0123_0213_1230_2301_3012
Counting sequence:
1, 1, 2, 6, 19, 54, 136, 310, 657, 1326, 2598, 5014, 9631, 18534, 35860, 69846, 136925, 269918, 534418, 1061558, 2113547, 4214742, 8413792, 16807926, 33591529, 67153294, 134270526, 268497750, 536943927, 1073826886, 2147582188, 4295080854, 8590064821, 17180017854, 34359907370, 68719668086, 137439169315, 274878149558, 549756085688, 1099511931318, 2199023593537, 4398046886382, 8796093437782, 17592186503446, 35184372594639, 70368744733734, 140737488965316, 281474977378390, 562949954150797, 1125899907638046, 2251799814550978, 4503599628311094, 9007199255761211, 18014398510586774, 36028797020158480, 72057594039217526, 144115188077246105, 288230376153208398, 576460752305032558, 1152921504608574678, 2305843009215546727, 4611686018429372422, 9223372036856898972, 18446744073711820566, 36893488147421525349, 73786976294840789374, 147573952589679164506, 295147905179355754230, 590295810358708765267, 1180591620717414610806, 2361183241434826116968, 4722366482869648935734, 9444732965739294370801, 18889465931478585029294, 37778931862957166125190, 75557863725914328086166, 151115727451828651767295, 302231454903657298878438, 604462909807314592839028, 1208925819614629180487638, 2417851639229258355501117, 4835703278458516705232862, 9671406556917033404389362, 19342813113834066802383286, 38685626227668133598039659, 77371252455336267189008214, 154742504910672534370588096, 309485009821345068733377270, 618970019642690137458571337, 1237940039285380274908561166, 2475880078570760549808128158, 4951760157141521099606834774, 9903520314283042199203805591, 19807040628566084398397289414, 39614081257132168796783783500, 79228162514264337593556282006, 158456325028528675187100772885, 316912650057057350374189231678, 633825300114114700748365609098, 1267650600228229401496717806198, 2535301200456458802993421624707
Generating function in Maple syntax:
-(2*x^7+x^6+5*x^5-12*x^4+18*x^3-15*x^2+6*x-1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
-\frac{2 x^{7}+x^{6}+5 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-2*x**7 - x**6 - 5*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)+2*x^7+x^6+5*x^5-12*x^4+18*x^3-15*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{7}+x^{6}+5 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,5+2^(n+1)-21/2*n+29/6*n^2-3/2*n^3+1/6*n^4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 5+2^{n +1}-\frac{21 n}{2}+\frac{29 n^{2}}{6}-\frac{3 n^{3}}{2}+\frac{n^{4}}{6} & \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) = 54
a(6) = 136
a(7) = 310
a(n+1) = -1/6*n^4+13/6*n^3-25/3*n^2+2*a(n)+49/3*n-12, 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) = 19
a \! \left(5\right) = 54
a \! \left(6\right) = 136
a \! \left(7\right) = 310
a \! \left(n +1\right) = -\frac{n^{4}}{6}+\frac{13 n^{3}}{6}-\frac{25 n^{2}}{3}+2 a \! \left(n \right)+\frac{49 n}{3}-12, \quad n \geq 8
Specification 1
Strategy pack name: insertion_point_placements
Tree: http://permpal.com/tree/21527/
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[18,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]+F[17,x]
F[17,x] = F[15,x]
F[18,x] = F[19,x]+F[41,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[24,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[19,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[10,x]+F[24,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[10,x]*F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[36,x]
F[29,x] = F[10,x]+F[30,x]
F[30,x] = F[11,x]+F[31,x]
F[31,x] = F[32,x]+F[33,x]+F[35,x]
F[32,x] = 0
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[11,x]+F[31,x]
F[35,x] = F[30,x]*F[4,x]
F[36,x] = F[11,x]+F[37,x]
F[37,x] = F[32,x]+F[38,x]+F[39,x]
F[38,x] = F[36,x]*F[4,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[11,x]+F[37,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[10,x]*F[11,x]
F[45,x] = F[129,x]+F[46,x]
F[46,x] = F[47,x]+F[76,x]
F[47,x] = F[48,x]+F[59,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[1,x]+F[4,x]
F[52,x] = F[48,x]+F[53,x]
F[53,x] = F[32,x]+F[54,x]+F[58,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[4,x]
F[57,x] = F[53,x]
F[58,x] = F[11,x]*F[4,x]
F[59,x] = F[32,x]+F[39,x]+F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[66,x]
F[62,x] = F[11,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[4,x]+F[63,x]
F[66,x] = F[59,x]+F[67,x]
F[67,x] = 2*F[32,x]+F[68,x]+F[72,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[71,x]
F[70,x] = F[63,x]
F[71,x] = F[67,x]
F[72,x] = F[4,x]*F[73,x]
F[73,x] = F[74,x]+F[75,x]
F[74,x] = F[58,x]
F[75,x] = F[72,x]
F[76,x] = F[112,x]+F[77,x]
F[77,x] = F[103,x]+F[32,x]+F[78,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[80,x]+F[87,x]
F[80,x] = F[7,x]+F[81,x]
F[81,x] = F[32,x]+F[82,x]+F[83,x]
F[82,x] = F[11,x]*F[4,x]
F[83,x] = F[4,x]*F[84,x]
F[84,x] = F[85,x]+F[86,x]
F[85,x] = F[4,x]
F[86,x] = F[81,x]
F[87,x] = F[77,x]+F[88,x]
F[88,x] = F[32,x]+F[89,x]+F[93,x]+F[97,x]
F[89,x] = F[4,x]*F[90,x]
F[90,x] = F[91,x]+F[92,x]
F[91,x] = F[81,x]
F[92,x] = F[88,x]
F[93,x] = F[4,x]*F[94,x]
F[94,x] = F[95,x]
F[95,x] = F[4,x]*F[96,x]
F[96,x] = F[11,x]+F[94,x]
F[97,x] = F[4,x]*F[98,x]
F[98,x] = F[101,x]+F[99,x]
F[99,x] = F[100,x]
F[100,x] = F[11,x]*F[4,x]
F[101,x] = F[102,x]
F[102,x] = 2*F[32,x]+F[93,x]+F[97,x]
F[103,x] = F[104,x]*F[4,x]
F[104,x] = F[105,x]+F[106,x]
F[105,x] = F[11,x]+F[94,x]
F[106,x] = F[107,x]+F[108,x]
F[107,x] = F[103,x]+F[32,x]+F[33,x]
F[108,x] = F[109,x]
F[109,x] = F[110,x]*F[4,x]
F[110,x] = F[111,x]+F[17,x]
F[111,x] = F[109,x]
F[112,x] = F[113,x]
F[113,x] = F[114,x]*F[4,x]
F[114,x] = F[115,x]+F[124,x]
F[115,x] = F[116,x]+F[122,x]
F[116,x] = F[117,x]+F[119,x]
F[117,x] = F[11,x]*F[118,x]
F[118,x] = F[11,x]+F[44,x]
F[119,x] = F[112,x]+F[120,x]
F[120,x] = F[121,x]
F[121,x] = F[11,x]^2*F[40,x]
F[122,x] = F[123,x]
F[123,x] = F[11,x]^2*F[29,x]
F[124,x] = F[125,x]
F[125,x] = F[11,x]^2*F[126,x]
F[126,x] = F[127,x]+F[128,x]
F[127,x] = F[10,x]*F[11,x]
F[128,x] = F[10,x]*F[11,x]
F[129,x] = F[130,x]+F[88,x]
F[130,x] = F[100,x]+F[131,x]+F[32,x]
F[131,x] = F[132,x]*F[4,x]
F[132,x] = F[133,x]+F[134,x]
F[133,x] = F[4,x]
F[134,x] = F[130,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_{18}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{41}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
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_{24}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right) F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{36}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{11}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{11}\! \left(x \right)+F_{31}\! \left(x \right)
F_{35}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{11}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{32}\! \left(x \right)+F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{11}\! \left(x \right)+F_{37}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right)
F_{45}\! \left(x \right) = F_{129}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{76}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{59}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{52}\! \left(x \right) = F_{48}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{32}\! \left(x \right)+F_{54}\! \left(x \right)+F_{58}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{59}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{66}\! \left(x \right)
F_{62}\! \left(x \right) = F_{11}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{59}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{68}\! \left(x \right)+F_{72}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{71}\! \left(x \right)
F_{70}\! \left(x \right) = F_{63}\! \left(x \right)
F_{71}\! \left(x \right) = F_{67}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{75}\! \left(x \right)
F_{74}\! \left(x \right) = F_{58}\! \left(x \right)
F_{75}\! \left(x \right) = F_{72}\! \left(x \right)
F_{76}\! \left(x \right) = F_{112}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{103}\! \left(x \right)+F_{32}\! \left(x \right)+F_{78}\! \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_{87}\! \left(x \right)
F_{80}\! \left(x \right) = F_{7}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{32}\! \left(x \right)+F_{82}\! \left(x \right)+F_{83}\! \left(x \right)
F_{82}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \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_{86}\! \left(x \right)
F_{85}\! \left(x \right) = F_{4}\! \left(x \right)
F_{86}\! \left(x \right) = F_{81}\! \left(x \right)
F_{87}\! \left(x \right) = F_{77}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{32}\! \left(x \right)+F_{89}\! \left(x \right)+F_{93}\! \left(x \right)+F_{97}\! \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_{81}\! \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_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{4}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{11}\! \left(x \right)+F_{94}\! \left(x \right)
F_{97}\! \left(x \right) = F_{4}\! \left(x \right) F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{101}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)
F_{100}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)
F_{102}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{93}\! \left(x \right)+F_{97}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{4}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{106}\! \left(x \right)
F_{105}\! \left(x \right) = F_{11}\! \left(x \right)+F_{94}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{108}\! \left(x \right)
F_{107}\! \left(x \right) = F_{103}\! \left(x \right)+F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right) F_{4}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{17}\! \left(x \right)
F_{111}\! \left(x \right) = F_{109}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \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_{124}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{122}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)+F_{119}\! \left(x \right)
F_{117}\! \left(x \right) = F_{11}\! \left(x \right) F_{118}\! \left(x \right)
F_{118}\! \left(x \right) = F_{11}\! \left(x \right)+F_{44}\! \left(x \right)
F_{119}\! \left(x \right) = F_{112}\! \left(x \right)+F_{120}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{40}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)
F_{123}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{29}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)
F_{125}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{126}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)+F_{128}\! \left(x \right)
F_{127}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right)
F_{128}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)+F_{88}\! \left(x \right)
F_{130}\! \left(x \right) = F_{100}\! \left(x \right)+F_{131}\! \left(x \right)+F_{32}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right) F_{4}\! \left(x \right)
F_{132}\! \left(x \right) = F_{133}\! \left(x \right)+F_{134}\! \left(x \right)
F_{133}\! \left(x \right) = F_{4}\! \left(x \right)
F_{134}\! \left(x \right) = F_{130}\! \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_18(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x) + F_17(x))
Eq(F_17(x), F_15(x))
Eq(F_18(x), F_19(x) + F_41(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_24(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_19(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_10(x) + F_24(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_10(x)*F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_36(x))
Eq(F_29(x), F_10(x) + F_30(x))
Eq(F_30(x), F_11(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_35(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_11(x) + F_31(x))
Eq(F_35(x), F_30(x)*F_4(x))
Eq(F_36(x), F_11(x) + F_37(x))
Eq(F_37(x), F_32(x) + F_38(x) + F_39(x))
Eq(F_38(x), F_36(x)*F_4(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_11(x) + F_37(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_10(x)*F_11(x))
Eq(F_45(x), F_129(x) + F_46(x))
Eq(F_46(x), F_47(x) + F_76(x))
Eq(F_47(x), F_48(x) + F_59(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_1(x) + F_4(x))
Eq(F_52(x), F_48(x) + F_53(x))
Eq(F_53(x), F_32(x) + F_54(x) + F_58(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_4(x))
Eq(F_57(x), F_53(x))
Eq(F_58(x), F_11(x)*F_4(x))
Eq(F_59(x), F_32(x) + F_39(x) + F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_66(x))
Eq(F_62(x), F_11(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_4(x) + F_63(x))
Eq(F_66(x), F_59(x) + F_67(x))
Eq(F_67(x), 2*F_32(x) + F_68(x) + F_72(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_71(x))
Eq(F_70(x), F_63(x))
Eq(F_71(x), F_67(x))
Eq(F_72(x), F_4(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_75(x))
Eq(F_74(x), F_58(x))
Eq(F_75(x), F_72(x))
Eq(F_76(x), F_112(x) + F_77(x))
Eq(F_77(x), F_103(x) + F_32(x) + F_78(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_80(x) + F_87(x))
Eq(F_80(x), F_7(x) + F_81(x))
Eq(F_81(x), F_32(x) + F_82(x) + F_83(x))
Eq(F_82(x), F_11(x)*F_4(x))
Eq(F_83(x), F_4(x)*F_84(x))
Eq(F_84(x), F_85(x) + F_86(x))
Eq(F_85(x), F_4(x))
Eq(F_86(x), F_81(x))
Eq(F_87(x), F_77(x) + F_88(x))
Eq(F_88(x), F_32(x) + F_89(x) + F_93(x) + F_97(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_81(x))
Eq(F_92(x), F_88(x))
Eq(F_93(x), F_4(x)*F_94(x))
Eq(F_94(x), F_95(x))
Eq(F_95(x), F_4(x)*F_96(x))
Eq(F_96(x), F_11(x) + F_94(x))
Eq(F_97(x), F_4(x)*F_98(x))
Eq(F_98(x), F_101(x) + F_99(x))
Eq(F_99(x), F_100(x))
Eq(F_100(x), F_11(x)*F_4(x))
Eq(F_101(x), F_102(x))
Eq(F_102(x), 2*F_32(x) + F_93(x) + F_97(x))
Eq(F_103(x), F_104(x)*F_4(x))
Eq(F_104(x), F_105(x) + F_106(x))
Eq(F_105(x), F_11(x) + F_94(x))
Eq(F_106(x), F_107(x) + F_108(x))
Eq(F_107(x), F_103(x) + F_32(x) + F_33(x))
Eq(F_108(x), F_109(x))
Eq(F_109(x), F_110(x)*F_4(x))
Eq(F_110(x), F_111(x) + F_17(x))
Eq(F_111(x), F_109(x))
Eq(F_112(x), F_113(x))
Eq(F_113(x), F_114(x)*F_4(x))
Eq(F_114(x), F_115(x) + F_124(x))
Eq(F_115(x), F_116(x) + F_122(x))
Eq(F_116(x), F_117(x) + F_119(x))
Eq(F_117(x), F_11(x)*F_118(x))
Eq(F_118(x), F_11(x) + F_44(x))
Eq(F_119(x), F_112(x) + F_120(x))
Eq(F_120(x), F_121(x))
Eq(F_121(x), F_11(x)**2*F_40(x))
Eq(F_122(x), F_123(x))
Eq(F_123(x), F_11(x)**2*F_29(x))
Eq(F_124(x), F_125(x))
Eq(F_125(x), F_11(x)**2*F_126(x))
Eq(F_126(x), F_127(x) + F_128(x))
Eq(F_127(x), F_10(x)*F_11(x))
Eq(F_128(x), F_10(x)*F_11(x))
Eq(F_129(x), F_130(x) + F_88(x))
Eq(F_130(x), F_100(x) + F_131(x) + F_32(x))
Eq(F_131(x), F_132(x)*F_4(x))
Eq(F_132(x), F_133(x) + F_134(x))
Eq(F_133(x), F_4(x))
Eq(F_134(x), F_130(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, 2, 1, 3], "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, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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": [1, 2, 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]]}, {"patt": [3, 0, 1, 2], "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": [1, 2, 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]]}, {"patt": [3, 0, 1, 2], "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": "BasicVerificationStrategy"}}, {"children": [{"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]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}], "requirements": []}], "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": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 2], [1, 0], [1, 2]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [3, 0, 1, 2], "pos": [[1, 2], [1, 0], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[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", 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