0132_1230_2130_2301
Counting sequence:
1, 1, 2, 6, 20, 63, 183, 498, 1291, 3228, 7853, 18702, 43791, 101136, 230929, 522258, 1171475, 2609172, 5775381, 12714006, 27852823, 60751896, 131989529, 285736986, 616562715, 1326448668, 2845835293, 6090129438, 13002342431, 27699183648, 58888028193, 124956704802, 264677359651, 559687925796, 1181652877349, 2491081031718, 5244155068455, 11025181048872, 23149873725481, 48550310314026, 101704825569323, 212824219451436, 444889892388909, 929087325470766, 1938438999769135, 4040705232076848, 8415661998997553, 17513021207216178, 36415825111941171, 75663992177033268, 157098221376634933, 325948023030939702, 675821419082285111, 1400338009135513656, 2899755210073178169, 6001046503471185978, 12411920573033087035, 25657007077129715772, 53007367614150737981, 109455485343612534846, 225900557308904079423, 465996460643279962176, 960815958901731098689, 1980142684162259681346, 4079036283299024601155, 8399033161060880220228, 17286905040075063558213, 35565322574112015515718, 73141340252258372157511, 150359410944806555222088, 308982963234634989568073, 634715570088198252003402, 1303373149272022078980171, 2675515760450833366384716, 5490341332146321266573389, 11262844061644103834665038, 23097094467715434740187215, 47351168723733932557729872, 97044631222971209141452881, 198830518394743542077456466, 407256885482678203229143123, 833932141942916387577004116, 1707154373023308223331958869, 3493795618686278314900848726, 7148378371381302310037618775, 14621957788238939868071198808, 29901571222348238007182557273, 61132960846272568106541908058, 124954572715368071497630351451, 255344475915723515764739670108, 521675669680104781469209067613, 1065556888814891071619421175902, 2175989104053877178203935604831, 4442657315985408461544232058976, 9068529757785053203773534503009, 18507203587316435109741907148898, 37762122958361239905522885328995, 77034532764650643746422702211172, 157119349786100464490196846510181, 320398689207684981228291735158886, 653236199930109463458769870520423
Generating function in Maple syntax:
-(x^7-x^6-3*x^5+12*x^4-23*x^3+19*x^2-7*x+1)/(x-1)^2/(2*x-1)^3
Generating function in latex syntax:
-\frac{x^{7}-x^{6}-3 x^{5}+12 x^{4}-23 x^{3}+19 x^{2}-7 x +1}{\left(x -1\right)^{2} \left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(-x**7 + x**6 + 3*x**5 - 12*x**4 + 23*x**3 - 19*x**2 + 7*x - 1)/((x - 1)**2*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
(x-1)^2*(2*x-1)^3*F(x)+x^7-x^6-3*x^5+12*x^4-23*x^3+19*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} \left(2 x -1\right)^{3} F \! \left(x \right)+x^{7}-x^{6}-3 x^{5}+12 x^{4}-23 x^{3}+19 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/64*(3*n^2+31*n-120)*2^n+n+3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(3 n^{2}+31 n -120\right) 2^{n}}{64}+n +3 & \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) = 183
a(7) = 498
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)-n, 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) = 183
a \! \left(7\right) = 498
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-n, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15639/
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[12,x]*F[4,x]
F[4,x] = F[39,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[24,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[1,x]+F[21,x]
F[24,x] = F[25,x]+F[26,x]+F[37,x]
F[25,x] = 0
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[10,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[10,x]*F[12,x]
F[31,x] = F[24,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[25,x]+F[35,x]+F[37,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[10,x]+F[34,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[21,x]+F[34,x]
F[39,x] = F[40,x]+F[48,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[23,x]+F[43,x]
F[43,x] = F[40,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[21,x]+F[47,x]
F[47,x] = F[45,x]
F[48,x] = F[114,x]+F[25,x]+F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[81,x]
F[51,x] = F[52,x]+F[6,x]
F[52,x] = F[25,x]+F[35,x]+F[53,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[38,x]+F[55,x]
F[55,x] = F[56,x]+F[66,x]
F[56,x] = F[25,x]+F[57,x]+F[60,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[12,x]+F[59,x]
F[59,x] = F[20,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[21,x]+F[59,x]
F[63,x] = F[56,x]+F[64,x]
F[64,x] = F[65,x]
F[65,x] = F[12,x]*F[47,x]
F[66,x] = F[25,x]+F[67,x]+F[70,x]+F[79,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[29,x]+F[69,x]
F[69,x] = F[33,x]
F[70,x] = F[12,x]*F[71,x]
F[71,x] = F[72,x]+F[73,x]
F[72,x] = F[34,x]+F[69,x]
F[73,x] = F[66,x]+F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = 2*F[25,x]+F[77,x]+F[79,x]
F[77,x] = F[12,x]*F[78,x]
F[78,x] = F[34,x]+F[76,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[47,x]+F[76,x]
F[81,x] = F[48,x]+F[82,x]
F[82,x] = 2*F[25,x]+F[77,x]+F[83,x]
F[83,x] = F[12,x]*F[84,x]
F[84,x] = F[80,x]+F[85,x]
F[85,x] = F[86,x]+F[99,x]
F[86,x] = 2*F[25,x]+F[87,x]+F[90,x]
F[87,x] = F[12,x]*F[88,x]
F[88,x] = F[59,x]+F[89,x]
F[89,x] = F[65,x]
F[90,x] = F[12,x]*F[91,x]
F[91,x] = F[92,x]+F[93,x]
F[92,x] = F[47,x]+F[89,x]
F[93,x] = F[86,x]+F[94,x]
F[94,x] = F[95,x]
F[95,x] = F[12,x]*F[96,x]
F[96,x] = F[97,x]
F[97,x] = F[12,x]*F[98,x]
F[98,x] = F[47,x]+F[96,x]
F[99,x] = 2*F[25,x]+F[100,x]+F[103,x]+F[112,x]
F[100,x] = F[101,x]*F[12,x]
F[101,x] = F[102,x]+F[69,x]
F[102,x] = F[75,x]
F[103,x] = F[104,x]*F[12,x]
F[104,x] = F[105,x]+F[106,x]
F[105,x] = F[102,x]+F[76,x]
F[106,x] = F[107,x]+F[99,x]
F[107,x] = F[108,x]
F[108,x] = F[109,x]*F[12,x]
F[109,x] = 3*F[25,x]+F[110,x]+F[112,x]
F[110,x] = F[111,x]*F[12,x]
F[111,x] = F[109,x]+F[76,x]
F[112,x] = F[113,x]*F[12,x]
F[113,x] = F[109,x]+F[96,x]
F[114,x] = F[115,x]*F[12,x]
F[115,x] = F[116,x]+F[38,x]
F[116,x] = F[117,x]+F[124,x]
F[117,x] = F[118,x]+F[120,x]+F[25,x]
F[118,x] = F[119,x]*F[12,x]
F[119,x] = F[21,x]+F[47,x]
F[120,x] = F[12,x]*F[121,x]
F[121,x] = F[122,x]+F[62,x]
F[122,x] = F[117,x]+F[123,x]
F[123,x] = F[65,x]
F[124,x] = F[125,x]+F[127,x]+F[25,x]+F[79,x]
F[125,x] = F[12,x]*F[126,x]
F[126,x] = F[34,x]+F[76,x]
F[127,x] = F[12,x]*F[128,x]
F[128,x] = F[129,x]+F[72,x]
F[129,x] = F[124,x]+F[130,x]
F[130,x] = F[75,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{39}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{24}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \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_{37}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{31}\! \left(x \right) = F_{24}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{25}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{21}\! \left(x \right)+F_{34}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{48}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{23}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{21}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{114}\! \left(x \right)+F_{25}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{81}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{6}\! \left(x \right)
F_{52}\! \left(x \right) = F_{25}\! \left(x \right)+F_{35}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{38}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{66}\! \left(x \right)
F_{56}\! \left(x \right) = F_{25}\! \left(x \right)+F_{57}\! \left(x \right)+F_{60}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{20}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{21}\! \left(x \right)+F_{59}\! \left(x \right)
F_{63}\! \left(x \right) = F_{56}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{66}\! \left(x \right) = F_{25}\! \left(x \right)+F_{67}\! \left(x \right)+F_{70}\! \left(x \right)+F_{79}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{29}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{33}\! \left(x \right)
F_{70}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)+F_{73}\! \left(x \right)
F_{72}\! \left(x \right) = F_{34}\! \left(x \right)+F_{69}\! \left(x \right)
F_{73}\! \left(x \right) = F_{66}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{77}\! \left(x \right)+F_{79}\! \left(x \right)
F_{77}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{34}\! \left(x \right)+F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{47}\! \left(x \right)+F_{76}\! \left(x \right)
F_{81}\! \left(x \right) = F_{48}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{77}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{80}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{86}\! \left(x \right)+F_{99}\! \left(x \right)
F_{86}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{87}\! \left(x \right)+F_{90}\! \left(x \right)
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{59}\! \left(x \right)+F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{65}\! \left(x \right)
F_{90}\! \left(x \right) = F_{12}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)+F_{93}\! \left(x \right)
F_{92}\! \left(x \right) = F_{47}\! \left(x \right)+F_{89}\! \left(x \right)
F_{93}\! \left(x \right) = F_{86}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{12}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{12}\! \left(x \right) F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{47}\! \left(x \right)+F_{96}\! \left(x \right)
F_{99}\! \left(x \right) = 2 F_{25}\! \left(x \right)+F_{100}\! \left(x \right)+F_{103}\! \left(x \right)+F_{112}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{12}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{69}\! \left(x \right)
F_{102}\! \left(x \right) = F_{75}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{12}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{106}\! \left(x \right)
F_{105}\! \left(x \right) = F_{102}\! \left(x \right)+F_{76}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{99}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{12}\! \left(x \right)
F_{109}\! \left(x \right) = 3 F_{25}\! \left(x \right)+F_{110}\! \left(x \right)+F_{112}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right) F_{12}\! \left(x \right)
F_{111}\! \left(x \right) = F_{109}\! \left(x \right)+F_{76}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{12}\! \left(x \right)
F_{113}\! \left(x \right) = F_{109}\! \left(x \right)+F_{96}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right) F_{12}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{38}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)+F_{124}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)+F_{120}\! \left(x \right)+F_{25}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right) F_{12}\! \left(x \right)
F_{119}\! \left(x \right) = F_{21}\! \left(x \right)+F_{47}\! \left(x \right)
F_{120}\! \left(x \right) = F_{12}\! \left(x \right) F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)+F_{62}\! \left(x \right)
F_{122}\! \left(x \right) = F_{117}\! \left(x \right)+F_{123}\! \left(x \right)
F_{123}\! \left(x \right) = F_{65}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)+F_{127}\! \left(x \right)+F_{25}\! \left(x \right)+F_{79}\! \left(x \right)
F_{125}\! \left(x \right) = F_{12}\! \left(x \right) F_{126}\! \left(x \right)
F_{126}\! \left(x \right) = F_{34}\! \left(x \right)+F_{76}\! \left(x \right)
F_{127}\! \left(x \right) = F_{12}\! \left(x \right) F_{128}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right)+F_{72}\! \left(x \right)
F_{129}\! \left(x \right) = F_{124}\! \left(x \right)+F_{130}\! \left(x \right)
F_{130}\! \left(x \right) = F_{75}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_39(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_24(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(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_37(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_10(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_10(x)*F_12(x))
Eq(F_31(x), F_24(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_25(x) + F_35(x) + F_37(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_10(x) + F_34(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_21(x) + F_34(x))
Eq(F_39(x), F_40(x) + F_48(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_23(x) + F_43(x))
Eq(F_43(x), F_40(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_21(x) + F_47(x))
Eq(F_47(x), F_45(x))
Eq(F_48(x), F_114(x) + F_25(x) + F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_81(x))
Eq(F_51(x), F_52(x) + F_6(x))
Eq(F_52(x), F_25(x) + F_35(x) + F_53(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_38(x) + F_55(x))
Eq(F_55(x), F_56(x) + F_66(x))
Eq(F_56(x), F_25(x) + F_57(x) + F_60(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_12(x) + F_59(x))
Eq(F_59(x), F_20(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_21(x) + F_59(x))
Eq(F_63(x), F_56(x) + F_64(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_12(x)*F_47(x))
Eq(F_66(x), F_25(x) + F_67(x) + F_70(x) + F_79(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_29(x) + F_69(x))
Eq(F_69(x), F_33(x))
Eq(F_70(x), F_12(x)*F_71(x))
Eq(F_71(x), F_72(x) + F_73(x))
Eq(F_72(x), F_34(x) + F_69(x))
Eq(F_73(x), F_66(x) + F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), 2*F_25(x) + F_77(x) + F_79(x))
Eq(F_77(x), F_12(x)*F_78(x))
Eq(F_78(x), F_34(x) + F_76(x))
Eq(F_79(x), F_12(x)*F_80(x))
Eq(F_80(x), F_47(x) + F_76(x))
Eq(F_81(x), F_48(x) + F_82(x))
Eq(F_82(x), 2*F_25(x) + F_77(x) + F_83(x))
Eq(F_83(x), F_12(x)*F_84(x))
Eq(F_84(x), F_80(x) + F_85(x))
Eq(F_85(x), F_86(x) + F_99(x))
Eq(F_86(x), 2*F_25(x) + F_87(x) + F_90(x))
Eq(F_87(x), F_12(x)*F_88(x))
Eq(F_88(x), F_59(x) + F_89(x))
Eq(F_89(x), F_65(x))
Eq(F_90(x), F_12(x)*F_91(x))
Eq(F_91(x), F_92(x) + F_93(x))
Eq(F_92(x), F_47(x) + F_89(x))
Eq(F_93(x), F_86(x) + F_94(x))
Eq(F_94(x), F_95(x))
Eq(F_95(x), F_12(x)*F_96(x))
Eq(F_96(x), F_97(x))
Eq(F_97(x), F_12(x)*F_98(x))
Eq(F_98(x), F_47(x) + F_96(x))
Eq(F_99(x), F_100(x) + F_103(x) + F_112(x) + 2*F_25(x))
Eq(F_100(x), F_101(x)*F_12(x))
Eq(F_101(x), F_102(x) + F_69(x))
Eq(F_102(x), F_75(x))
Eq(F_103(x), F_104(x)*F_12(x))
Eq(F_104(x), F_105(x) + F_106(x))
Eq(F_105(x), F_102(x) + F_76(x))
Eq(F_106(x), F_107(x) + F_99(x))
Eq(F_107(x), F_108(x))
Eq(F_108(x), F_109(x)*F_12(x))
Eq(F_109(x), F_110(x) + F_112(x) + 3*F_25(x))
Eq(F_110(x), F_111(x)*F_12(x))
Eq(F_111(x), F_109(x) + F_76(x))
Eq(F_112(x), F_113(x)*F_12(x))
Eq(F_113(x), F_109(x) + F_96(x))
Eq(F_114(x), F_115(x)*F_12(x))
Eq(F_115(x), F_116(x) + F_38(x))
Eq(F_116(x), F_117(x) + F_124(x))
Eq(F_117(x), F_118(x) + F_120(x) + F_25(x))
Eq(F_118(x), F_119(x)*F_12(x))
Eq(F_119(x), F_21(x) + F_47(x))
Eq(F_120(x), F_12(x)*F_121(x))
Eq(F_121(x), F_122(x) + F_62(x))
Eq(F_122(x), F_117(x) + F_123(x))
Eq(F_123(x), F_65(x))
Eq(F_124(x), F_125(x) + F_127(x) + F_25(x) + F_79(x))
Eq(F_125(x), F_12(x)*F_126(x))
Eq(F_126(x), F_34(x) + F_76(x))
Eq(F_127(x), F_12(x)*F_128(x))
Eq(F_128(x), F_129(x) + F_72(x))
Eq(F_129(x), F_124(x) + F_130(x))
Eq(F_130(x), F_75(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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], 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