 1)**2*(3*x**2/2 - 3*x/2 - sqrt(1 - 4*x)*(x**2 - x + 1)/2 + 1/2)/(x**2*(x**3 - x + 1))
Implicit equation for the generating function in Maple syntax:
(x^3-x+1)*x^2*F(x)^2-(3*x^2-3*x+1)*(x-1)^2*F(x)+(x-1)^4 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-x +1\right) x^{2} F \! \left(x \right)^{2}-\left(3 x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(x -1\right)^{4} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 67
a(6) = 225
a(n+7) = -2*(1+2*n)/(9+n)*a(n)-(47+6*n)/(9+n)*a(2+n)+(9*n+26)/(9+n)*a(n+1)-(-15+7*n)/(9+n)*a(n+3)+6*(3*n+11)/(9+n)*a(n+4)-4*(4*n+23)/(9+n)*a(n+5)+(7*n+52)/(9+n)*a(n+6), n >= 7
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) = 225
a \! \left(n +7\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{9+n}-\frac{\left(47+6 n \right) a \! \left(2+n \right)}{9+n}+\frac{\left(9 n +26\right) a \! \left(n +1\right)}{9+n}-\frac{\left(-15+7 n \right) a \! \left(n +3\right)}{9+n}+\frac{6 \left(3 n +11\right) a \! \left(n +4\right)}{9+n}-\frac{4 \left(4 n +23\right) a \! \left(n +5\right)}{9+n}+\frac{\left(7 n +52\right) a \! \left(n +6\right)}{9+n}, \quad n \geq 7
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21229/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[22,x]*F[3,x]
F[3,x] = F[1,x]+F[126,x]+F[4,x]
F[4,x] = F[22,x]*F[5,x]
F[5,x] = F[1,x]+F[45,x]+F[52,x]+F[6,x]
F[6,x] = F[22,x]*F[7,x]
F[7,x] = F[1,x]+F[36,x]+F[45,x]+F[52,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[22,x]*F[3,x]
F[10,x] = F[11,x]+F[35,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[22,x]
F[14,x] = F[15,x]+F[31,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[22,x]
F[18,x] = F[19,x]+F[24,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[19,x]*F[22,x]
F[22,x] = F[23,x,1]
F[23,x,k[0]] = k[0]*x
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[26,x]
F[26,x] = F[22,x]*F[27,x]
F[27,x] = F[1,x]+F[25,x]
F[28,x] = F[29,x]
F[29,x] = F[22,x]*F[30,x]
F[30,x] = F[20,x]+F[28,x]
F[31,x] = F[25,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[22,x]*F[34,x]
F[34,x] = F[16,x]+F[32,x]
F[35,x] = F[25,x]*F[27,x]
F[36,x] = F[22,x]*F[37,x]
F[37,x] = F[38,x,1]
F[38,x,k[0]] = F[1,x]+F[39,x,k[0]]+F[44,x,k[0]]+F[45,x]+F[52,x]
F[39,x,k[0]] = F[40,x,k[0]]
F[40,x,k[0]] = F[23,x,k[0]]*F[3,x]*F[41,x,k[0]]
F[41,x,k[0]] = F[1,x]+F[42,x,k[0]]
F[42,x,k[0]] = F[43,x,k[0]]
F[43,x,k[0]] = F[23,x,k[0]]*F[41,x,k[0]]
F[44,x,k[0]] = F[22,x]*F[38,x,k[0]]
F[45,x] = F[46,x,1]
F[46,x,k[0]] = F[22,x]*F[47,x,k[0]]
F[47,x,k[0]] = F[1,x]+F[48,x,k[0]]+F[94,x,k[0]]+F[96,x,k[0]]+F[99,x,k[0]]
F[48,x,k[0]] = F[23,x,k[0]]*F[49,x,k[0]]
F[49,x,k[0]] = F[1,x]+F[48,x,k[0]]+F[50,x]+F[94,x,k[0]]+F[96,x,k[0]]
F[50,x] = F[22,x]*F[51,x]
F[51,x] = F[1,x]+F[45,x]+F[50,x]+F[52,x]
F[52,x] = F[53,x,1]
F[53,x,k[0]] = F[54,x,k[0]]
F[54,x,k[0]] = F[22,x]*F[55,x,k[0]]
F[55,x,k[0]] = (F[56,x,k[0]]*k[0]-F[56,x,1])/(-1+k[0])
F[56,x,k[0]] = F[57,x,k[0]]+F[91,x,k[0]]
F[57,x,k[0]] = F[58,x]+F[70,x,k[0]]
F[58,x] = F[1,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[22,x]*F[61,x]
F[61,x] = F[19,x]+F[62,x]
F[62,x] = F[25,x]+F[63,x]
F[63,x] = F[64,x]+F[65,x]+F[67,x]
F[64,x] = 0
F[65,x] = F[22,x]*F[66,x]
F[66,x] = F[20,x]+F[63,x]
F[67,x] = F[22,x]*F[68,x]
F[68,x] = F[25,x]+F[69,x]
F[69,x] = F[67,x]
F[70,x,k[0]] = F[42,x,k[0]]+F[71,x,k[0]]
F[71,x,k[0]] = F[64,x]+F[72,x,k[0]]+F[74,x,k[0]]
F[72,x,k[0]] = F[23,x,k[0]]*F[73,x,k[0]]
F[73,x,k[0]] = F[59,x]+F[71,x,k[0]]
F[74,x,k[0]] = F[22,x]*F[75,x,k[0]]
F[75,x,k[0]] = F[76,x,k[0]]+F[79,x,k[0]]
F[76,x,k[0]] = F[42,x,k[0]]+F[77,x,k[0]]
F[77,x,k[0]] = F[78,x,k[0]]
F[78,x,k[0]] = F[22,x]*F[76,x,k[0]]
F[79,x,k[0]] = F[80,x,k[0]]+F[85,x,k[0]]
F[80,x,k[0]] = F[64,x]+F[81,x,k[0]]+F[83,x,k[0]]
F[81,x,k[0]] = F[23,x,k[0]]*F[82,x,k[0]]
F[82,x,k[0]] = F[25,x]+F[80,x,k[0]]
F[83,x,k[0]] = F[22,x]*F[84,x,k[0]]
F[84,x,k[0]] = F[42,x,k[0]]+F[80,x,k[0]]
F[85,x,k[0]] = 2*F[64,x]+F[86,x,k[0]]+F[88,x,k[0]]
F[86,x,k[0]] = F[22,x]*F[87,x,k[0]]
F[87,x,k[0]] = F[77,x,k[0]]+F[85,x,k[0]]
F[88,x,k[0]] = F[22,x]*F[89,x,k[0]]
F[89,x,k[0]] = F[80,x,k[0]]+F[90,x,k[0]]
F[90,x,k[0]] = F[88,x,k[0]]
F[91,x,k[0]] = F[92,x,k[0]]
F[92,x,k[0]] = F[22,x]*F[41,x,k[0]]*F[93,x]
F[93,x] = F[16,x]+F[27,x]
F[94,x,k[0]] = F[22,x]*F[95,x,k[0]]
F[95,x,k[0]] = (F[47,x,k[0]]*k[0]-F[47,x,1])/(-1+k[0])
F[96,x,k[0]] = F[97,x,k[0]]
F[97,x,k[0]] = F[22,x]*F[98,x,k[0]]
F[98,x,k[0]] = (F[55,x,k[0]]*k[0]-F[55,x,1])/(-1+k[0])
F[99,x,k[0]] = F[100,x,k[0]]*F[22,x]
F[100,x,k[0]] = F[1,x]+F[101,x,k[0]]+F[103,x,k[0]]+F[124,x,k[0]]+F[45,x]+F[52,x]
F[101,x,k[0]] = F[102,x,k[0]]
F[102,x,k[0]] = F[23,x,k[0]]*F[27,x]*F[3,x]*F[41,x,k[0]]
F[103,x,k[0]] = F[104,x,k[0]]
F[104,x,k[0]] = F[105,x,k[0]]*F[22,x]*F[3,x]
F[105,x,k[0]] = F[106,x,k[0]]+F[122,x,k[0]]
F[106,x,k[0]] = F[107,x,k[0]]+F[11,x]
F[107,x,k[0]] = F[108,x,k[0]]+F[42,x,k[0]]
F[108,x,k[0]] = F[109,x,k[0]]+F[112,x,k[0]]+F[64,x]
F[109,x,k[0]] = F[110,x,k[0]]*F[23,x,k[0]]
F[110,x,k[0]] = F[111,x,k[0]]+F[16,x]
F[111,x,k[0]] = F[109,x,k[0]]
F[112,x,k[0]] = F[113,x,k[0]]*F[22,x]
F[113,x,k[0]] = F[114,x,k[0]]+F[115,x,k[0]]
F[114,x,k[0]] = F[111,x,k[0]]+F[42,x,k[0]]
F[115,x,k[0]] = F[116,x,k[0]]+F[119,x,k[0]]
F[116,x,k[0]] = F[117,x,k[0]]
F[117,x,k[0]] = F[118,x,k[0]]*F[22,x]
F[118,x,k[0]] = F[116,x,k[0]]+F[42,x,k[0]]
F[119,x,k[0]] = F[120,x,k[0]]
F[120,x,k[0]] = F[121,x,k[0]]*F[22,x]
F[121,x,k[0]] = F[111,x,k[0]]+F[119,x,k[0]]
F[122,x,k[0]] = F[123,x,k[0]]
F[123,x,k[0]] = F[27,x]^2*F[22,x]*F[41,x,k[0]]
F[124,x,k[0]] = F[125,x,k[0]]*F[22,x]
F[125,x,k[0]] = (F[38,x,k[0]]*k[0]-F[38,x,1])/(-1+k[0])
F[126,x] = F[127,x,1]
F[127,x,k[0]] = F[128,x,k[0]]
F[128,x,k[0]] = F[22,x]*F[56,x,k[0]]
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_{22}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{126}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{22}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{22}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{36}\! \left(x \right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{22}\! \left(x \right) F_{3}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{35}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{22}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{22}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{19}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x , 1\right)
F_{23}\! \left(x , y\right) = y x
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{22}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{22}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{16}\! \left(x \right)+F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right) F_{27}\! \left(x \right)
F_{36}\! \left(x \right) = F_{22}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x , 1\right)
F_{38}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{39}\! \left(x , y\right)+F_{44}\! \left(x , y\right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)
F_{39}\! \left(x , y\right) = F_{40}\! \left(x , y\right)
F_{40}\! \left(x , y\right) = F_{23}\! \left(x , y\right) F_{3}\! \left(x \right) F_{41}\! \left(x , y\right)
F_{41}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{42}\! \left(x , y\right)
F_{42}\! \left(x , y\right) = F_{43}\! \left(x , y\right)
F_{43}\! \left(x , y\right) = F_{23}\! \left(x , y\right) F_{41}\! \left(x , y\right)
F_{44}\! \left(x , y\right) = F_{22}\! \left(x \right) F_{38}\! \left(x , y\right)
F_{45}\! \left(x \right) = F_{46}\! \left(x , 1\right)
F_{46}\! \left(x , y\right) = F_{22}\! \left(x \right) F_{47}\! \left(x , y\right)
F_{47}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{48}\! \left(x , y\right)+F_{94}\! \left(x , y\right)+F_{96}\! \left(x , y\right)+F_{99}\! \left(x , y\right)
F_{48}\! \left(x , y\right) = F_{23}\! \left(x , y\right) F_{49}\! \left(x , y\right)
F_{49}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{48}\! \left(x , y\right)+F_{50}\! \left(x \right)+F_{94}\! \left(x , y\right)+F_{96}\! \left(x , y\right)
F_{50}\! \left(x \right) = F_{22}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{1}\! \left(x \right)+F_{45}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x , 1\right)
F_{53}\! \left(x , y\right) = F_{54}\! \left(x , y\right)
F_{54}\! \left(x , y\right) = F_{22}\! \left(x \right) F_{55}\! \left(x , y\right)
F_{55}\! \left(x , y\right) = -\frac{-y F_{56}\! \left(x , y\right)+F_{56}\! \left(x , 1\right)}{-1+y}
F_{56}\! \left(x , y\right) = F_{57}\! \left(x , y\right)+F_{91}\! \left(x , y\right)
F_{57}\! \left(x , y\right) = F_{58}\! \left(x \right)+F_{70}\! \left(x , y\right)
F_{58}\! \left(x \right) = F_{1}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{22}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{19}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{25}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{64}\! \left(x \right) = 0
F_{65}\! \left(x \right) = F_{22}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{20}\! \left(x \right)+F_{63}\! \left(x \right)
F_{67}\! \left(x \right) = F_{22}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{25}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{67}\! \left(x \right)
F_{70}\! \left(x , y\right) = F_{42}\! \left(x , y\right)+F_{71}\! \left(x , y\right)
F_{71}\! \left(x , y\right) = F_{64}\! \left(x \right)+F_{72}\! \left(x , y\right)+F_{74}\! \left(x , y\right)
F_{72}\! \left(x , y\right) = F_{23}\! \left(x , y\right) F_{73}\! \left(x , y\right)
F_{73}\! \left(x , y\right) = F_{59}\! \left(x \right)+F_{71}\! \left(x , y\right)
F_{74}\! \left(x , y\right) = F_{22}\! \left(x \right) F_{75}\! \left(x , y\right)
F_{75}\! \left(x , y\right) = F_{76}\! \left(x , y\right)+F_{79}\! \left(x , y\right)
F_{76}\! \left(x , y\right) = F_{42}\! \left(x , y\right)+F_{77}\! \left(x , y\right)
F_{77}\! \left(x , y\right) = F_{78}\! \left(x , y\right)
F_{78}\! \left(x , y\right) = F_{22}\! \left(x \right) F_{76}\! \left(x , y\right)
F_{79}\! \left(x , y\right) = F_{80}\! \left(x , y\right)+F_{85}\! \left(x , y\right)
F_{80}\! \left(x , y\right) = F_{64}\! \left(x \right)+F_{81}\! \left(x , y\right)+F_{83}\! \left(x , y\right)
F_{81}\! \left(x , y\right) = F_{23}\! \left(x , y\right) F_{82}\! \left(x , y\right)
F_{82}\! \left(x , y\right) = F_{25}\! \left(x \right)+F_{80}\! \left(x , y\right)
F_{83}\! \left(x , y\right) = F_{22}\! \left(x \right) F_{84}\! \left(x , y\right)
F_{84}\! \left(x , y\right) = F_{42}\! \left(x , y\right)+F_{80}\! \left(x , y\right)
F_{85}\! \left(x , y\right) = 2 F_{64}\! \left(x \right)+F_{86}\! \left(x , y\right)+F_{88}\! \left(x , y\right)
F_{86}\! \left(x , y\right) = F_{22}\! \left(x \right) F_{87}\! \left(x , y\right)
F_{87}\! \left(x , y\right) = F_{77}\! \left(x , y\right)+F_{85}\! \left(x , y\right)
F_{88}\! \left(x , y\right) = F_{22}\! \left(x \right) F_{89}\! \left(x , y\right)
F_{89}\! \left(x , y\right) = F_{80}\! \left(x , y\right)+F_{90}\! \left(x , y\right)
F_{90}\! \left(x , y\right) = F_{88}\! \left(x , y\right)
F_{91}\! \left(x , y\right) = F_{92}\! \left(x , y\right)
F_{92}\! \left(x , y\right) = F_{22}\! \left(x \right) F_{41}\! \left(x , y\right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{16}\! \left(x \right)+F_{27}\! \left(x \right)
F_{94}\! \left(x , y\right) = F_{22}\! \left(x \right) F_{95}\! \left(x , y\right)
F_{95}\! \left(x , y\right) = -\frac{-y F_{47}\! \left(x , y\right)+F_{47}\! \left(x , 1\right)}{-1+y}
F_{96}\! \left(x , y\right) = F_{97}\! \left(x , y\right)
F_{97}\! \left(x , y\right) = F_{22}\! \left(x \right) F_{98}\! \left(x , y\right)
F_{98}\! \left(x , y\right) = -\frac{-y F_{55}\! \left(x , y\right)+F_{55}\! \left(x , 1\right)}{-1+y}
F_{99}\! \left(x , y\right) = F_{100}\! \left(x , y\right) F_{22}\! \left(x \right)
F_{100}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{101}\! \left(x , y\right)+F_{103}\! \left(x , y\right)+F_{124}\! \left(x , y\right)+F_{45}\! \left(x \right)+F_{52}\! \left(x \right)
F_{101}\! \left(x , y\right) = F_{102}\! \left(x , y\right)
F_{102}\! \left(x , y\right) = F_{23}\! \left(x , y\right) F_{27}\! \left(x \right) F_{3}\! \left(x \right) F_{41}\! \left(x , y\right)
F_{103}\! \left(x , y\right) = F_{104}\! \left(x , y\right)
F_{104}\! \left(x , y\right) = F_{105}\! \left(x , y\right) F_{22}\! \left(x \right) F_{3}\! \left(x \right)
F_{105}\! \left(x , y\right) = F_{106}\! \left(x , y\right)+F_{122}\! \left(x , y\right)
F_{106}\! \left(x , y\right) = F_{107}\! \left(x , y\right)+F_{11}\! \left(x \right)
F_{107}\! \left(x , y\right) = F_{108}\! \left(x , y\right)+F_{42}\! \left(x , y\right)
F_{108}\! \left(x , y\right) = F_{109}\! \left(x , y\right)+F_{112}\! \left(x , y\right)+F_{64}\! \left(x \right)
F_{109}\! \left(x , y\right) = F_{110}\! \left(x , y\right) F_{23}\! \left(x , y\right)
F_{110}\! \left(x , y\right) = F_{111}\! \left(x , y\right)+F_{16}\! \left(x \right)
F_{111}\! \left(x , y\right) = F_{109}\! \left(x , y\right)
F_{112}\! \left(x , y\right) = F_{113}\! \left(x , y\right) F_{22}\! \left(x \right)
F_{113}\! \left(x , y\right) = F_{114}\! \left(x , y\right)+F_{115}\! \left(x , y\right)
F_{114}\! \left(x , y\right) = F_{111}\! \left(x , y\right)+F_{42}\! \left(x , y\right)
F_{115}\! \left(x , y\right) = F_{116}\! \left(x , y\right)+F_{119}\! \left(x , y\right)
F_{116}\! \left(x , y\right) = F_{117}\! \left(x , y\right)
F_{117}\! \left(x , y\right) = F_{118}\! \left(x , y\right) F_{22}\! \left(x \right)
F_{118}\! \left(x , y\right) = F_{116}\! \left(x , y\right)+F_{42}\! \left(x , y\right)
F_{119}\! \left(x , y\right) = F_{120}\! \left(x , y\right)
F_{120}\! \left(x , y\right) = F_{121}\! \left(x , y\right) F_{22}\! \left(x \right)
F_{121}\! \left(x , y\right) = F_{111}\! \left(x , y\right)+F_{119}\! \left(x , y\right)
F_{122}\! \left(x , y\right) = F_{123}\! \left(x , y\right)
F_{123}\! \left(x , y\right) = F_{27} \left(x \right)^{2} F_{22}\! \left(x \right) F_{41}\! \left(x , y\right)
F_{124}\! \left(x , y\right) = F_{125}\! \left(x , y\right) F_{22}\! \left(x \right)
F_{125}\! \left(x , y\right) = -\frac{-y F_{38}\! \left(x , y\right)+F_{38}\! \left(x , 1\right)}{-1+y}
F_{126}\! \left(x \right) = F_{127}\! \left(x , 1\right)
F_{127}\! \left(x , y\right) = F_{128}\! \left(x , y\right)
F_{128}\! \left(x , y\right) = F_{22}\! \left(x \right) F_{56}\! \left(x , y\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_22(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_126(x) + F_4(x))
Eq(F_4(x), F_22(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_45(x) + F_52(x) + F_6(x))
Eq(F_6(x), F_22(x)*F_7(x))
Eq(F_7(x), F_1(x) + F_36(x) + F_45(x) + F_52(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_22(x)*F_3(x))
Eq(F_10(x), F_11(x) + F_35(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_22(x))
Eq(F_14(x), F_15(x) + F_31(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_22(x))
Eq(F_18(x), F_19(x) + F_24(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_19(x)*F_22(x))
Eq(F_22(x), F_23(x, 1))
Eq(F_23(x, k_0), k_0*x)
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_22(x)*F_27(x))
Eq(F_27(x), F_1(x) + F_25(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_22(x)*F_30(x))
Eq(F_30(x), F_20(x) + F_28(x))
Eq(F_31(x), F_25(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_22(x)*F_34(x))
Eq(F_34(x), F_16(x) + F_32(x))
Eq(F_35(x), F_25(x)*F_27(x))
Eq(F_36(x), F_22(x)*F_37(x))
Eq(F_37(x), F_38(x, 1))
Eq(F_38(x, k_0), F_1(x) + F_39(x, k_0) + F_44(x, k_0) + F_45(x) + F_52(x))
Eq(F_39(x, k_0), F_40(x, k_0))
Eq(F_40(x, k_0), F_23(x, k_0)*F_3(x)*F_41(x, k_0))
Eq(F_41(x, k_0), F_1(x) + F_42(x, k_0))
Eq(F_42(x, k_0), F_43(x, k_0))
Eq(F_43(x, k_0), F_23(x, k_0)*F_41(x, k_0))
Eq(F_44(x, k_0), F_22(x)*F_38(x, k_0))
Eq(F_45(x), F_46(x, 1))
Eq(F_46(x, k_0), F_22(x)*F_47(x, k_0))
Eq(F_47(x, k_0), F_1(x) + F_48(x, k_0) + F_94(x, k_0) + F_96(x, k_0) + F_99(x, k_0))
Eq(F_48(x, k_0), F_23(x, k_0)*F_49(x, k_0))
Eq(F_49(x, k_0), F_1(x) + F_48(x, k_0) + F_50(x) + F_94(x, k_0) + F_96(x, k_0))
Eq(F_50(x), F_22(x)*F_51(x))
Eq(F_51(x), F_1(x) + F_45(x) + F_50(x) + F_52(x))
Eq(F_52(x), F_53(x, 1))
Eq(F_53(x, k_0), F_54(x, k_0))
Eq(F_54(x, k_0), F_22(x)*F_55(x, k_0))
Eq(F_55(x, k_0), (-k_0*F_56(x, k_0) + F_56(x, 1))/(1 - k_0))
Eq(F_56(x, k_0), F_57(x, k_0) + F_91(x, k_0))
Eq(F_57(x, k_0), F_58(x) + F_70(x, k_0))
Eq(F_58(x), F_1(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_22(x)*F_61(x))
Eq(F_61(x), F_19(x) + F_62(x))
Eq(F_62(x), F_25(x) + F_63(x))
Eq(F_63(x), F_64(x) + F_65(x) + F_67(x))
Eq(F_64(x), 0)
Eq(F_65(x), F_22(x)*F_66(x))
Eq(F_66(x), F_20(x) + F_63(x))
Eq(F_67(x), F_22(x)*F_68(x))
Eq(F_68(x), F_25(x) + F_69(x))
Eq(F_69(x), F_67(x))
Eq(F_70(x, k_0), F_42(x, k_0) + F_71(x, k_0))
Eq(F_71(x, k_0), F_64(x) + F_72(x, k_0) + F_74(x, k_0))
Eq(F_72(x, k_0), F_23(x, k_0)*F_73(x, k_0))
Eq(F_73(x, k_0), F_59(x) + F_71(x, k_0))
Eq(F_74(x, k_0), F_22(x)*F_75(x, k_0))
Eq(F_75(x, k_0), F_76(x, k_0) + F_79(x, k_0))
Eq(F_76(x, k_0), F_42(x, k_0) + F_77(x, k_0))
Eq(F_77(x, k_0), F_78(x, k_0))
Eq(F_78(x, k_0), F_22(x)*F_76(x, k_0))
Eq(F_79(x, k_0), F_80(x, k_0) + F_85(x, k_0))
Eq(F_80(x, k_0), F_64(x) + F_81(x, k_0) + F_83(x, k_0))
Eq(F_81(x, k_0), F_23(x, k_0)*F_82(x, k_0))
Eq(F_82(x, k_0), F_25(x) + F_80(x, k_0))
Eq(F_83(x, k_0), F_22(x)*F_84(x, k_0))
Eq(F_84(x, k_0), F_42(x, k_0) + F_80(x, k_0))
Eq(F_85(x, k_0), 2*F_64(x) + F_86(x, k_0) + F_88(x, k_0))
Eq(F_86(x, k_0), F_22(x)*F_87(x, k_0))
Eq(F_87(x, k_0), F_77(x, k_0) + F_85(x, k_0))
Eq(F_88(x, k_0), F_22(x)*F_89(x, k_0))
Eq(F_89(x, k_0), F_80(x, k_0) + F_90(x, k_0))
Eq(F_90(x, k_0), F_88(x, k_0))
Eq(F_91(x, k_0), F_92(x, k_0))
Eq(F_92(x, k_0), F_22(x)*F_41(x, k_0)*F_93(x))
Eq(F_93(x), F_16(x) + F_27(x))
Eq(F_94(x, k_0), F_22(x)*F_95(x, k_0))
Eq(F_95(x, k_0), (-k_0*F_47(x, k_0) + F_47(x, 1))/(1 - k_0))
Eq(F_96(x, k_0), F_97(x, k_0))
Eq(F_97(x, k_0), F_22(x)*F_98(x, k_0))
Eq(F_98(x, k_0), (-k_0*F_55(x, k_0) + F_55(x, 1))/(1 - k_0))
Eq(F_99(x, k_0), F_100(x, k_0)*F_22(x))
Eq(F_100(x, k_0), F_1(x) + F_101(x, k_0) + F_103(x, k_0) + F_124(x, k_0) + F_45(x) + F_52(x))
Eq(F_101(x, k_0), F_102(x, k_0))
Eq(F_102(x, k_0), F_23(x, k_0)*F_27(x)*F_3(x)*F_41(x, k_0))
Eq(F_103(x, k_0), F_104(x, k_0))
Eq(F_104(x, k_0), F_105(x, k_0)*F_22(x)*F_3(x))
Eq(F_105(x, k_0), F_106(x, k_0) + F_122(x, k_0))
Eq(F_106(x, k_0), F_107(x, k_0) + F_11(x))
Eq(F_107(x, k_0), F_108(x, k_0) + F_42(x, k_0))
Eq(F_108(x, k_0), F_109(x, k_0) + F_112(x, k_0) + F_64(x))
Eq(F_109(x, k_0), F_110(x, k_0)*F_23(x, k_0))
Eq(F_110(x, k_0), F_111(x, k_0) + F_16(x))
Eq(F_111(x, k_0), F_109(x, k_0))
Eq(F_112(x, k_0), F_113(x, k_0)*F_22(x))
Eq(F_113(x, k_0), F_114(x, k_0) + F_115(x, k_0))
Eq(F_114(x, k_0), F_111(x, k_0) + F_42(x, k_0))
Eq(F_115(x, k_0), F_116(x, k_0) + F_119(x, k_0))
Eq(F_116(x, k_0), F_117(x, k_0))
Eq(F_117(x, k_0), F_118(x, k_0)*F_22(x))
Eq(F_118(x, k_0), F_116(x, k_0) + F_42(x, k_0))
Eq(F_119(x, k_0), F_120(x, k_0))
Eq(F_120(x, k_0), F_121(x, k_0)*F_22(x))
Eq(F_121(x, k_0), F_111(x, k_0) + F_119(x, k_0))
Eq(F_122(x, k_0), F_123(x, k_0))
Eq(F_123(x, k_0), F_22(x)*F_27(x)**2*F_41(x, k_0))
Eq(F_124(x, k_0), F_125(x, k_0)*F_22(x))
Eq(F_125(x, k_0), (-k_0*F_38(x, k_0) + F_38(x, 1))/(1 - k_0))
Eq(F_126(x), F_127(x, 1))
Eq(F_127(x, k_0), F_128(x, k_0))
Eq(F_128(x, k_0), F_22(x)*F_56(x, k_0))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "row_placements_tracked_fusion", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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{"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": 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