0123_0132_0213_2301
Counting sequence:
1, 1, 2, 6, 20, 63, 180, 468, 1126, 2553, 5542, 11666, 24048, 48899, 98592, 197808, 395826, 791101, 1580418, 3157198, 6308108, 12606279, 25197740, 50374284, 100719198, 201398721, 402744958, 805421706, 1610756104, 3221401931, 6442666200, 12885162344, 25770116586, 51539980677, 103079657370, 206158951366, 412317471204, 824634433039, 1649268268196, 3298535838276, 6597070865366, 13194140792457, 26388280504278, 52776559768962, 105553118121376, 211106234629779, 422212467429136, 844424932787744, 1688849863240482, 3377699723855309, 6755399444766258, 13510798886239422, 27021597768804924, 54043195533520855, 108086391062501148, 216172782119971324, 432345564234379982, 864691128462621777, 1729382256918483374, 3458764513829535354, 6917529027650916024, 13835058055292899035, 27670116110576028616, 55340232221141390040, 110680464442271150554, 221360928884529641237, 442721857769045520714, 885443715538076102582, 1770887431076136010260, 3541774862152254486687, 7083549724304490013716, 14167099448608959550900, 28334198897217897013062, 56668397794435770225433, 113336795588871514833926, 226673591177743002125682, 453347182355485974670160, 906694364710971917601315, 1813388729421943801181952, 3626777458843887565932432, 7253554917687775092888082, 14507109835375550144114013, 29014219670751100243734754, 58028439341502200439993518, 116056878683004400829370732, 232113757366008801604821095, 464227514732017603152247692, 928455029464035206243450220, 1856910058928070412422021438, 3713820117856140824775140065, 7427640235712281649477156574, 14855280471424563298876764778, 29710560942849126597671345000, 59421121885698253195255650411, 118842243771396506390419179704, 237684487542793012780740922440, 475368975085586025561378849738, 950737950171172051122648895653, 1901475900342344102245182919930, 3802951800684688204490244633510, 7605903601369376408980361449540
Generating function in Maple syntax:
(2*x^7-3*x^6+16*x^5-31*x^4+33*x^3-21*x^2+7*x-1)/(2*x-1)/(-1+x)^6
Generating function in latex syntax:
\frac{2 x^{7}-3 x^{6}+16 x^{5}-31 x^{4}+33 x^{3}-21 x^{2}+7 x -1}{\left(2 x -1\right) \left(-1+x \right)^{6}}
Generating function in sympy syntax:
(2*x**7 - 3*x**6 + 16*x**5 - 31*x**4 + 33*x**3 - 21*x**2 + 7*x - 1)/((x - 1)**6*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(-1+x)^6*F(x)-2*x^7+3*x^6-16*x^5+31*x^4-33*x^3+21*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(-1+x \right)^{6} F \! \left(x \right)-2 x^{7}+3 x^{6}-16 x^{5}+31 x^{4}-33 x^{3}+21 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-6+6*2^n+1/60*n^5-1/4*n^4+7/12*n^3-13/4*n^2-21/10*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -6+6 \,2^{n}+\frac{n^{5}}{60}-\frac{n^{4}}{4}+\frac{7 n^{3}}{12}-\frac{13 n^{2}}{4}-\frac{21 n}{10} & \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) = 63
a(6) = 180
a(7) = 468
a(n+1) = 2*a(n)-1/60*(n-1)*(n^4-19*n^3+66*n^2-154*n+60), n >= 8
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) = 63
a \! \left(6\right) = 180
a \! \left(7\right) = 468
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{4}-19 n^{3}+66 n^{2}-154 n +60\right)}{60}, \quad n \geq 8
Specification 1
Strategy pack name: insertion_point_placements
Tree: http://permpal.com/tree/21126/
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[8,x]
F[4,x] = F[17,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[11,x]
F[10,x] = F[1,x]+F[8,x]
F[11,x] = F[12,x]+F[6,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[8,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[8,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[8,x]
F[20,x] = F[21,x]+F[62,x]
F[21,x] = F[22,x]+F[43,x]
F[22,x] = F[23,x]+F[6,x]
F[23,x] = F[24,x]+F[6,x]
F[24,x] = F[25,x]+F[26,x]+F[30,x]
F[25,x] = 0
F[26,x] = F[27,x]*F[8,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[8,x]
F[29,x] = F[12,x]
F[30,x] = F[31,x]*F[8,x]
F[31,x] = F[32,x]+F[34,x]
F[32,x] = F[14,x]+F[33,x]
F[33,x] = F[13,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[25,x]+F[30,x]+F[36,x]
F[36,x] = F[37,x]*F[8,x]
F[37,x] = F[33,x]+F[8,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]*F[8,x]
F[40,x] = F[41,x]
F[41,x] = F[42,x]*F[8,x]
F[42,x] = F[14,x]+F[40,x]
F[43,x] = F[25,x]+F[44,x]+F[49,x]
F[44,x] = F[45,x]*F[8,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[8,x]
F[47,x] = F[48,x]
F[48,x] = F[14,x]*F[8,x]
F[49,x] = F[50,x]*F[8,x]
F[50,x] = F[51,x]+F[53,x]
F[51,x] = F[14,x]+F[52,x]
F[52,x] = F[48,x]
F[53,x] = F[54,x]+F[57,x]
F[54,x] = F[25,x]+F[49,x]+F[55,x]
F[55,x] = F[56,x]*F[8,x]
F[56,x] = F[52,x]+F[8,x]
F[57,x] = F[58,x]
F[58,x] = F[59,x]*F[8,x]
F[59,x] = F[60,x]
F[60,x] = F[61,x]*F[8,x]
F[61,x] = F[14,x]+F[59,x]
F[62,x] = F[119,x]+F[63,x]
F[63,x] = F[64,x]+F[69,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[14,x]*F[6,x]
F[66,x] = F[67,x]*F[8,x]
F[67,x] = F[68,x]
F[68,x] = F[16,x]^2*F[14,x]*F[8,x]
F[69,x] = F[18,x]+F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[72,x]*F[8,x]
F[72,x] = F[114,x]+F[73,x]
F[73,x] = F[113,x]+F[74,x]
F[74,x] = F[75,x]+F[96,x]
F[75,x] = F[25,x]+F[76,x]+F[91,x]
F[76,x] = F[77,x]*F[8,x]
F[77,x] = F[78,x]+F[82,x]
F[78,x] = F[14,x]+F[79,x]
F[79,x] = F[80,x]
F[80,x] = F[8,x]*F[81,x]
F[81,x] = F[79,x]+F[8,x]
F[82,x] = F[75,x]+F[83,x]
F[83,x] = 2*F[25,x]+F[84,x]+F[88,x]
F[84,x] = F[8,x]*F[85,x]
F[85,x] = F[86,x]+F[87,x]
F[86,x] = F[79,x]
F[87,x] = F[83,x]
F[88,x] = F[8,x]*F[89,x]
F[89,x] = F[52,x]+F[90,x]
F[90,x] = F[88,x]
F[91,x] = F[8,x]*F[92,x]
F[92,x] = F[14,x]+F[93,x]
F[93,x] = F[25,x]+F[91,x]+F[94,x]
F[94,x] = F[8,x]*F[95,x]
F[95,x] = F[14,x]+F[93,x]
F[96,x] = F[101,x]+F[97,x]
F[97,x] = F[25,x]+F[48,x]+F[98,x]
F[98,x] = F[8,x]*F[99,x]
F[99,x] = F[100,x]+F[46,x]
F[100,x] = F[97,x]
F[101,x] = 2*F[25,x]+F[102,x]+F[105,x]
F[102,x] = F[103,x]*F[8,x]
F[103,x] = F[104,x]+F[86,x]
F[104,x] = F[101,x]
F[105,x] = F[106,x]*F[8,x]
F[106,x] = F[107,x]+F[110,x]
F[107,x] = F[108,x]+F[25,x]+F[48,x]
F[108,x] = F[109,x]*F[8,x]
F[109,x] = F[107,x]+F[8,x]
F[110,x] = 2*F[25,x]+F[105,x]+F[111,x]
F[111,x] = F[112,x]*F[8,x]
F[112,x] = F[110,x]+F[79,x]
F[113,x] = F[8,x]*F[93,x]
F[114,x] = F[115,x]+F[118,x]
F[115,x] = F[116,x]+F[117,x]
F[116,x] = F[6,x]*F[93,x]
F[117,x] = F[106,x]*F[14,x]
F[118,x] = F[14,x]*F[8,x]*F[93,x]
F[119,x] = F[120,x]
F[120,x] = F[121,x]*F[8,x]
F[121,x] = F[122,x]+F[147,x]
F[122,x] = F[123,x]+F[146,x]
F[123,x] = F[124,x]+F[145,x]
F[124,x] = F[125,x]+F[140,x]+F[25,x]
F[125,x] = F[126,x]*F[8,x]
F[126,x] = F[127,x]+F[131,x]
F[127,x] = F[128,x]+F[14,x]
F[128,x] = F[129,x]
F[129,x] = F[130,x]*F[8,x]
F[130,x] = F[128,x]+F[8,x]
F[131,x] = F[124,x]+F[132,x]
F[132,x] = 2*F[25,x]+F[133,x]+F[137,x]
F[133,x] = F[134,x]*F[8,x]
F[134,x] = F[135,x]+F[136,x]
F[135,x] = F[128,x]
F[136,x] = F[132,x]
F[137,x] = F[138,x]*F[8,x]
F[138,x] = F[139,x]+F[33,x]
F[139,x] = F[137,x]
F[140,x] = F[141,x]*F[8,x]
F[141,x] = F[14,x]+F[142,x]
F[142,x] = F[140,x]+F[143,x]+F[25,x]
F[143,x] = F[144,x]*F[8,x]
F[144,x] = F[14,x]+F[142,x]
F[145,x] = F[142,x]*F[6,x]
F[146,x] = F[142,x]*F[16,x]*F[8,x]
F[147,x] = F[148,x]+F[157,x]
F[148,x] = F[149,x]+F[153,x]
F[149,x] = F[13,x]+F[150,x]+F[25,x]
F[150,x] = F[151,x]*F[8,x]
F[151,x] = F[152,x]+F[28,x]
F[152,x] = F[149,x]
F[153,x] = F[14,x]*F[154,x]
F[154,x] = F[13,x]+F[155,x]+F[25,x]
F[155,x] = F[156,x]*F[8,x]
F[156,x] = F[154,x]+F[8,x]
F[157,x] = F[158,x]+F[167,x]
F[158,x] = 2*F[25,x]+F[159,x]+F[162,x]
F[159,x] = F[160,x]*F[8,x]
F[160,x] = F[135,x]+F[161,x]
F[161,x] = F[158,x]
F[162,x] = F[163,x]*F[8,x]
F[163,x] = F[154,x]+F[164,x]
F[164,x] = 2*F[25,x]+F[162,x]+F[165,x]
F[165,x] = F[166,x]*F[8,x]
F[166,x] = F[128,x]+F[164,x]
F[167,x] = F[14,x]*F[164,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{17}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{6}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{62}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{43}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{6}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{6}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{8}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{8}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{8}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{8}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right)+F_{8}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right) F_{8}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right) F_{8}\! \left(x \right)
F_{42}\! \left(x \right) = F_{14}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{25}\! \left(x \right)+F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right) F_{8}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{8}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right) F_{8}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{14}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{48}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{57}\! \left(x \right)
F_{54}\! \left(x \right) = F_{25}\! \left(x \right)+F_{49}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right) F_{8}\! \left(x \right)
F_{56}\! \left(x \right) = F_{52}\! \left(x \right)+F_{8}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right) F_{8}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right) F_{8}\! \left(x \right)
F_{61}\! \left(x \right) = F_{14}\! \left(x \right)+F_{59}\! \left(x \right)
F_{62}\! \left(x \right) = F_{119}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{69}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{14}\! \left(x \right) F_{6}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right) F_{8}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{16} \left(x \right)^{2} F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{69}\! \left(x \right) = F_{18}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right) F_{8}\! \left(x \right)
F_{72}\! \left(x \right) = F_{114}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{113}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{96}\! \left(x \right)
F_{75}\! \left(x \right) = F_{25}\! \left(x \right)+F_{76}\! \left(x \right)+F_{91}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right) F_{8}\! \left(x \right)
F_{77}\! \left(x \right) = F_{78}\! \left(x \right)+F_{82}\! \left(x \right)
F_{78}\! \left(x \right) = F_{14}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{8}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{79}\! \left(x \right)+F_{8}\! \left(x \right)
F_{82}\! \left(x \right) = F_{75}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{84}\! \left(x \right)+F_{88}\! \left(x \right)
F_{84}\! \left(x \right) = F_{8}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)+F_{87}\! \left(x \right)
F_{86}\! \left(x \right) = F_{79}\! \left(x \right)
F_{87}\! \left(x \right) = F_{83}\! \left(x \right)
F_{88}\! \left(x \right) = F_{8}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{52}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{88}\! \left(x \right)
F_{91}\! \left(x \right) = F_{8}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{14}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{25}\! \left(x \right)+F_{91}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{8}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{14}\! \left(x \right)+F_{93}\! \left(x \right)
F_{96}\! \left(x \right) = F_{101}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{25}\! \left(x \right)+F_{48}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{8}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{46}\! \left(x \right)
F_{100}\! \left(x \right) = F_{97}\! \left(x \right)
F_{101}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{102}\! \left(x \right)+F_{105}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right) F_{8}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{86}\! \left(x \right)
F_{104}\! \left(x \right) = F_{101}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{8}\! \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_{25}\! \left(x \right)+F_{48}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{8}\! \left(x \right)
F_{109}\! \left(x \right) = F_{107}\! \left(x \right)+F_{8}\! \left(x \right)
F_{110}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{105}\! \left(x \right)+F_{111}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right) F_{8}\! \left(x \right)
F_{112}\! \left(x \right) = F_{110}\! \left(x \right)+F_{79}\! \left(x \right)
F_{113}\! \left(x \right) = F_{8}\! \left(x \right) F_{93}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{118}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{117}\! \left(x \right)
F_{116}\! \left(x \right) = F_{6}\! \left(x \right) F_{93}\! \left(x \right)
F_{117}\! \left(x \right) = F_{106}\! \left(x \right) F_{14}\! \left(x \right)
F_{118}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right) F_{93}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right) F_{8}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)+F_{147}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)+F_{146}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)+F_{145}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)+F_{140}\! \left(x \right)+F_{25}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right) F_{8}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)+F_{131}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)+F_{14}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right) F_{8}\! \left(x \right)
F_{130}\! \left(x \right) = F_{128}\! \left(x \right)+F_{8}\! \left(x \right)
F_{131}\! \left(x \right) = F_{124}\! \left(x \right)+F_{132}\! \left(x \right)
F_{132}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{133}\! \left(x \right)+F_{137}\! \left(x \right)
F_{133}\! \left(x \right) = F_{134}\! \left(x \right) F_{8}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)+F_{136}\! \left(x \right)
F_{135}\! \left(x \right) = F_{128}\! \left(x \right)
F_{136}\! \left(x \right) = F_{132}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right) F_{8}\! \left(x \right)
F_{138}\! \left(x \right) = F_{139}\! \left(x \right)+F_{33}\! \left(x \right)
F_{139}\! \left(x \right) = F_{137}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right) F_{8}\! \left(x \right)
F_{141}\! \left(x \right) = F_{14}\! \left(x \right)+F_{142}\! \left(x \right)
F_{142}\! \left(x \right) = F_{140}\! \left(x \right)+F_{143}\! \left(x \right)+F_{25}\! \left(x \right)
F_{143}\! \left(x \right) = F_{144}\! \left(x \right) F_{8}\! \left(x \right)
F_{144}\! \left(x \right) = F_{14}\! \left(x \right)+F_{142}\! \left(x \right)
F_{145}\! \left(x \right) = F_{142}\! \left(x \right) F_{6}\! \left(x \right)
F_{146}\! \left(x \right) = F_{142}\! \left(x \right) F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{147}\! \left(x \right) = F_{148}\! \left(x \right)+F_{157}\! \left(x \right)
F_{148}\! \left(x \right) = F_{149}\! \left(x \right)+F_{153}\! \left(x \right)
F_{149}\! \left(x \right) = F_{13}\! \left(x \right)+F_{150}\! \left(x \right)+F_{25}\! \left(x \right)
F_{150}\! \left(x \right) = F_{151}\! \left(x \right) F_{8}\! \left(x \right)
F_{151}\! \left(x \right) = F_{152}\! \left(x \right)+F_{28}\! \left(x \right)
F_{152}\! \left(x \right) = F_{149}\! \left(x \right)
F_{153}\! \left(x \right) = F_{14}\! \left(x \right) F_{154}\! \left(x \right)
F_{154}\! \left(x \right) = F_{13}\! \left(x \right)+F_{155}\! \left(x \right)+F_{25}\! \left(x \right)
F_{155}\! \left(x \right) = F_{156}\! \left(x \right) F_{8}\! \left(x \right)
F_{156}\! \left(x \right) = F_{154}\! \left(x \right)+F_{8}\! \left(x \right)
F_{157}\! \left(x \right) = F_{158}\! \left(x \right)+F_{167}\! \left(x \right)
F_{158}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{159}\! \left(x \right)+F_{162}\! \left(x \right)
F_{159}\! \left(x \right) = F_{160}\! \left(x \right) F_{8}\! \left(x \right)
F_{160}\! \left(x \right) = F_{135}\! \left(x \right)+F_{161}\! \left(x \right)
F_{161}\! \left(x \right) = F_{158}\! \left(x \right)
F_{162}\! \left(x \right) = F_{163}\! \left(x \right) F_{8}\! \left(x \right)
F_{163}\! \left(x \right) = F_{154}\! \left(x \right)+F_{164}\! \left(x \right)
F_{164}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{162}\! \left(x \right)+F_{165}\! \left(x \right)
F_{165}\! \left(x \right) = F_{166}\! \left(x \right) F_{8}\! \left(x \right)
F_{166}\! \left(x \right) = F_{128}\! \left(x \right)+F_{164}\! \left(x \right)
F_{167}\! \left(x \right) = F_{14}\! \left(x \right) F_{164}\! \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_8(x))
Eq(F_4(x), F_17(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_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), F_1(x) + F_8(x))
Eq(F_11(x), F_12(x) + F_6(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_8(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_8(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_8(x))
Eq(F_20(x), F_21(x) + F_62(x))
Eq(F_21(x), F_22(x) + F_43(x))
Eq(F_22(x), F_23(x) + F_6(x))
Eq(F_23(x), F_24(x) + F_6(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_30(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_27(x)*F_8(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_8(x))
Eq(F_29(x), F_12(x))
Eq(F_30(x), F_31(x)*F_8(x))
Eq(F_31(x), F_32(x) + F_34(x))
Eq(F_32(x), F_14(x) + F_33(x))
Eq(F_33(x), F_13(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_25(x) + F_30(x) + F_36(x))
Eq(F_36(x), F_37(x)*F_8(x))
Eq(F_37(x), F_33(x) + F_8(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x)*F_8(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_42(x)*F_8(x))
Eq(F_42(x), F_14(x) + F_40(x))
Eq(F_43(x), F_25(x) + F_44(x) + F_49(x))
Eq(F_44(x), F_45(x)*F_8(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_8(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_14(x)*F_8(x))
Eq(F_49(x), F_50(x)*F_8(x))
Eq(F_50(x), F_51(x) + F_53(x))
Eq(F_51(x), F_14(x) + F_52(x))
Eq(F_52(x), F_48(x))
Eq(F_53(x), F_54(x) + F_57(x))
Eq(F_54(x), F_25(x) + F_49(x) + F_55(x))
Eq(F_55(x), F_56(x)*F_8(x))
Eq(F_56(x), F_52(x) + F_8(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_59(x)*F_8(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_61(x)*F_8(x))
Eq(F_61(x), F_14(x) + F_59(x))
Eq(F_62(x), F_119(x) + F_63(x))
Eq(F_63(x), F_64(x) + F_69(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_14(x)*F_6(x))
Eq(F_66(x), F_67(x)*F_8(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_14(x)*F_16(x)**2*F_8(x))
Eq(F_69(x), F_18(x) + F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_72(x)*F_8(x))
Eq(F_72(x), F_114(x) + F_73(x))
Eq(F_73(x), F_113(x) + F_74(x))
Eq(F_74(x), F_75(x) + F_96(x))
Eq(F_75(x), F_25(x) + F_76(x) + F_91(x))
Eq(F_76(x), F_77(x)*F_8(x))
Eq(F_77(x), F_78(x) + F_82(x))
Eq(F_78(x), F_14(x) + F_79(x))
Eq(F_79(x), F_80(x))
Eq(F_80(x), F_8(x)*F_81(x))
Eq(F_81(x), F_79(x) + F_8(x))
Eq(F_82(x), F_75(x) + F_83(x))
Eq(F_83(x), 2*F_25(x) + F_84(x) + F_88(x))
Eq(F_84(x), F_8(x)*F_85(x))
Eq(F_85(x), F_86(x) + F_87(x))
Eq(F_86(x), F_79(x))
Eq(F_87(x), F_83(x))
Eq(F_88(x), F_8(x)*F_89(x))
Eq(F_89(x), F_52(x) + F_90(x))
Eq(F_90(x), F_88(x))
Eq(F_91(x), F_8(x)*F_92(x))
Eq(F_92(x), F_14(x) + F_93(x))
Eq(F_93(x), F_25(x) + F_91(x) + F_94(x))
Eq(F_94(x), F_8(x)*F_95(x))
Eq(F_95(x), F_14(x) + F_93(x))
Eq(F_96(x), F_101(x) + F_97(x))
Eq(F_97(x), F_25(x) + F_48(x) + F_98(x))
Eq(F_98(x), F_8(x)*F_99(x))
Eq(F_99(x), F_100(x) + F_46(x))
Eq(F_100(x), F_97(x))
Eq(F_101(x), F_102(x) + F_105(x) + 2*F_25(x))
Eq(F_102(x), F_103(x)*F_8(x))
Eq(F_103(x), F_104(x) + F_86(x))
Eq(F_104(x), F_101(x))
Eq(F_105(x), F_106(x)*F_8(x))
Eq(F_106(x), F_107(x) + F_110(x))
Eq(F_107(x), F_108(x) + F_25(x) + F_48(x))
Eq(F_108(x), F_109(x)*F_8(x))
Eq(F_109(x), F_107(x) + F_8(x))
Eq(F_110(x), F_105(x) + F_111(x) + 2*F_25(x))
Eq(F_111(x), F_112(x)*F_8(x))
Eq(F_112(x), F_110(x) + F_79(x))
Eq(F_113(x), F_8(x)*F_93(x))
Eq(F_114(x), F_115(x) + F_118(x))
Eq(F_115(x), F_116(x) + F_117(x))
Eq(F_116(x), F_6(x)*F_93(x))
Eq(F_117(x), F_106(x)*F_14(x))
Eq(F_118(x), F_14(x)*F_8(x)*F_93(x))
Eq(F_119(x), F_120(x))
Eq(F_120(x), F_121(x)*F_8(x))
Eq(F_121(x), F_122(x) + F_147(x))
Eq(F_122(x), F_123(x) + F_146(x))
Eq(F_123(x), F_124(x) + F_145(x))
Eq(F_124(x), F_125(x) + F_140(x) + F_25(x))
Eq(F_125(x), F_126(x)*F_8(x))
Eq(F_126(x), F_127(x) + F_131(x))
Eq(F_127(x), F_128(x) + F_14(x))
Eq(F_128(x), F_129(x))
Eq(F_129(x), F_130(x)*F_8(x))
Eq(F_130(x), F_128(x) + F_8(x))
Eq(F_131(x), F_124(x) + F_132(x))
Eq(F_132(x), F_133(x) + F_137(x) + 2*F_25(x))
Eq(F_133(x), F_134(x)*F_8(x))
Eq(F_134(x), F_135(x) + F_136(x))
Eq(F_135(x), F_128(x))
Eq(F_136(x), F_132(x))
Eq(F_137(x), F_138(x)*F_8(x))
Eq(F_138(x), F_139(x) + F_33(x))
Eq(F_139(x), F_137(x))
Eq(F_140(x), F_141(x)*F_8(x))
Eq(F_141(x), F_14(x) + F_142(x))
Eq(F_142(x), F_140(x) + F_143(x) + F_25(x))
Eq(F_143(x), F_144(x)*F_8(x))
Eq(F_144(x), F_14(x) + F_142(x))
Eq(F_145(x), F_142(x)*F_6(x))
Eq(F_146(x), F_142(x)*F_16(x)*F_8(x))
Eq(F_147(x), F_148(x) + F_157(x))
Eq(F_148(x), F_149(x) + F_153(x))
Eq(F_149(x), F_13(x) + F_150(x) + F_25(x))
Eq(F_150(x), F_151(x)*F_8(x))
Eq(F_151(x), F_152(x) + F_28(x))
Eq(F_152(x), F_149(x))
Eq(F_153(x), F_14(x)*F_154(x))
Eq(F_154(x), F_13(x) + F_155(x) + F_25(x))
Eq(F_155(x), F_156(x)*F_8(x))
Eq(F_156(x), F_154(x) + F_8(x))
Eq(F_157(x), F_158(x) + F_167(x))
Eq(F_158(x), F_159(x) + F_162(x) + 2*F_25(x))
Eq(F_159(x), F_160(x)*F_8(x))
Eq(F_160(x), F_135(x) + F_161(x))
Eq(F_161(x), F_158(x))
Eq(F_162(x), F_163(x)*F_8(x))
Eq(F_163(x), F_154(x) + F_164(x))
Eq(F_164(x), F_162(x) + F_165(x) + 2*F_25(x))
Eq(F_165(x), F_166(x)*F_8(x))
Eq(F_166(x), F_128(x) + F_164(x))
Eq(F_167(x), F_14(x)*F_164(x))
Pack JSON:
{"expansion_strats": [[{"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}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "insertion_point_placements", "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:
{"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, 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": [2, 3, 0, 1], "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, 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": [2, 3, 0, 1], "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, 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": [2, 3, 0, 1], "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"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": 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