0123_0213_0231_0321_1230
Counting sequence:
1, 1, 2, 6, 19, 58, 170, 483, 1342, 3670, 9923, 26610, 70930, 188227, 497846, 1313502, 3459043, 9096394, 23895674, 62721699, 164531566, 431397286, 1130708867, 2962826466, 7761964834, 20331456643, 53249182310, 139449644718, 365166860707, 956185155130, 2503657040138, 6555322836195, 17163385210270, 44936980278262, 117651850591811, 308027161431762, 806446813572658, 2111347639024579, 5527664822977814, 14471784268862334, 37887962861516131, 99192654071499946, 259691098864611482, 679882841545590051, 1779961823818669774, 4660011426003441478, 12200090046377699075, 31940293897501744578, 83620862014871712322, 218922432884601747715, 573146718113910241478, 1500518284407082398030, 3928409261007243795235, 10284711750414462672922, 26925730493835771594026, 70492488738292106850147, 184551753735439058438398, 483162808496822087429014, 1264936743812621241776579, 3311647567056229713756594, 8670006245586444051204946, 22698371746163854743281731, 59425110145826624785487222, 155576960997159028826873886, 407305777457336480122522339, 1066340380598222448395468938, 2791715382784074938773436090, 7308805804647490515343942563, 19134702104945372902096598062, 50095300657762580780622264550, 131151200163490274619123021443, 343358300423004053435452451490, 898923702286113506404645636450, 2353412808796519707213307064707, 6161314728825812098104920771366, 16130531387125649552840745676782, 42230279451440602491895897113763, 110560307004975089785804107374074, 289450641639042530591430748427594, 757791618063268229440316784746979, 1983924212852993612633176899489886, 5193981021100175518266528501075766, 13598018851656458761781037778443587, 35600075536287052406305843183667346, 93202207762040401735595008471383154, 244006547759505559357396215628131523, 638817435535819089450427705208310230, 1672445758886637335221555033587396798, 4378519841201464168669573662735075427, 11463113764872497675697838488980220010, 30010821453725513868245286872930365658, 78569350596923013948680712267260439075, 205697230338281468017082230203750075790, 538522340420397270181136738893788036742, 1409869790927862102683469507577210531331, 3691087032373092558183554826037036551042, 9663391306211222612495761054932285109378, 25299086886300189360560860507556590752259, 66233869352768573631701084805331031097734, 173402521172163987859570922583623590441614, 453973694164040302597068740295913916028451
Generating function in Maple syntax:
-(3*x^3-8*x^2+5*x-1)/(x-1)/(2*x-1)/(x^2-3*x+1)
Generating function in latex syntax:
-\frac{3 x^{3}-8 x^{2}+5 x -1}{\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(-3*x**3 + 8*x**2 - 5*x + 1)/((x - 1)*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x-1)*(x^2-3*x+1)*F(x)+3*x^3-8*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+3 x^{3}-8 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
1/10*(5+5^(1/2))*(3/2-1/2*5^(1/2))^(-n)+1/10*(5-5^(1/2))*(3/2+1/2*5^(1/2))^(-n)-2^n+1
Explicit closed form in latex syntax:
\frac{\left(5+\sqrt{5}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(5-\sqrt{5}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}-2^{n}+1
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)+1, n >= 4
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(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+1, \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16688/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[17,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[172,x]+F[19,x]+F[20,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[35,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[19,x]+F[24,x]+F[30,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[11,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[4,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[14,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[4,x]
F[35,x] = F[18,x]+F[36,x]
F[36,x] = F[113,x]+F[19,x]+F[37,x]+F[95,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[46,x]
F[39,x] = F[23,x]+F[40,x]
F[40,x] = F[19,x]+F[41,x]+F[43,x]+F[44,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[27,x]
F[43,x] = 0
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[32,x]
F[46,x] = F[36,x]+F[47,x]
F[47,x] = F[19,x]+F[48,x]+F[52,x]+F[69,x]+F[70,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[40,x]
F[51,x] = F[47,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[61,x]
F[57,x] = F[27,x]+F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[27,x]
F[61,x] = F[54,x]+F[62,x]
F[62,x] = 3*F[19,x]+F[63,x]+F[67,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[58,x]
F[66,x] = F[62,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[54,x]
F[69,x] = 0
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[72,x]
F[72,x] = F[73,x]
F[73,x] = 2*F[19,x]+F[74,x]+F[78,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[76,x]+F[77,x]
F[76,x] = F[33,x]
F[77,x] = F[73,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[80,x]
F[80,x] = F[81,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[83,x]+F[87,x]
F[83,x] = F[4,x]+F[84,x]
F[84,x] = F[85,x]
F[85,x] = F[4,x]*F[86,x]
F[86,x] = F[4,x]
F[87,x] = F[80,x]+F[88,x]
F[88,x] = 2*F[19,x]+F[89,x]+F[93,x]
F[89,x] = F[4,x]*F[90,x]
F[90,x] = F[91,x]+F[92,x]
F[91,x] = F[84,x]
F[92,x] = F[88,x]
F[93,x] = F[4,x]*F[94,x]
F[94,x] = F[80,x]
F[95,x] = F[4,x]*F[96,x]
F[96,x] = F[97,x]
F[97,x] = F[54,x]+F[98,x]
F[98,x] = F[99,x]
F[99,x] = F[100,x]*F[4,x]
F[100,x] = F[101,x]+F[105,x]
F[101,x] = F[102,x]+F[11,x]
F[102,x] = F[103,x]
F[103,x] = F[104,x]*F[4,x]
F[104,x] = F[11,x]
F[105,x] = F[106,x]+F[98,x]
F[106,x] = 2*F[19,x]+F[107,x]+F[111,x]
F[107,x] = F[108,x]*F[4,x]
F[108,x] = F[109,x]+F[110,x]
F[109,x] = F[102,x]
F[110,x] = F[106,x]
F[111,x] = F[112,x]*F[4,x]
F[112,x] = F[98,x]
F[113,x] = F[114,x]*F[4,x]
F[114,x] = F[115,x]+F[72,x]
F[115,x] = F[116,x]
F[116,x] = F[117,x]+F[170,x]+F[19,x]
F[117,x] = F[118,x]*F[4,x]
F[118,x] = F[119,x]+F[128,x]
F[119,x] = F[120,x]+F[15,x]
F[120,x] = F[121,x]+F[126,x]+F[19,x]
F[121,x] = F[122,x]*F[4,x]
F[122,x] = F[123,x]
F[123,x] = F[124,x]+F[4,x]
F[124,x] = F[125,x]
F[125,x] = x^2
F[126,x] = F[127,x]*F[4,x]
F[127,x] = F[15,x]
F[128,x] = F[116,x]+F[129,x]
F[129,x] = F[130,x]+F[165,x]+F[168,x]+F[19,x]
F[130,x] = F[131,x]*F[4,x]
F[131,x] = F[132,x]+F[139,x]
F[132,x] = F[120,x]+F[133,x]
F[133,x] = F[134,x]+F[136,x]+F[137,x]+F[19,x]
F[134,x] = F[135,x]*F[4,x]
F[135,x] = F[124,x]
F[136,x] = 0
F[137,x] = F[138,x]*F[4,x]
F[138,x] = F[33,x]
F[139,x] = F[129,x]+F[140,x]
F[140,x] = F[141,x]+F[145,x]+F[162,x]+F[163,x]+F[19,x]
F[141,x] = F[142,x]*F[4,x]
F[142,x] = F[143,x]+F[144,x]
F[143,x] = F[133,x]
F[144,x] = F[140,x]
F[145,x] = F[146,x]*F[4,x]
F[146,x] = F[147,x]
F[147,x] = F[148,x]
F[148,x] = F[149,x]*F[4,x]
F[149,x] = F[150,x]+F[154,x]
F[150,x] = F[124,x]+F[151,x]
F[151,x] = F[152,x]
F[152,x] = F[153,x]*F[4,x]
F[153,x] = F[124,x]
F[154,x] = F[147,x]+F[155,x]
F[155,x] = 3*F[19,x]+F[156,x]+F[160,x]
F[156,x] = F[157,x]*F[4,x]
F[157,x] = F[158,x]+F[159,x]
F[158,x] = F[151,x]
F[159,x] = F[155,x]
F[160,x] = F[161,x]*F[4,x]
F[161,x] = F[147,x]
F[162,x] = 0
F[163,x] = F[164,x]*F[4,x]
F[164,x] = F[73,x]
F[165,x] = F[166,x]*F[4,x]
F[166,x] = F[167,x]
F[167,x] = F[147,x]+F[80,x]
F[168,x] = F[169,x]*F[4,x]
F[169,x] = F[116,x]
F[170,x] = F[171,x]*F[4,x]
F[171,x] = F[2,x]+F[80,x]
F[172,x] = F[173,x]*F[4,x]
F[173,x] = F[115,x]+F[174,x]
F[174,x] = F[2,x]+F[98,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{17}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{172}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{35}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{30}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{18}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{113}\! \left(x \right)+F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{95}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{46}\! \left(x \right)
F_{39}\! \left(x \right) = F_{23}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{19}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{27}\! \left(x \right)
F_{43}\! \left(x \right) = 0
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{32}\! \left(x \right)
F_{46}\! \left(x \right) = F_{36}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{52}\! \left(x \right)+F_{69}\! \left(x \right)+F_{70}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{40}\! \left(x \right)
F_{51}\! \left(x \right) = F_{47}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{57}\! \left(x \right) = F_{27}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{27}\! \left(x \right)
F_{61}\! \left(x \right) = F_{54}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = 3 F_{19}\! \left(x \right)+F_{63}\! \left(x \right)+F_{67}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{58}\! \left(x \right)
F_{66}\! \left(x \right) = F_{62}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{54}\! \left(x \right)
F_{69}\! \left(x \right) = 0
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{74}\! \left(x \right)+F_{78}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{76}\! \left(x \right) = F_{33}\! \left(x \right)
F_{77}\! \left(x \right) = F_{73}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{87}\! \left(x \right)
F_{83}\! \left(x \right) = F_{4}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right)
F_{87}\! \left(x \right) = F_{80}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{89}\! \left(x \right)+F_{93}\! \left(x \right)
F_{89}\! \left(x \right) = F_{4}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)+F_{92}\! \left(x \right)
F_{91}\! \left(x \right) = F_{84}\! \left(x \right)
F_{92}\! \left(x \right) = F_{88}\! \left(x \right)
F_{93}\! \left(x \right) = F_{4}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{80}\! \left(x \right)
F_{95}\! \left(x \right) = F_{4}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{54}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right) F_{4}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{105}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{11}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{4}\! \left(x \right)
F_{104}\! \left(x \right) = F_{11}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{98}\! \left(x \right)
F_{106}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{107}\! \left(x \right)+F_{111}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{4}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{110}\! \left(x \right)
F_{109}\! \left(x \right) = F_{102}\! \left(x \right)
F_{110}\! \left(x \right) = F_{106}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right) F_{4}\! \left(x \right)
F_{112}\! \left(x \right) = F_{98}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right) F_{4}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{72}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)+F_{170}\! \left(x \right)+F_{19}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right) F_{4}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right)+F_{128}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)+F_{15}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)+F_{126}\! \left(x \right)+F_{19}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right) F_{4}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)+F_{4}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)
F_{125}\! \left(x \right) = x^{2}
F_{126}\! \left(x \right) = F_{127}\! \left(x \right) F_{4}\! \left(x \right)
F_{127}\! \left(x \right) = F_{15}\! \left(x \right)
F_{128}\! \left(x \right) = F_{116}\! \left(x \right)+F_{129}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)+F_{165}\! \left(x \right)+F_{168}\! \left(x \right)+F_{19}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right) F_{4}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)+F_{139}\! \left(x \right)
F_{132}\! \left(x \right) = F_{120}\! \left(x \right)+F_{133}\! \left(x \right)
F_{133}\! \left(x \right) = F_{134}\! \left(x \right)+F_{136}\! \left(x \right)+F_{137}\! \left(x \right)+F_{19}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right) F_{4}\! \left(x \right)
F_{135}\! \left(x \right) = F_{124}\! \left(x \right)
F_{136}\! \left(x \right) = 0
F_{137}\! \left(x \right) = F_{138}\! \left(x \right) F_{4}\! \left(x \right)
F_{138}\! \left(x \right) = F_{33}\! \left(x \right)
F_{139}\! \left(x \right) = F_{129}\! \left(x \right)+F_{140}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right)+F_{145}\! \left(x \right)+F_{162}\! \left(x \right)+F_{163}\! \left(x \right)+F_{19}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right) F_{4}\! \left(x \right)
F_{142}\! \left(x \right) = F_{143}\! \left(x \right)+F_{144}\! \left(x \right)
F_{143}\! \left(x \right) = F_{133}\! \left(x \right)
F_{144}\! \left(x \right) = F_{140}\! \left(x \right)
F_{145}\! \left(x \right) = F_{146}\! \left(x \right) F_{4}\! \left(x \right)
F_{146}\! \left(x \right) = F_{147}\! \left(x \right)
F_{147}\! \left(x \right) = F_{148}\! \left(x \right)
F_{148}\! \left(x \right) = F_{149}\! \left(x \right) F_{4}\! \left(x \right)
F_{149}\! \left(x \right) = F_{150}\! \left(x \right)+F_{154}\! \left(x \right)
F_{150}\! \left(x \right) = F_{124}\! \left(x \right)+F_{151}\! \left(x \right)
F_{151}\! \left(x \right) = F_{152}\! \left(x \right)
F_{152}\! \left(x \right) = F_{153}\! \left(x \right) F_{4}\! \left(x \right)
F_{153}\! \left(x \right) = F_{124}\! \left(x \right)
F_{154}\! \left(x \right) = F_{147}\! \left(x \right)+F_{155}\! \left(x \right)
F_{155}\! \left(x \right) = 3 F_{19}\! \left(x \right)+F_{156}\! \left(x \right)+F_{160}\! \left(x \right)
F_{156}\! \left(x \right) = F_{157}\! \left(x \right) F_{4}\! \left(x \right)
F_{157}\! \left(x \right) = F_{158}\! \left(x \right)+F_{159}\! \left(x \right)
F_{158}\! \left(x \right) = F_{151}\! \left(x \right)
F_{159}\! \left(x \right) = F_{155}\! \left(x \right)
F_{160}\! \left(x \right) = F_{161}\! \left(x \right) F_{4}\! \left(x \right)
F_{161}\! \left(x \right) = F_{147}\! \left(x \right)
F_{162}\! \left(x \right) = 0
F_{163}\! \left(x \right) = F_{164}\! \left(x \right) F_{4}\! \left(x \right)
F_{164}\! \left(x \right) = F_{73}\! \left(x \right)
F_{165}\! \left(x \right) = F_{166}\! \left(x \right) F_{4}\! \left(x \right)
F_{166}\! \left(x \right) = F_{167}\! \left(x \right)
F_{167}\! \left(x \right) = F_{147}\! \left(x \right)+F_{80}\! \left(x \right)
F_{168}\! \left(x \right) = F_{169}\! \left(x \right) F_{4}\! \left(x \right)
F_{169}\! \left(x \right) = F_{116}\! \left(x \right)
F_{170}\! \left(x \right) = F_{171}\! \left(x \right) F_{4}\! \left(x \right)
F_{171}\! \left(x \right) = F_{2}\! \left(x \right)+F_{80}\! \left(x \right)
F_{172}\! \left(x \right) = F_{173}\! \left(x \right) F_{4}\! \left(x \right)
F_{173}\! \left(x \right) = F_{115}\! \left(x \right)+F_{174}\! \left(x \right)
F_{174}\! \left(x \right) = F_{2}\! \left(x \right)+F_{98}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_17(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_172(x) + F_19(x) + F_20(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_35(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_19(x) + F_24(x) + F_30(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_11(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_4(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_14(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_4(x))
Eq(F_35(x), F_18(x) + F_36(x))
Eq(F_36(x), F_113(x) + F_19(x) + F_37(x) + F_95(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_46(x))
Eq(F_39(x), F_23(x) + F_40(x))
Eq(F_40(x), F_19(x) + F_41(x) + F_43(x) + F_44(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_27(x))
Eq(F_43(x), 0)
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_32(x))
Eq(F_46(x), F_36(x) + F_47(x))
Eq(F_47(x), F_19(x) + F_48(x) + F_52(x) + F_69(x) + F_70(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_40(x))
Eq(F_51(x), F_47(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_61(x))
Eq(F_57(x), F_27(x) + F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_27(x))
Eq(F_61(x), F_54(x) + F_62(x))
Eq(F_62(x), 3*F_19(x) + F_63(x) + F_67(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_58(x))
Eq(F_66(x), F_62(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_54(x))
Eq(F_69(x), 0)
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), 2*F_19(x) + F_74(x) + F_78(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_77(x))
Eq(F_76(x), F_33(x))
Eq(F_77(x), F_73(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_87(x))
Eq(F_83(x), F_4(x) + F_84(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), F_4(x)*F_86(x))
Eq(F_86(x), F_4(x))
Eq(F_87(x), F_80(x) + F_88(x))
Eq(F_88(x), 2*F_19(x) + F_89(x) + F_93(x))
Eq(F_89(x), F_4(x)*F_90(x))
Eq(F_90(x), F_91(x) + F_92(x))
Eq(F_91(x), F_84(x))
Eq(F_92(x), F_88(x))
Eq(F_93(x), F_4(x)*F_94(x))
Eq(F_94(x), F_80(x))
Eq(F_95(x), F_4(x)*F_96(x))
Eq(F_96(x), F_97(x))
Eq(F_97(x), F_54(x) + F_98(x))
Eq(F_98(x), F_99(x))
Eq(F_99(x), F_100(x)*F_4(x))
Eq(F_100(x), F_101(x) + F_105(x))
Eq(F_101(x), F_102(x) + F_11(x))
Eq(F_102(x), F_103(x))
Eq(F_103(x), F_104(x)*F_4(x))
Eq(F_104(x), F_11(x))
Eq(F_105(x), F_106(x) + F_98(x))
Eq(F_106(x), F_107(x) + F_111(x) + 2*F_19(x))
Eq(F_107(x), F_108(x)*F_4(x))
Eq(F_108(x), F_109(x) + F_110(x))
Eq(F_109(x), F_102(x))
Eq(F_110(x), F_106(x))
Eq(F_111(x), F_112(x)*F_4(x))
Eq(F_112(x), F_98(x))
Eq(F_113(x), F_114(x)*F_4(x))
Eq(F_114(x), F_115(x) + F_72(x))
Eq(F_115(x), F_116(x))
Eq(F_116(x), F_117(x) + F_170(x) + F_19(x))
Eq(F_117(x), F_118(x)*F_4(x))
Eq(F_118(x), F_119(x) + F_128(x))
Eq(F_119(x), F_120(x) + F_15(x))
Eq(F_120(x), F_121(x) + F_126(x) + F_19(x))
Eq(F_121(x), F_122(x)*F_4(x))
Eq(F_122(x), F_123(x))
Eq(F_123(x), F_124(x) + F_4(x))
Eq(F_124(x), F_125(x))
Eq(F_125(x), x**2)
Eq(F_126(x), F_127(x)*F_4(x))
Eq(F_127(x), F_15(x))
Eq(F_128(x), F_116(x) + F_129(x))
Eq(F_129(x), F_130(x) + F_165(x) + F_168(x) + F_19(x))
Eq(F_130(x), F_131(x)*F_4(x))
Eq(F_131(x), F_132(x) + F_139(x))
Eq(F_132(x), F_120(x) + F_133(x))
Eq(F_133(x), F_134(x) + F_136(x) + F_137(x) + F_19(x))
Eq(F_134(x), F_135(x)*F_4(x))
Eq(F_135(x), F_124(x))
Eq(F_136(x), 0)
Eq(F_137(x), F_138(x)*F_4(x))
Eq(F_138(x), F_33(x))
Eq(F_139(x), F_129(x) + F_140(x))
Eq(F_140(x), F_141(x) + F_145(x) + F_162(x) + F_163(x) + F_19(x))
Eq(F_141(x), F_142(x)*F_4(x))
Eq(F_142(x), F_143(x) + F_144(x))
Eq(F_143(x), F_133(x))
Eq(F_144(x), F_140(x))
Eq(F_145(x), F_146(x)*F_4(x))
Eq(F_146(x), F_147(x))
Eq(F_147(x), F_148(x))
Eq(F_148(x), F_149(x)*F_4(x))
Eq(F_149(x), F_150(x) + F_154(x))
Eq(F_150(x), F_124(x) + F_151(x))
Eq(F_151(x), F_152(x))
Eq(F_152(x), F_153(x)*F_4(x))
Eq(F_153(x), F_124(x))
Eq(F_154(x), F_147(x) + F_155(x))
Eq(F_155(x), F_156(x) + F_160(x) + 3*F_19(x))
Eq(F_156(x), F_157(x)*F_4(x))
Eq(F_157(x), F_158(x) + F_159(x))
Eq(F_158(x), F_151(x))
Eq(F_159(x), F_155(x))
Eq(F_160(x), F_161(x)*F_4(x))
Eq(F_161(x), F_147(x))
Eq(F_162(x), 0)
Eq(F_163(x), F_164(x)*F_4(x))
Eq(F_164(x), F_73(x))
Eq(F_165(x), F_166(x)*F_4(x))
Eq(F_166(x), F_167(x))
Eq(F_167(x), F_147(x) + F_80(x))
Eq(F_168(x), F_169(x)*F_4(x))
Eq(F_169(x), F_116(x))
Eq(F_170(x), F_171(x)*F_4(x))
Eq(F_171(x), F_2(x) + F_80(x))
Eq(F_172(x), F_173(x)*F_4(x))
Eq(F_173(x), F_115(x) + F_174(x))
Eq(F_174(x), F_2(x) + F_98(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 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, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": 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