0132_0213_0231_0312_3120
Counting sequence:
1, 1, 2, 6, 19, 56, 151, 376, 880, 1971, 4289, 9171, 19420, 40922, 86039, 180714, 379320, 795637, 1667365, 3490269, 7296708, 15233172, 31756315, 66106940, 137421808, 285284935, 591485545, 1224844695, 2533489964, 5234647710, 10804713343, 22280363390, 45902722664, 94489584681, 194347624653, 399432369793, 820339228564, 1683627726456, 3453154332099, 7078106818608, 14499810404512, 29686814872091, 60748018477009, 124244815113499, 253987187336572, 518969489790114, 1059929210830439, 2163838885308114, 4415638699201048, 9007199257019549, 18366242232894197, 37436171905307109, 76279718691665508, 155374187147670940, 316377873826501867, 644014746718067172, 1310547491569289808, 2666130979408227375, 5422333951359419833, 11024811887808798047, 22409911745801927116, 45540399431977345510, 92521950744706948303, 187926205250924163558, 381617018024875124936, 774763251095810656593, 1572584932283749522333, 3191286724751763482633, 6474807169872064526132, 13134081780481213567488, 26637098442436606312851, 54012066647821581930328, 109499872821539914269952, 221951224694873342061699, 449805407493333724827681, 911416731193841545738851, 1846445294802031299393052, 3740114254432759031500394, 7574675838522910946512567, 15338246336360607679398586, 31054281991350786952230520, 62864142619960717113423429, 127239442514439720668352197, 257501199577916014244859117, 521047028253905174332816900, 1054183314703956640380350372, 2132545145800205864220471227, 4313447324384996895392730252, 8723608714339164124723287216, 17640645559816668917358581527, 35668147381910019170579735305, 72110007288373401012925483047, 145767439625853527369426277996, 294629729349920505426048998958, 595449158896267912226539352095, 1203277718185389627202012650830, 2431314237156486859901947328296, 4912146075884388930799795867513, 9923327354911608283591454472301, 20044725116108877411166698030225, 40485591044789076510301041280724
Generating function in Maple syntax:
-(x^10+x^9-4*x^8+7*x^7-33*x^6+75*x^5-96*x^4+75*x^3-35*x^2+9*x-1)/(2*x-1)^2/(x-1)^6
Generating function in latex syntax:
-\frac{x^{10}+x^{9}-4 x^{8}+7 x^{7}-33 x^{6}+75 x^{5}-96 x^{4}+75 x^{3}-35 x^{2}+9 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{6}}
Generating function in sympy syntax:
(-x**10 - x**9 + 4*x**8 - 7*x**7 + 33*x**6 - 75*x**5 + 96*x**4 - 75*x**3 + 35*x**2 - 9*x + 1)/((x - 1)**6*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^6*F(x)+x^10+x^9-4*x^8+7*x^7-33*x^6+75*x^5-96*x^4+75*x^3-35*x^2+9*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)+x^{10}+x^{9}-4 x^{8}+7 x^{7}-33 x^{6}+75 x^{5}-96 x^{4}+75 x^{3}-35 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,8+5/16*2^n*n+11/16*2^n-49/5*n+7/2*n^2-17/24*n^3+1/120*n^5)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 8+\frac{5 \,2^{n} n}{16}+\frac{11 \,2^{n}}{16}-\frac{49 n}{5}+\frac{7 n^{2}}{2}-\frac{17 n^{3}}{24}+\frac{n^{5}}{120} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 151
a(7) = 376
a(8) = 880
a(9) = 1971
a(10) = 4289
a(n+2) = 1/120*n^5-1/12*n^4-17/24*n^3+97/12*n^2-4*a(n)+4*a(n+1)-233/10*n+25, n >= 11
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) = 56
a \! \left(6\right) = 151
a \! \left(7\right) = 376
a \! \left(8\right) = 880
a \! \left(9\right) = 1971
a \! \left(10\right) = 4289
a \! \left(n +2\right) = \frac{n^{5}}{120}-\frac{n^{4}}{12}-\frac{17 n^{3}}{24}+\frac{97 n^{2}}{12}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{233 n}{10}+25, \quad n \geq 11
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/21548/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[16,x]*F[3,x]
F[3,x] = F[1,x]+F[110,x]+F[4,x]
F[4,x] = F[5,x]
F[5,x] = F[16,x]*F[6,x]
F[6,x] = F[103,x]+F[42,x]+F[7,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[16,x]
F[10,x] = F[11,x]+F[7,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[16,x]*F[18,x]*F[30,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[17,x]
F[16,x] = x
F[17,x] = F[18,x]+F[20,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[16,x]*F[18,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[22,x]
F[22,x] = F[16,x]*F[23,x]
F[23,x] = F[1,x]+F[21,x]
F[24,x] = F[25,x]+F[26,x]+F[29,x]
F[25,x] = 0
F[26,x] = F[16,x]*F[27,x]
F[27,x] = F[24,x]+F[28,x]
F[28,x] = F[19,x]
F[29,x] = F[16,x]*F[20,x]
F[30,x] = F[1,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[16,x]*F[33,x]
F[33,x] = F[30,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[16,x]*F[36,x]
F[36,x] = F[18,x]+F[37,x]
F[37,x] = F[21,x]+F[38,x]
F[38,x] = F[25,x]+F[39,x]+F[41,x]
F[39,x] = F[16,x]*F[40,x]
F[40,x] = F[28,x]+F[38,x]
F[41,x] = F[16,x]*F[37,x]
F[42,x] = F[43,x]
F[43,x] = F[13,x]*F[16,x]*F[44,x]
F[44,x] = F[31,x]+F[45,x]
F[45,x] = F[46,x]+F[65,x]
F[46,x] = F[1,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[16,x]*F[49,x]
F[49,x] = F[50,x]+F[55,x]
F[50,x] = F[1,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[16,x]*F[53,x]
F[53,x] = F[18,x]+F[54,x]
F[54,x] = F[21,x]
F[55,x] = F[21,x]+F[56,x]
F[56,x] = F[25,x]+F[57,x]+F[59,x]
F[57,x] = F[16,x]*F[58,x]
F[58,x] = F[51,x]+F[56,x]
F[59,x] = F[16,x]*F[60,x]
F[60,x] = F[37,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[16,x]*F[64,x]
F[64,x] = F[21,x]+F[62,x]
F[65,x] = F[34,x]+F[66,x]
F[66,x] = F[25,x]+F[67,x]+F[74,x]
F[67,x] = F[16,x]*F[68,x]
F[68,x] = F[69,x]+F[70,x]
F[69,x] = F[21,x]
F[70,x] = F[71,x]
F[71,x] = F[25,x]+F[63,x]+F[72,x]
F[72,x] = F[16,x]*F[73,x]
F[73,x] = F[21,x]+F[71,x]
F[74,x] = F[16,x]*F[75,x]
F[75,x] = F[76,x]+F[88,x]
F[76,x] = F[21,x]+F[77,x]
F[77,x] = F[25,x]+F[72,x]+F[78,x]
F[78,x] = F[16,x]*F[79,x]
F[79,x] = F[80,x]+F[87,x]
F[80,x] = F[21,x]+F[81,x]
F[81,x] = F[25,x]+F[82,x]+F[86,x]
F[82,x] = F[16,x]*F[83,x]
F[83,x] = F[16,x]+F[84,x]
F[84,x] = F[25,x]+F[82,x]+F[85,x]
F[85,x] = F[16,x]*F[21,x]
F[86,x] = F[16,x]*F[80,x]
F[87,x] = F[62,x]
F[88,x] = F[62,x]+F[89,x]
F[89,x] = 2*F[25,x]+F[90,x]+F[92,x]
F[90,x] = F[16,x]*F[91,x]
F[91,x] = F[77,x]+F[89,x]
F[92,x] = F[16,x]*F[93,x]
F[93,x] = F[94,x]+F[99,x]
F[94,x] = F[62,x]+F[95,x]
F[95,x] = 2*F[25,x]+F[96,x]+F[98,x]
F[96,x] = F[16,x]*F[97,x]
F[97,x] = F[81,x]+F[95,x]
F[98,x] = F[16,x]*F[94,x]
F[99,x] = F[100,x]
F[100,x] = F[101,x]
F[101,x] = F[102,x]*F[16,x]
F[102,x] = F[100,x]+F[62,x]
F[103,x] = F[104,x]
F[104,x] = F[105,x]*F[13,x]*F[16,x]
F[105,x] = F[106,x]+F[109,x]
F[106,x] = F[107,x]*F[23,x]
F[107,x] = F[108,x]+F[30,x]
F[108,x] = F[21,x]*F[23,x]
F[109,x] = F[57,x]
F[110,x] = F[111,x]
F[111,x] = F[112,x]*F[16,x]
F[112,x] = F[113,x]+F[161,x]
F[113,x] = F[114,x]+F[46,x]
F[114,x] = F[115,x]+F[21,x]
F[115,x] = F[116,x]+F[125,x]+F[25,x]
F[116,x] = F[117,x]*F[16,x]
F[117,x] = F[118,x]+F[21,x]
F[118,x] = F[116,x]+F[119,x]+F[25,x]
F[119,x] = F[120,x]*F[16,x]
F[120,x] = F[121,x]+F[21,x]
F[121,x] = F[119,x]+F[122,x]+F[25,x]
F[122,x] = F[123,x]*F[16,x]
F[123,x] = F[124,x]+F[21,x]
F[124,x] = F[122,x]
F[125,x] = F[126,x]*F[16,x]
F[126,x] = F[127,x]+F[132,x]
F[127,x] = F[128,x]+F[21,x]
F[128,x] = F[116,x]+F[129,x]+F[25,x]
F[129,x] = F[130,x]*F[16,x]
F[130,x] = F[131,x]+F[80,x]
F[131,x] = F[121,x]
F[132,x] = F[121,x]+F[133,x]
F[133,x] = F[134,x]+F[143,x]+F[145,x]+F[25,x]
F[134,x] = F[135,x]*F[16,x]
F[135,x] = F[136,x]+F[62,x]
F[136,x] = 2*F[25,x]+F[134,x]+F[137,x]
F[137,x] = F[138,x]*F[16,x]
F[138,x] = F[124,x]+F[139,x]
F[139,x] = 2*F[25,x]+F[137,x]+F[140,x]
F[140,x] = F[141,x]*F[16,x]
F[141,x] = F[142,x]+F[62,x]
F[142,x] = F[140,x]
F[143,x] = F[144,x]*F[16,x]
F[144,x] = F[128,x]+F[133,x]
F[145,x] = F[146,x]*F[16,x]
F[146,x] = F[147,x]+F[157,x]
F[147,x] = F[121,x]+F[148,x]
F[148,x] = F[149,x]+F[154,x]+F[156,x]+F[25,x]
F[149,x] = F[150,x]*F[16,x]
F[150,x] = F[151,x]+F[152,x]
F[151,x] = F[85,x]
F[152,x] = 2*F[25,x]+F[149,x]+F[153,x]
F[153,x] = F[124,x]*F[16,x]
F[154,x] = F[155,x]*F[16,x]
F[155,x] = F[148,x]+F[81,x]
F[156,x] = F[147,x]*F[16,x]
F[157,x] = F[158,x]
F[158,x] = 2*F[25,x]+F[140,x]+F[159,x]
F[159,x] = F[16,x]*F[160,x]
F[160,x] = F[121,x]+F[158,x]
F[161,x] = F[16,x]*F[162,x]
F[162,x] = F[161,x]+F[163,x]+F[165,x]
F[163,x] = F[164,x]+F[23,x]
F[164,x] = F[118,x]+F[21,x]
F[165,x] = F[16,x]*F[23,x]*F[36,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_{16}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{110}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{16}\! \left(x \right) F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{103}\! \left(x \right)+F_{42}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \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_{16}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{7}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{18}\! \left(x \right) F_{30}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{20}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{18}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{19}\! \left(x \right)
F_{29}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{16}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{18}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{21}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{25}\! \left(x \right)+F_{39}\! \left(x \right)+F_{41}\! \left(x \right)
F_{39}\! \left(x \right) = F_{16}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{28}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{16}\! \left(x \right) F_{37}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{31}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{65}\! \left(x \right)
F_{46}\! \left(x \right) = F_{1}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{16}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{55}\! \left(x \right)
F_{50}\! \left(x \right) = F_{1}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{16}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{18}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{21}\! \left(x \right)
F_{55}\! \left(x \right) = F_{21}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{25}\! \left(x \right)+F_{57}\! \left(x \right)+F_{59}\! \left(x \right)
F_{57}\! \left(x \right) = F_{16}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{16}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{37}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{16}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{21}\! \left(x \right)+F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{34}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{25}\! \left(x \right)+F_{67}\! \left(x \right)+F_{74}\! \left(x \right)
F_{67}\! \left(x \right) = F_{16}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{70}\! \left(x \right)
F_{69}\! \left(x \right) = F_{21}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{25}\! \left(x \right)+F_{63}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{16}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{21}\! \left(x \right)+F_{71}\! \left(x \right)
F_{74}\! \left(x \right) = F_{16}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{88}\! \left(x \right)
F_{76}\! \left(x \right) = F_{21}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{25}\! \left(x \right)+F_{72}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{16}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{80}\! \left(x \right)+F_{87}\! \left(x \right)
F_{80}\! \left(x \right) = F_{21}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{25}\! \left(x \right)+F_{82}\! \left(x \right)+F_{86}\! \left(x \right)
F_{82}\! \left(x \right) = F_{16}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{16}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{25}\! \left(x \right)+F_{82}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{16}\! \left(x \right) F_{21}\! \left(x \right)
F_{86}\! \left(x \right) = F_{16}\! \left(x \right) F_{80}\! \left(x \right)
F_{87}\! \left(x \right) = F_{62}\! \left(x \right)
F_{88}\! \left(x \right) = F_{62}\! \left(x \right)+F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{90}\! \left(x \right)+F_{92}\! \left(x \right)
F_{90}\! \left(x \right) = F_{16}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{77}\! \left(x \right)+F_{89}\! \left(x \right)
F_{92}\! \left(x \right) = F_{16}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)+F_{99}\! \left(x \right)
F_{94}\! \left(x \right) = F_{62}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{96}\! \left(x \right)+F_{98}\! \left(x \right)
F_{96}\! \left(x \right) = F_{16}\! \left(x \right) F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{81}\! \left(x \right)+F_{95}\! \left(x \right)
F_{98}\! \left(x \right) = F_{16}\! \left(x \right) F_{94}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{16}\! \left(x \right)
F_{102}\! \left(x \right) = F_{100}\! \left(x \right)+F_{62}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right) F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{109}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right) F_{23}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)+F_{30}\! \left(x \right)
F_{108}\! \left(x \right) = F_{21}\! \left(x \right) F_{23}\! \left(x \right)
F_{109}\! \left(x \right) = F_{57}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right) F_{16}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)+F_{161}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)+F_{46}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{21}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{125}\! \left(x \right)+F_{25}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right) F_{16}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)+F_{21}\! \left(x \right)
F_{118}\! \left(x \right) = F_{116}\! \left(x \right)+F_{119}\! \left(x \right)+F_{25}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right) F_{16}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)+F_{21}\! \left(x \right)
F_{121}\! \left(x \right) = F_{119}\! \left(x \right)+F_{122}\! \left(x \right)+F_{25}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right) F_{16}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)+F_{21}\! \left(x \right)
F_{124}\! \left(x \right) = F_{122}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right) F_{16}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)+F_{132}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)+F_{21}\! \left(x \right)
F_{128}\! \left(x \right) = F_{116}\! \left(x \right)+F_{129}\! \left(x \right)+F_{25}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right) F_{16}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right)+F_{80}\! \left(x \right)
F_{131}\! \left(x \right) = F_{121}\! \left(x \right)
F_{132}\! \left(x \right) = F_{121}\! \left(x \right)+F_{133}\! \left(x \right)
F_{133}\! \left(x \right) = F_{134}\! \left(x \right)+F_{143}\! \left(x \right)+F_{145}\! \left(x \right)+F_{25}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right) F_{16}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right)+F_{62}\! \left(x \right)
F_{136}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{134}\! \left(x \right)+F_{137}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right) F_{16}\! \left(x \right)
F_{138}\! \left(x \right) = F_{124}\! \left(x \right)+F_{139}\! \left(x \right)
F_{139}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{137}\! \left(x \right)+F_{140}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right) F_{16}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right)+F_{62}\! \left(x \right)
F_{142}\! \left(x \right) = F_{140}\! \left(x \right)
F_{143}\! \left(x \right) = F_{144}\! \left(x \right) F_{16}\! \left(x \right)
F_{144}\! \left(x \right) = F_{128}\! \left(x \right)+F_{133}\! \left(x \right)
F_{145}\! \left(x \right) = F_{146}\! \left(x \right) F_{16}\! \left(x \right)
F_{146}\! \left(x \right) = F_{147}\! \left(x \right)+F_{157}\! \left(x \right)
F_{147}\! \left(x \right) = F_{121}\! \left(x \right)+F_{148}\! \left(x \right)
F_{148}\! \left(x \right) = F_{149}\! \left(x \right)+F_{154}\! \left(x \right)+F_{156}\! \left(x \right)+F_{25}\! \left(x \right)
F_{149}\! \left(x \right) = F_{150}\! \left(x \right) F_{16}\! \left(x \right)
F_{150}\! \left(x \right) = F_{151}\! \left(x \right)+F_{152}\! \left(x \right)
F_{151}\! \left(x \right) = F_{85}\! \left(x \right)
F_{152}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{149}\! \left(x \right)+F_{153}\! \left(x \right)
F_{153}\! \left(x \right) = F_{124}\! \left(x \right) F_{16}\! \left(x \right)
F_{154}\! \left(x \right) = F_{155}\! \left(x \right) F_{16}\! \left(x \right)
F_{155}\! \left(x \right) = F_{148}\! \left(x \right)+F_{81}\! \left(x \right)
F_{156}\! \left(x \right) = F_{147}\! \left(x \right) F_{16}\! \left(x \right)
F_{157}\! \left(x \right) = F_{158}\! \left(x \right)
F_{158}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{140}\! \left(x \right)+F_{159}\! \left(x \right)
F_{159}\! \left(x \right) = F_{16}\! \left(x \right) F_{160}\! \left(x \right)
F_{160}\! \left(x \right) = F_{121}\! \left(x \right)+F_{158}\! \left(x \right)
F_{161}\! \left(x \right) = F_{16}\! \left(x \right) F_{162}\! \left(x \right)
F_{162}\! \left(x \right) = F_{161}\! \left(x \right)+F_{163}\! \left(x \right)+F_{165}\! \left(x \right)
F_{163}\! \left(x \right) = F_{164}\! \left(x \right)+F_{23}\! \left(x \right)
F_{164}\! \left(x \right) = F_{118}\! \left(x \right)+F_{21}\! \left(x \right)
F_{165}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right) F_{36}\! \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_16(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_110(x) + F_4(x))
Eq(F_4(x), F_5(x))
Eq(F_5(x), F_16(x)*F_6(x))
Eq(F_6(x), F_103(x) + F_42(x) + F_7(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_16(x))
Eq(F_10(x), F_11(x) + F_7(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_16(x)*F_18(x)*F_30(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_17(x))
Eq(F_16(x), x)
Eq(F_17(x), F_18(x) + F_20(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_16(x)*F_18(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_16(x)*F_23(x))
Eq(F_23(x), F_1(x) + F_21(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_29(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_16(x)*F_27(x))
Eq(F_27(x), F_24(x) + F_28(x))
Eq(F_28(x), F_19(x))
Eq(F_29(x), F_16(x)*F_20(x))
Eq(F_30(x), F_1(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_16(x)*F_33(x))
Eq(F_33(x), F_30(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_16(x)*F_36(x))
Eq(F_36(x), F_18(x) + F_37(x))
Eq(F_37(x), F_21(x) + F_38(x))
Eq(F_38(x), F_25(x) + F_39(x) + F_41(x))
Eq(F_39(x), F_16(x)*F_40(x))
Eq(F_40(x), F_28(x) + F_38(x))
Eq(F_41(x), F_16(x)*F_37(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_13(x)*F_16(x)*F_44(x))
Eq(F_44(x), F_31(x) + F_45(x))
Eq(F_45(x), F_46(x) + F_65(x))
Eq(F_46(x), F_1(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_16(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_55(x))
Eq(F_50(x), F_1(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_16(x)*F_53(x))
Eq(F_53(x), F_18(x) + F_54(x))
Eq(F_54(x), F_21(x))
Eq(F_55(x), F_21(x) + F_56(x))
Eq(F_56(x), F_25(x) + F_57(x) + F_59(x))
Eq(F_57(x), F_16(x)*F_58(x))
Eq(F_58(x), F_51(x) + F_56(x))
Eq(F_59(x), F_16(x)*F_60(x))
Eq(F_60(x), F_37(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_16(x)*F_64(x))
Eq(F_64(x), F_21(x) + F_62(x))
Eq(F_65(x), F_34(x) + F_66(x))
Eq(F_66(x), F_25(x) + F_67(x) + F_74(x))
Eq(F_67(x), F_16(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_70(x))
Eq(F_69(x), F_21(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_25(x) + F_63(x) + F_72(x))
Eq(F_72(x), F_16(x)*F_73(x))
Eq(F_73(x), F_21(x) + F_71(x))
Eq(F_74(x), F_16(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_88(x))
Eq(F_76(x), F_21(x) + F_77(x))
Eq(F_77(x), F_25(x) + F_72(x) + F_78(x))
Eq(F_78(x), F_16(x)*F_79(x))
Eq(F_79(x), F_80(x) + F_87(x))
Eq(F_80(x), F_21(x) + F_81(x))
Eq(F_81(x), F_25(x) + F_82(x) + F_86(x))
Eq(F_82(x), F_16(x)*F_83(x))
Eq(F_83(x), F_16(x) + F_84(x))
Eq(F_84(x), F_25(x) + F_82(x) + F_85(x))
Eq(F_85(x), F_16(x)*F_21(x))
Eq(F_86(x), F_16(x)*F_80(x))
Eq(F_87(x), F_62(x))
Eq(F_88(x), F_62(x) + F_89(x))
Eq(F_89(x), 2*F_25(x) + F_90(x) + F_92(x))
Eq(F_90(x), F_16(x)*F_91(x))
Eq(F_91(x), F_77(x) + F_89(x))
Eq(F_92(x), F_16(x)*F_93(x))
Eq(F_93(x), F_94(x) + F_99(x))
Eq(F_94(x), F_62(x) + F_95(x))
Eq(F_95(x), 2*F_25(x) + F_96(x) + F_98(x))
Eq(F_96(x), F_16(x)*F_97(x))
Eq(F_97(x), F_81(x) + F_95(x))
Eq(F_98(x), F_16(x)*F_94(x))
Eq(F_99(x), F_100(x))
Eq(F_100(x), F_101(x))
Eq(F_101(x), F_102(x)*F_16(x))
Eq(F_102(x), F_100(x) + F_62(x))
Eq(F_103(x), F_104(x))
Eq(F_104(x), F_105(x)*F_13(x)*F_16(x))
Eq(F_105(x), F_106(x) + F_109(x))
Eq(F_106(x), F_107(x)*F_23(x))
Eq(F_107(x), F_108(x) + F_30(x))
Eq(F_108(x), F_21(x)*F_23(x))
Eq(F_109(x), F_57(x))
Eq(F_110(x), F_111(x))
Eq(F_111(x), F_112(x)*F_16(x))
Eq(F_112(x), F_113(x) + F_161(x))
Eq(F_113(x), F_114(x) + F_46(x))
Eq(F_114(x), F_115(x) + F_21(x))
Eq(F_115(x), F_116(x) + F_125(x) + F_25(x))
Eq(F_116(x), F_117(x)*F_16(x))
Eq(F_117(x), F_118(x) + F_21(x))
Eq(F_118(x), F_116(x) + F_119(x) + F_25(x))
Eq(F_119(x), F_120(x)*F_16(x))
Eq(F_120(x), F_121(x) + F_21(x))
Eq(F_121(x), F_119(x) + F_122(x) + F_25(x))
Eq(F_122(x), F_123(x)*F_16(x))
Eq(F_123(x), F_124(x) + F_21(x))
Eq(F_124(x), F_122(x))
Eq(F_125(x), F_126(x)*F_16(x))
Eq(F_126(x), F_127(x) + F_132(x))
Eq(F_127(x), F_128(x) + F_21(x))
Eq(F_128(x), F_116(x) + F_129(x) + F_25(x))
Eq(F_129(x), F_130(x)*F_16(x))
Eq(F_130(x), F_131(x) + F_80(x))
Eq(F_131(x), F_121(x))
Eq(F_132(x), F_121(x) + F_133(x))
Eq(F_133(x), F_134(x) + F_143(x) + F_145(x) + F_25(x))
Eq(F_134(x), F_135(x)*F_16(x))
Eq(F_135(x), F_136(x) + F_62(x))
Eq(F_136(x), F_134(x) + F_137(x) + 2*F_25(x))
Eq(F_137(x), F_138(x)*F_16(x))
Eq(F_138(x), F_124(x) + F_139(x))
Eq(F_139(x), F_137(x) + F_140(x) + 2*F_25(x))
Eq(F_140(x), F_141(x)*F_16(x))
Eq(F_141(x), F_142(x) + F_62(x))
Eq(F_142(x), F_140(x))
Eq(F_143(x), F_144(x)*F_16(x))
Eq(F_144(x), F_128(x) + F_133(x))
Eq(F_145(x), F_146(x)*F_16(x))
Eq(F_146(x), F_147(x) + F_157(x))
Eq(F_147(x), F_121(x) + F_148(x))
Eq(F_148(x), F_149(x) + F_154(x) + F_156(x) + F_25(x))
Eq(F_149(x), F_150(x)*F_16(x))
Eq(F_150(x), F_151(x) + F_152(x))
Eq(F_151(x), F_85(x))
Eq(F_152(x), F_149(x) + F_153(x) + 2*F_25(x))
Eq(F_153(x), F_124(x)*F_16(x))
Eq(F_154(x), F_155(x)*F_16(x))
Eq(F_155(x), F_148(x) + F_81(x))
Eq(F_156(x), F_147(x)*F_16(x))
Eq(F_157(x), F_158(x))
Eq(F_158(x), F_140(x) + F_159(x) + 2*F_25(x))
Eq(F_159(x), F_16(x)*F_160(x))
Eq(F_160(x), F_121(x) + F_158(x))
Eq(F_161(x), F_16(x)*F_162(x))
Eq(F_162(x), F_161(x) + F_163(x) + F_165(x))
Eq(F_163(x), F_164(x) + F_23(x))
Eq(F_164(x), F_118(x) + F_21(x))
Eq(F_165(x), F_16(x)*F_23(x)*F_36(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.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "row_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, 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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [2, 0], 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