0123_0132_0231_1230_3012
Counting sequence:
1, 1, 2, 6, 19, 60, 189, 602, 1957, 6510, 22123, 76549, 268778, 954928, 3425439, 12385562, 45085161, 165067858, 607414407, 2245163861, 8331907706, 31031435804, 115950927579, 434546371339, 1632972149304, 6151867325690, 23229333027709, 87900971096667, 333281636883712, 1265990436836776, 4817229255614147, 18359623615060570, 70078703063758257, 267870199584258074, 1025280181151907903, 3929229554236900613, 15076083915127492490, 57910602779201671924, 222684667249067396531, 857159234791860407991, 3302552414556223722316, 12736045635013901869794, 49158348505741365347949, 189897909121387842130139, 734153598806878137770000, 2840417973687836952385184, 10997446565645605835387835, 42609102468794589020652391, 165196671999501149852775588, 640880143422067978509309950, 2487813902936340741631803113, 9663047328271602938761431959, 37553952616289194135186917588, 146027038401559563800323632508, 568117589483744912897418094863, 2211374334636790356838598865893, 8611871544453728397339157302710, 33553409784619797459388917303314, 130789413145897300052931615578961, 510033455416699577994517913889223, 1989795959723697604878168525827724, 7765978923180976846095358233100376, 30321865701630484782041548607181707, 118435588301817611225332861920039386, 462775020434588648490316464514700865, 1808897743026736346798243010666961834, 7073100157711232440759893747512111535, 27666406606516830272139444614744299301, 108252649020916803337118145867434469098, 423704426509176261578117339298112556708, 1658907610930498246637021781003993818595, 6496992033066387831089473991902706561887, 25452460314355505424709122003684529346340, 99740379754786777708859602683473555121266, 390960367676180698988189262116897693336429, 1532892532741482132985186479458249845670267, 6011804584793682723194925862265783753522256, 23583551657173188486615305351208323977690768, 92538386338342343035052628861186703079420907, 363195324786797357914276229339041753797401291, 1425809659122601171386521185943965538519356096, 5598649627935656754396083100330415269341971614, 21988875065605097317200799667061018078245298049, 86381026148396971542628028255862761296080261679, 339411892172175312736540186730394130597054132460, 1333911153961668954001698464145758859210569980204, 5243433776478798570487771171840611585285445312815, 20615394908221951531583918160186248938063522900781, 81068575420428341167565937189413978278583488042638, 318857418889763710894450387749997649894297632064738, 1254358759104355578453353310285403154139145911535505, 4935446834384359467347254061239930659516767669094727, 19422670056075894092546183012421771312548512606103660, 76448234357832305158590895496368149265504759631766216, 300954222053417420920378350107826686181292755588657019, 1184967143495505213021521666885456948409595984109116767, 4666417002383382593416476451448267513678906252385568956, 18379372560535529822580748467264441935686799241822395078, 72401287425730933219204578613156214124262361762744262017, 285252188863830707444007241320644360347119764600473454831, 1124028777884360846348989611103614786490993904888638554396
Generating function in Maple syntax:
1/2*(-2*(x-1/2)*(x+1)*(x-1)^3*(1-4*x)^(1/2)+8*x^5-15*x^4+10*x^3+2*x^2-4*x+1)/x/(2*x-1)/(x-1)^3
Generating function in latex syntax:
\frac{-2 \left(x -\frac{1}{2}\right) \left(x +1\right) \left(x -1\right)^{3} \sqrt{1-4 x}+8 x^{5}-15 x^{4}+10 x^{3}+2 x^{2}-4 x +1}{2 x \left(2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(4*x**5 - 15*x**4/2 + 5*x**3 + x**2 - 2*x + sqrt(1 - 4*x)*(1 - 2*x)*(x - 1)**3*(x + 1)/2 + 1/2)/(x*(x - 1)**3*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(x-1)^6*F(x)^2-(2*x-1)*(8*x^5-15*x^4+10*x^3+2*x^2-4*x+1)*(x-1)^3*F(x)+4*x^10-5*x^9-22*x^8+84*x^7-118*x^6+67*x^5+19*x^4-52*x^3+32*x^2-9*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)^{2}-\left(2 x -1\right) \left(8 x^{5}-15 x^{4}+10 x^{3}+2 x^{2}-4 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+4 x^{10}-5 x^{9}-22 x^{8}+84 x^{7}-118 x^{6}+67 x^{5}+19 x^{4}-52 x^{3}+32 x^{2}-9 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 60
a(6) = 189
a(7) = 602
a(8) = 1957
a(n+4) = 8*(-1+2*n)/(n+5)*a(n)-4*(n-3)/(n+5)*a(n+1)-2*(19+6*n)/(n+5)*a(n+2)+(27+7*n)/(n+5)*a(n+3)+(3*n^3-20*n^2+18*n-11)/(n+5), 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) = 60
a \! \left(6\right) = 189
a \! \left(7\right) = 602
a \! \left(8\right) = 1957
a \! \left(n +4\right) = \frac{8 \left(-1+2 n \right) a \! \left(n \right)}{n +5}-\frac{4 \left(n -3\right) a \! \left(n +1\right)}{n +5}-\frac{2 \left(19+6 n \right) a \! \left(n +2\right)}{n +5}+\frac{\left(27+7 n \right) a \! \left(n +3\right)}{n +5}+\frac{3 n^{3}-20 n^{2}+18 n -11}{n +5}, \quad n \geq 9
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21504/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[14,x]*F[3,x]
F[3,x] = F[1,x]+F[4,x]+F[86,x]
F[4,x] = F[14,x]*F[5,x]
F[5,x] = F[1,x]+F[46,x]+F[54,x]+F[6,x]
F[6,x] = F[14,x]*F[7,x]
F[7,x] = F[8,x,1]
F[8,x,k[0]] = F[1,x]+F[15,x,k[0]]+F[44,x,k[0]]+F[46,x]+F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]*F[11,x,k[0]]
F[10,x,k[0]] = k[0]*x
F[11,x,k[0]] = F[1,x]+F[12,x,k[0]]+F[15,x,k[0]]+F[9,x,k[0]]
F[12,x,k[0]] = F[13,x,k[0]]*F[14,x]
F[13,x,k[0]] = 1/(-1+k[0])*(F[11,x,k[0]]*k[0]-F[11,x,1])
F[14,x] = F[10,x,1]
F[15,x,k[0]] = F[14,x]*F[16,x,k[0]]
F[16,x,k[0]] = F[17,x]+F[27,x,k[0]]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[18,x]
F[21,x] = F[19,x]+F[22,x]
F[22,x] = F[23,x]+F[24,x]+F[26,x]
F[23,x] = 0
F[24,x] = F[14,x]*F[25,x]
F[25,x] = F[19,x]+F[22,x]
F[26,x] = F[14,x]*F[21,x]
F[27,x,k[0]] = F[28,x,k[0]]+F[36,x,k[0]]
F[28,x,k[0]] = F[29,x,k[0]]+F[32,x,k[0]]
F[29,x,k[0]] = F[30,x,k[0]]
F[30,x,k[0]] = F[10,x,k[0]]*F[31,x,k[0]]
F[31,x,k[0]] = F[1,x]+F[29,x,k[0]]
F[32,x,k[0]] = F[23,x]+F[33,x,k[0]]+F[35,x,k[0]]
F[33,x,k[0]] = F[10,x,k[0]]*F[34,x,k[0]]
F[34,x,k[0]] = F[19,x]+F[32,x,k[0]]
F[35,x,k[0]] = F[14,x]*F[28,x,k[0]]
F[36,x,k[0]] = F[37,x,k[0]]+F[40,x,k[0]]
F[37,x,k[0]] = F[38,x,k[0]]
F[38,x,k[0]] = F[14,x]*F[39,x,k[0]]
F[39,x,k[0]] = F[29,x,k[0]]+F[37,x,k[0]]
F[40,x,k[0]] = 2*F[23,x]+F[41,x,k[0]]+F[43,x,k[0]]
F[41,x,k[0]] = F[14,x]*F[42,x,k[0]]
F[42,x,k[0]] = F[32,x,k[0]]+F[40,x,k[0]]
F[43,x,k[0]] = F[14,x]*F[36,x,k[0]]
F[44,x,k[0]] = F[14,x]*F[45,x,k[0]]
F[45,x,k[0]] = (F[8,x,k[0]]*k[0]-F[8,x,1])/(-1+k[0])
F[46,x] = F[47,x,1]
F[47,x,k[0]] = F[10,x,k[0]]*F[48,x,k[0]]
F[48,x,k[0]] = F[31,x,k[0]]+F[49,x,k[0]]
F[49,x,k[0]] = F[19,x]+F[50,x,k[0]]
F[50,x,k[0]] = F[23,x]+F[51,x,k[0]]+F[53,x,k[0]]
F[51,x,k[0]] = F[14,x]*F[52,x,k[0]]
F[52,x,k[0]] = F[29,x,k[0]]+F[50,x,k[0]]
F[53,x,k[0]] = F[10,x,k[0]]*F[49,x,k[0]]
F[54,x] = F[14,x]*F[55,x]
F[55,x] = F[56,x]+F[68,x]
F[56,x] = F[57,x]+F[67,x]
F[57,x] = F[1,x]+F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[14,x]*F[60,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[1,x]+F[14,x]
F[62,x] = F[19,x]+F[63,x]
F[63,x] = F[23,x]+F[64,x]+F[66,x]
F[64,x] = F[14,x]*F[65,x]
F[65,x] = F[14,x]+F[63,x]
F[66,x] = F[14,x]*F[19,x]
F[67,x] = F[19,x]
F[68,x] = F[69,x]+F[85,x]
F[69,x] = F[19,x]+F[70,x]
F[70,x] = F[23,x]+F[71,x]+F[73,x]
F[71,x] = F[14,x]*F[72,x]
F[72,x] = F[58,x]+F[70,x]
F[73,x] = F[14,x]*F[74,x]
F[74,x] = F[75,x]+F[79,x]
F[75,x] = F[19,x]+F[76,x]
F[76,x] = F[77,x]
F[77,x] = F[14,x]*F[78,x]
F[78,x] = F[14,x]+F[76,x]
F[79,x] = F[22,x]+F[80,x]
F[80,x] = 2*F[23,x]+F[81,x]+F[83,x]
F[81,x] = F[14,x]*F[82,x]
F[82,x] = F[63,x]+F[80,x]
F[83,x] = F[14,x]*F[84,x]
F[84,x] = F[76,x]+F[80,x]
F[85,x] = F[22,x]
F[86,x] = F[14,x]*F[87,x]
F[87,x] = F[1,x]+F[137,x]+F[88,x]+F[89,x]
F[88,x] = F[9,x,1]
F[89,x] = F[14,x]*F[90,x]
F[90,x] = F[91,x,1]
F[91,x,k[0]] = F[1,x]+F[9,x,k[0]]+F[92,x,k[0]]+F[94,x,k[0]]+F[96,x,k[0]]
F[92,x,k[0]] = F[14,x]*F[93,x,k[0]]
F[93,x,k[0]] = 1/(-1+k[0])*(F[11,x,k[0]]*k[0]-F[11,x,1])
F[94,x,k[0]] = F[14,x]*F[95,x,k[0]]
F[95,x,k[0]] = (F[91,x,k[0]]*k[0]-F[91,x,1])/(-1+k[0])
F[96,x,k[0]] = F[14,x]*F[97,x,k[0]]
F[97,x,k[0]] = F[16,x,k[0]]+F[98,x,k[0]]
F[98,x,k[0]] = F[113,x,k[0]]+F[99,x]
F[99,x] = F[100,x]+F[21,x]
F[100,x] = F[101,x]+F[22,x]
F[101,x] = F[102,x]+F[103,x]+F[105,x]+F[23,x]
F[102,x] = 0
F[103,x] = F[104,x]*F[14,x]
F[104,x] = F[101,x]+F[22,x]
F[105,x] = F[106,x]*F[14,x]
F[106,x] = F[107,x]+F[110,x]
F[107,x] = F[108,x]
F[108,x] = F[109,x]*F[14,x]
F[109,x] = F[107,x]+F[19,x]
F[110,x] = 2*F[23,x]+F[105,x]+F[111,x]
F[111,x] = F[112,x]*F[14,x]
F[112,x] = F[110,x]+F[22,x]
F[113,x,k[0]] = F[114,x,k[0]]+F[122,x,k[0]]
F[114,x,k[0]] = F[115,x,k[0]]+F[118,x,k[0]]
F[115,x,k[0]] = F[116,x,k[0]]
F[116,x,k[0]] = F[10,x,k[0]]*F[117,x,k[0]]
F[117,x,k[0]] = F[115,x,k[0]]+F[19,x]
F[118,x,k[0]] = 2*F[23,x]+F[119,x,k[0]]+F[121,x,k[0]]
F[119,x,k[0]] = F[10,x,k[0]]*F[120,x,k[0]]
F[120,x,k[0]] = F[118,x,k[0]]+F[22,x]
F[121,x,k[0]] = F[114,x,k[0]]*F[14,x]
F[122,x,k[0]] = F[123,x,k[0]]+F[126,x,k[0]]
F[123,x,k[0]] = F[124,x,k[0]]
F[124,x,k[0]] = F[125,x,k[0]]*F[14,x]
F[125,x,k[0]] = F[115,x,k[0]]+F[123,x,k[0]]
F[126,x,k[0]] = 3*F[23,x]+F[127,x,k[0]]+F[129,x,k[0]]
F[127,x,k[0]] = F[128,x,k[0]]*F[14,x]
F[128,x,k[0]] = F[118,x,k[0]]+F[126,x,k[0]]
F[129,x,k[0]] = F[130,x,k[0]]*F[14,x]
F[130,x,k[0]] = F[131,x,k[0]]+F[134,x,k[0]]
F[131,x,k[0]] = F[132,x,k[0]]
F[132,x,k[0]] = F[133,x,k[0]]*F[14,x]
F[133,x,k[0]] = F[115,x,k[0]]+F[131,x,k[0]]
F[134,x,k[0]] = 3*F[23,x]+F[129,x,k[0]]+F[135,x,k[0]]
F[135,x,k[0]] = F[136,x,k[0]]*F[14,x]
F[136,x,k[0]] = F[118,x,k[0]]+F[134,x,k[0]]
F[137,x] = F[138,x]*F[14,x]
F[138,x] = F[17,x]+F[99,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_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{86}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{46}\! \left(x \right)+F_{54}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{14}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x , 1\right)
F_{8}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{44}\! \left(x , y\right)+F_{46}\! \left(x \right)+F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{11}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = y x
F_{11}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{15}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{13}\! \left(x , y\right) = -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}
F_{14}\! \left(x \right) = F_{10}\! \left(x , 1\right)
F_{15}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{16}\! \left(x , y\right)
F_{16}\! \left(x , y\right) = F_{17}\! \left(x \right)+F_{27}\! \left(x , y\right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{18}\! \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 \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{14}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{27}\! \left(x , y\right) = F_{28}\! \left(x , y\right)+F_{36}\! \left(x , y\right)
F_{28}\! \left(x , y\right) = F_{29}\! \left(x , y\right)+F_{32}\! \left(x , y\right)
F_{29}\! \left(x , y\right) = F_{30}\! \left(x , y\right)
F_{30}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{31}\! \left(x , y\right)
F_{31}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{29}\! \left(x , y\right)
F_{32}\! \left(x , y\right) = F_{23}\! \left(x \right)+F_{33}\! \left(x , y\right)+F_{35}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{34}\! \left(x , y\right)
F_{34}\! \left(x , y\right) = F_{19}\! \left(x \right)+F_{32}\! \left(x , y\right)
F_{35}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{28}\! \left(x , y\right)
F_{36}\! \left(x , y\right) = F_{37}\! \left(x , y\right)+F_{40}\! \left(x , y\right)
F_{37}\! \left(x , y\right) = F_{38}\! \left(x , y\right)
F_{38}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{39}\! \left(x , y\right)
F_{39}\! \left(x , y\right) = F_{29}\! \left(x , y\right)+F_{37}\! \left(x , y\right)
F_{40}\! \left(x , y\right) = 2 F_{23}\! \left(x \right)+F_{41}\! \left(x , y\right)+F_{43}\! \left(x , y\right)
F_{41}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{42}\! \left(x , y\right)
F_{42}\! \left(x , y\right) = F_{32}\! \left(x , y\right)+F_{40}\! \left(x , y\right)
F_{43}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{36}\! \left(x , y\right)
F_{44}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{45}\! \left(x , y\right)
F_{45}\! \left(x , y\right) = -\frac{-y F_{8}\! \left(x , y\right)+F_{8}\! \left(x , 1\right)}{-1+y}
F_{46}\! \left(x \right) = F_{47}\! \left(x , 1\right)
F_{47}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{48}\! \left(x , y\right)
F_{48}\! \left(x , y\right) = F_{31}\! \left(x , y\right)+F_{49}\! \left(x , y\right)
F_{49}\! \left(x , y\right) = F_{19}\! \left(x \right)+F_{50}\! \left(x , y\right)
F_{50}\! \left(x , y\right) = F_{23}\! \left(x \right)+F_{51}\! \left(x , y\right)+F_{53}\! \left(x , y\right)
F_{51}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{52}\! \left(x , y\right)
F_{52}\! \left(x , y\right) = F_{29}\! \left(x , y\right)+F_{50}\! \left(x , y\right)
F_{53}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{49}\! \left(x , y\right)
F_{54}\! \left(x \right) = F_{14}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{68}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{67}\! \left(x \right)
F_{57}\! \left(x \right) = F_{1}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{14}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{62}\! \left(x \right) = F_{19}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{23}\! \left(x \right)+F_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = F_{14}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{14}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{14}\! \left(x \right) F_{19}\! \left(x \right)
F_{67}\! \left(x \right) = F_{19}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{85}\! \left(x \right)
F_{69}\! \left(x \right) = F_{19}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{23}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)
F_{71}\! \left(x \right) = F_{14}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{58}\! \left(x \right)+F_{70}\! \left(x \right)
F_{73}\! \left(x \right) = F_{14}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{79}\! \left(x \right)
F_{75}\! \left(x \right) = F_{19}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{14}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{14}\! \left(x \right)+F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{22}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{81}\! \left(x \right)+F_{83}\! \left(x \right)
F_{81}\! \left(x \right) = F_{14}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{63}\! \left(x \right)+F_{80}\! \left(x \right)
F_{83}\! \left(x \right) = F_{14}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{76}\! \left(x \right)+F_{80}\! \left(x \right)
F_{85}\! \left(x \right) = F_{22}\! \left(x \right)
F_{86}\! \left(x \right) = F_{14}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = F_{1}\! \left(x \right)+F_{137}\! \left(x \right)+F_{88}\! \left(x \right)+F_{89}\! \left(x \right)
F_{88}\! \left(x \right) = F_{9}\! \left(x , 1\right)
F_{89}\! \left(x \right) = F_{14}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x , 1\right)
F_{91}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{9}\! \left(x , y\right)+F_{92}\! \left(x , y\right)+F_{94}\! \left(x , y\right)+F_{96}\! \left(x , y\right)
F_{92}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{93}\! \left(x , y\right)
F_{93}\! \left(x , y\right) = -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}
F_{94}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{95}\! \left(x , y\right)
F_{95}\! \left(x , y\right) = -\frac{-y F_{91}\! \left(x , y\right)+F_{91}\! \left(x , 1\right)}{-1+y}
F_{96}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{97}\! \left(x , y\right)
F_{97}\! \left(x , y\right) = F_{16}\! \left(x , y\right)+F_{98}\! \left(x , y\right)
F_{98}\! \left(x , y\right) = F_{113}\! \left(x , y\right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{21}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{22}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{103}\! \left(x \right)+F_{105}\! \left(x \right)+F_{23}\! \left(x \right)
F_{102}\! \left(x \right) = 0
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{14}\! \left(x \right)
F_{104}\! \left(x \right) = F_{101}\! \left(x \right)+F_{22}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{14}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{110}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{14}\! \left(x \right)
F_{109}\! \left(x \right) = F_{107}\! \left(x \right)+F_{19}\! \left(x \right)
F_{110}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{105}\! \left(x \right)+F_{111}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right) F_{14}\! \left(x \right)
F_{112}\! \left(x \right) = F_{110}\! \left(x \right)+F_{22}\! \left(x \right)
F_{113}\! \left(x , y\right) = F_{114}\! \left(x , y\right)+F_{122}\! \left(x , y\right)
F_{114}\! \left(x , y\right) = F_{115}\! \left(x , y\right)+F_{118}\! \left(x , y\right)
F_{115}\! \left(x , y\right) = F_{116}\! \left(x , y\right)
F_{116}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{117}\! \left(x , y\right)
F_{117}\! \left(x , y\right) = F_{115}\! \left(x , y\right)+F_{19}\! \left(x \right)
F_{118}\! \left(x , y\right) = 2 F_{23}\! \left(x \right)+F_{119}\! \left(x , y\right)+F_{121}\! \left(x , y\right)
F_{119}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{120}\! \left(x , y\right)
F_{120}\! \left(x , y\right) = F_{118}\! \left(x , y\right)+F_{22}\! \left(x \right)
F_{121}\! \left(x , y\right) = F_{114}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{122}\! \left(x , y\right) = F_{123}\! \left(x , y\right)+F_{126}\! \left(x , y\right)
F_{123}\! \left(x , y\right) = F_{124}\! \left(x , y\right)
F_{124}\! \left(x , y\right) = F_{125}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{125}\! \left(x , y\right) = F_{115}\! \left(x , y\right)+F_{123}\! \left(x , y\right)
F_{126}\! \left(x , y\right) = 3 F_{23}\! \left(x \right)+F_{127}\! \left(x , y\right)+F_{129}\! \left(x , y\right)
F_{127}\! \left(x , y\right) = F_{128}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{128}\! \left(x , y\right) = F_{118}\! \left(x , y\right)+F_{126}\! \left(x , y\right)
F_{129}\! \left(x , y\right) = F_{130}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{130}\! \left(x , y\right) = F_{131}\! \left(x , y\right)+F_{134}\! \left(x , y\right)
F_{131}\! \left(x , y\right) = F_{132}\! \left(x , y\right)
F_{132}\! \left(x , y\right) = F_{133}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{133}\! \left(x , y\right) = F_{115}\! \left(x , y\right)+F_{131}\! \left(x , y\right)
F_{134}\! \left(x , y\right) = 3 F_{23}\! \left(x \right)+F_{129}\! \left(x , y\right)+F_{135}\! \left(x , y\right)
F_{135}\! \left(x , y\right) = F_{136}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{136}\! \left(x , y\right) = F_{118}\! \left(x , y\right)+F_{134}\! \left(x , y\right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right) F_{14}\! \left(x \right)
F_{138}\! \left(x \right) = F_{17}\! \left(x \right)+F_{99}\! \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_14(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_86(x))
Eq(F_4(x), F_14(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_46(x) + F_54(x) + F_6(x))
Eq(F_6(x), F_14(x)*F_7(x))
Eq(F_7(x), F_8(x, 1))
Eq(F_8(x, k_0), F_1(x) + F_15(x, k_0) + F_44(x, k_0) + F_46(x) + F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0)*F_11(x, k_0))
Eq(F_10(x, k_0), k_0*x)
Eq(F_11(x, k_0), F_1(x) + F_12(x, k_0) + F_15(x, k_0) + F_9(x, k_0))
Eq(F_12(x, k_0), F_13(x, k_0)*F_14(x))
Eq(F_13(x, k_0), (-k_0*F_11(x, k_0) + F_11(x, 1))/(1 - k_0))
Eq(F_14(x), F_10(x, 1))
Eq(F_15(x, k_0), F_14(x)*F_16(x, k_0))
Eq(F_16(x, k_0), F_17(x) + F_27(x, k_0))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_18(x))
Eq(F_21(x), F_19(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_26(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_14(x)*F_25(x))
Eq(F_25(x), F_19(x) + F_22(x))
Eq(F_26(x), F_14(x)*F_21(x))
Eq(F_27(x, k_0), F_28(x, k_0) + F_36(x, k_0))
Eq(F_28(x, k_0), F_29(x, k_0) + F_32(x, k_0))
Eq(F_29(x, k_0), F_30(x, k_0))
Eq(F_30(x, k_0), F_10(x, k_0)*F_31(x, k_0))
Eq(F_31(x, k_0), F_1(x) + F_29(x, k_0))
Eq(F_32(x, k_0), F_23(x) + F_33(x, k_0) + F_35(x, k_0))
Eq(F_33(x, k_0), F_10(x, k_0)*F_34(x, k_0))
Eq(F_34(x, k_0), F_19(x) + F_32(x, k_0))
Eq(F_35(x, k_0), F_14(x)*F_28(x, k_0))
Eq(F_36(x, k_0), F_37(x, k_0) + F_40(x, k_0))
Eq(F_37(x, k_0), F_38(x, k_0))
Eq(F_38(x, k_0), F_14(x)*F_39(x, k_0))
Eq(F_39(x, k_0), F_29(x, k_0) + F_37(x, k_0))
Eq(F_40(x, k_0), 2*F_23(x) + F_41(x, k_0) + F_43(x, k_0))
Eq(F_41(x, k_0), F_14(x)*F_42(x, k_0))
Eq(F_42(x, k_0), F_32(x, k_0) + F_40(x, k_0))
Eq(F_43(x, k_0), F_14(x)*F_36(x, k_0))
Eq(F_44(x, k_0), F_14(x)*F_45(x, k_0))
Eq(F_45(x, k_0), (-k_0*F_8(x, k_0) + F_8(x, 1))/(1 - k_0))
Eq(F_46(x), F_47(x, 1))
Eq(F_47(x, k_0), F_10(x, k_0)*F_48(x, k_0))
Eq(F_48(x, k_0), F_31(x, k_0) + F_49(x, k_0))
Eq(F_49(x, k_0), F_19(x) + F_50(x, k_0))
Eq(F_50(x, k_0), F_23(x) + F_51(x, k_0) + F_53(x, k_0))
Eq(F_51(x, k_0), F_14(x)*F_52(x, k_0))
Eq(F_52(x, k_0), F_29(x, k_0) + F_50(x, k_0))
Eq(F_53(x, k_0), F_10(x, k_0)*F_49(x, k_0))
Eq(F_54(x), F_14(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_68(x))
Eq(F_56(x), F_57(x) + F_67(x))
Eq(F_57(x), F_1(x) + F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_14(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_1(x) + F_14(x))
Eq(F_62(x), F_19(x) + F_63(x))
Eq(F_63(x), F_23(x) + F_64(x) + F_66(x))
Eq(F_64(x), F_14(x)*F_65(x))
Eq(F_65(x), F_14(x) + F_63(x))
Eq(F_66(x), F_14(x)*F_19(x))
Eq(F_67(x), F_19(x))
Eq(F_68(x), F_69(x) + F_85(x))
Eq(F_69(x), F_19(x) + F_70(x))
Eq(F_70(x), F_23(x) + F_71(x) + F_73(x))
Eq(F_71(x), F_14(x)*F_72(x))
Eq(F_72(x), F_58(x) + F_70(x))
Eq(F_73(x), F_14(x)*F_74(x))
Eq(F_74(x), F_75(x) + F_79(x))
Eq(F_75(x), F_19(x) + F_76(x))
Eq(F_76(x), F_77(x))
Eq(F_77(x), F_14(x)*F_78(x))
Eq(F_78(x), F_14(x) + F_76(x))
Eq(F_79(x), F_22(x) + F_80(x))
Eq(F_80(x), 2*F_23(x) + F_81(x) + F_83(x))
Eq(F_81(x), F_14(x)*F_82(x))
Eq(F_82(x), F_63(x) + F_80(x))
Eq(F_83(x), F_14(x)*F_84(x))
Eq(F_84(x), F_76(x) + F_80(x))
Eq(F_85(x), F_22(x))
Eq(F_86(x), F_14(x)*F_87(x))
Eq(F_87(x), F_1(x) + F_137(x) + F_88(x) + F_89(x))
Eq(F_88(x), F_9(x, 1))
Eq(F_89(x), F_14(x)*F_90(x))
Eq(F_90(x), F_91(x, 1))
Eq(F_91(x, k_0), F_1(x) + F_9(x, k_0) + F_92(x, k_0) + F_94(x, k_0) + F_96(x, k_0))
Eq(F_92(x, k_0), F_14(x)*F_93(x, k_0))
Eq(F_93(x, k_0), (k_0*F_11(x, k_0) - F_11(x, 1))/(k_0 - 1))
Eq(F_94(x, k_0), F_14(x)*F_95(x, k_0))
Eq(F_95(x, k_0), (-k_0*F_91(x, k_0) + F_91(x, 1))/(1 - k_0))
Eq(F_96(x, k_0), F_14(x)*F_97(x, k_0))
Eq(F_97(x, k_0), F_16(x, k_0) + F_98(x, k_0))
Eq(F_98(x, k_0), F_113(x, k_0) + F_99(x))
Eq(F_99(x), F_100(x) + F_21(x))
Eq(F_100(x), F_101(x) + F_22(x))
Eq(F_101(x), F_102(x) + F_103(x) + F_105(x) + F_23(x))
Eq(F_102(x), 0)
Eq(F_103(x), F_104(x)*F_14(x))
Eq(F_104(x), F_101(x) + F_22(x))
Eq(F_105(x), F_106(x)*F_14(x))
Eq(F_106(x), F_107(x) + F_110(x))
Eq(F_107(x), F_108(x))
Eq(F_108(x), F_109(x)*F_14(x))
Eq(F_109(x), F_107(x) + F_19(x))
Eq(F_110(x), F_105(x) + F_111(x) + 2*F_23(x))
Eq(F_111(x), F_112(x)*F_14(x))
Eq(F_112(x), F_110(x) + F_22(x))
Eq(F_113(x, k_0), F_114(x, k_0) + F_122(x, k_0))
Eq(F_114(x, k_0), F_115(x, k_0) + F_118(x, k_0))
Eq(F_115(x, k_0), F_116(x, k_0))
Eq(F_116(x, k_0), F_10(x, k_0)*F_117(x, k_0))
Eq(F_117(x, k_0), F_115(x, k_0) + F_19(x))
Eq(F_118(x, k_0), F_119(x, k_0) + F_121(x, k_0) + 2*F_23(x))
Eq(F_119(x, k_0), F_10(x, k_0)*F_120(x, k_0))
Eq(F_120(x, k_0), F_118(x, k_0) + F_22(x))
Eq(F_121(x, k_0), F_114(x, k_0)*F_14(x))
Eq(F_122(x, k_0), F_123(x, k_0) + F_126(x, k_0))
Eq(F_123(x, k_0), F_124(x, k_0))
Eq(F_124(x, k_0), F_125(x, k_0)*F_14(x))
Eq(F_125(x, k_0), F_115(x, k_0) + F_123(x, k_0))
Eq(F_126(x, k_0), F_127(x, k_0) + F_129(x, k_0) + 3*F_23(x))
Eq(F_127(x, k_0), F_128(x, k_0)*F_14(x))
Eq(F_128(x, k_0), F_118(x, k_0) + F_126(x, k_0))
Eq(F_129(x, k_0), F_130(x, k_0)*F_14(x))
Eq(F_130(x, k_0), F_131(x, k_0) + F_134(x, k_0))
Eq(F_131(x, k_0), F_132(x, k_0))
Eq(F_132(x, k_0), F_133(x, k_0)*F_14(x))
Eq(F_133(x, k_0), F_115(x, k_0) + F_131(x, k_0))
Eq(F_134(x, k_0), F_129(x, k_0) + F_135(x, k_0) + 3*F_23(x))
Eq(F_135(x, k_0), F_136(x, k_0)*F_14(x))
Eq(F_136(x, k_0), F_118(x, k_0) + F_134(x, k_0))
Eq(F_137(x), F_138(x)*F_14(x))
Eq(F_138(x), F_17(x) + F_99(x))
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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