0123_0231_1032_1302_3021
Counting sequence:
1, 1, 2, 6, 19, 52, 135, 352, 918, 2396, 6263, 16385, 42883, 112254, 293868, 769338, 2014133, 5273047, 13804993, 36141916, 94620738, 247720280, 648540083, 1697899949, 4445159743, 11637579258, 30467578008, 79765154742, 208827886193, 546718503811, 1431327625213, 3747264371800, 9810465490158, 25684132098644, 67241930805743, 176041660318553, 460883050149883, 1206607490131062, 3158939420243268, 8270210770598706, 21651692891552813, 56684867904059695, 148402910820626233, 388523864557818964, 1017168682852830618, 2662982184000672848, 6971777869149187883, 18252351423446890757, 47785276401191484343, 125103477780127562226, 327525156939191202288, 857471993037446044590, 2244890822173146931433, 5877200473481994749659, 15386710598272837317493, 40282931321336517202768, 105462083365736714290758, 276103318775873625669452, 722847872961884162717543, 1892440300109778862483121, 4954473027367452424731763, 12970978781992578411712110, 33958463318610282810404508, 88904411173838270019501354, 232754770202904527248099493, 609359899434875311724797063, 1595324928101721407926291633, 4176614884870288912054077772, 10934519726509145328235941618, 28626944294657147072653747016, 74946313157462295889725299363, 196211995177729740596522151005, 513689672375726925899841153583, 1344857021949451037103001309674, 3520881393472626185409162775368, 9217787158468427519124487016358, 24132480081932656371964298273633, 63179653087329541596768407804467, 165406479180055968418340925139693, 433039784452838363658254367614536, 1133712874178459122556422177703838, 2968098838082539004011012165496900, 7770583640069157889476614318786783, 20343652082124934664418830790863369, 53260372606305646103779878053803243, 139437465736792003646920803370546278, 365052024604070364836982532057835508, 955718608075419090864026792802960162, 2502103799622186907755097846351044893, 6550592790791141632401266746250174431, 17149674572751237989448702392399478313, 44898430927462572335944840430948260420, 117545618209636479018385818900445302858, 307738423701446864719212616270387648064, 805669652894704115139252029910717641243, 2109270534982665480698543473461765275573, 5522141952053292326956378390474578185383, 14457155321177211500170591697961969280482, 37849324011478342173555396703411329655968, 99090816713257815020495598412272019687326, 259423126128295102887931398533404729405913
Generating function in Maple syntax:
(2*x^10-x^9-3*x^8+4*x^7-x^6-4*x^5+x^4-x^3+5*x^2-4*x+1)/(x^2-3*x+1)/(x-1)^2
Generating function in latex syntax:
\frac{2 x^{10}-x^{9}-3 x^{8}+4 x^{7}-x^{6}-4 x^{5}+x^{4}-x^{3}+5 x^{2}-4 x +1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(2*x**10 - x**9 - 3*x**8 + 4*x**7 - x**6 - 4*x**5 + x**4 - x**3 + 5*x**2 - 4*x + 1)/((x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)*(x-1)^2*F(x)+2*x^10-x^9-3*x^8+4*x^7-x^6-4*x^5+x^4-x^3+5*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+2 x^{10}-x^{9}-3 x^{8}+4 x^{7}-x^{6}-4 x^{5}+x^{4}-x^{3}+5 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1338,n = 1,511,n = 2,194,n = 3,72,n = 4,26,n = 5,9,n = 6,2,0)+1/10*(2987*5^(1/2)-6675)*(3/2-1/2*5^(1/2))^(-n)+1/10*(-2987*5^(1/2)-6675)*(3/2+1/2*5^(1/2))^(-n)+n-2
Explicit closed form in latex syntax:
\left(\left\{\begin{array}{cc}1338 & n =0 \\ 511 & n =1 \\ 194 & n =2 \\ 72 & n =3 \\ 26 & n =4 \\ 9 & n =5 \\ 2 & n =6 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)+\frac{\left(2987 \sqrt{5}-6675\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(-2987 \sqrt{5}-6675\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+n -2
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 52
a(6) = 135
a(7) = 352
a(8) = 918
a(9) = 2396
a(10) = 6263
a(n+2) = -a(n)+3*a(n+1)+1-n, 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) = 52
a \! \left(6\right) = 135
a \! \left(7\right) = 352
a \! \left(8\right) = 918
a \! \left(9\right) = 2396
a \! \left(10\right) = 6263
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)+1-n, \quad n \geq 11
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17235/
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[25,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[24,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[4,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[4,x]
F[23,x] = F[19,x]
F[24,x] = F[20,x]
F[25,x] = F[26,x]+F[62,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[1,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[17,x]*F[4,x]
F[32,x] = F[26,x]+F[33,x]
F[33,x] = F[34,x]+F[35,x]+F[60,x]
F[34,x] = 0
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[30,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[22,x]*F[4,x]
F[40,x] = F[33,x]+F[41,x]
F[41,x] = 2*F[34,x]+F[42,x]+F[46,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[38,x]
F[45,x] = F[41,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[34,x]+F[49,x]+F[59,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[53,x]
F[51,x] = F[4,x]+F[52,x]
F[52,x] = F[39,x]
F[53,x] = F[48,x]+F[54,x]
F[54,x] = 2*F[34,x]+F[46,x]+F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[52,x]
F[58,x] = F[54,x]
F[59,x] = F[26,x]*F[4,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[26,x]+F[48,x]
F[62,x] = F[112,x]+F[34,x]+F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[73,x]
F[65,x] = F[30,x]+F[66,x]
F[66,x] = F[34,x]+F[67,x]+F[71,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[4,x]+F[69,x]
F[69,x] = F[70,x]
F[70,x] = x^2
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[30,x]+F[52,x]
F[73,x] = F[101,x]+F[74,x]
F[74,x] = F[34,x]+F[63,x]+F[75,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[26,x]+F[77,x]
F[77,x] = F[34,x]+F[59,x]+F[78,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[80,x]+F[83,x]
F[80,x] = F[4,x]+F[81,x]
F[81,x] = F[34,x]+F[67,x]+F[82,x]
F[82,x] = F[30,x]*F[4,x]
F[83,x] = F[77,x]+F[84,x]
F[84,x] = F[34,x]+F[85,x]+F[92,x]+F[97,x]
F[85,x] = F[4,x]*F[86,x]
F[86,x] = F[87,x]+F[88,x]
F[87,x] = F[52,x]
F[88,x] = F[89,x]
F[89,x] = 2*F[34,x]+F[85,x]+F[90,x]
F[90,x] = F[4,x]*F[91,x]
F[91,x] = F[77,x]
F[92,x] = F[4,x]*F[93,x]
F[93,x] = F[77,x]+F[94,x]
F[94,x] = F[95,x]
F[95,x] = F[4,x]*F[96,x]
F[96,x] = F[59,x]
F[97,x] = F[4,x]*F[98,x]
F[98,x] = F[99,x]
F[99,x] = F[100,x]*F[4,x]
F[100,x] = F[26,x]+F[96,x]
F[101,x] = F[102,x]+F[107,x]+F[34,x]+F[92,x]
F[102,x] = F[103,x]*F[4,x]
F[103,x] = F[104,x]+F[105,x]
F[104,x] = F[38,x]
F[105,x] = F[106,x]
F[106,x] = 2*F[34,x]+F[102,x]+F[90,x]
F[107,x] = F[108,x]*F[4,x]
F[108,x] = F[109,x]+F[98,x]
F[109,x] = F[110,x]
F[110,x] = F[111,x]*F[4,x]
F[111,x] = F[77,x]
F[112,x] = F[113,x]*F[4,x]
F[113,x] = F[114,x]+F[119,x]
F[114,x] = F[115,x]+F[26,x]
F[115,x] = F[116,x]+F[34,x]+F[78,x]
F[116,x] = F[117,x]*F[4,x]
F[117,x] = F[118,x]+F[26,x]
F[118,x] = F[116,x]
F[119,x] = F[120,x]+F[129,x]
F[120,x] = F[121,x]
F[121,x] = F[122,x]*F[4,x]
F[122,x] = F[100,x]+F[123,x]
F[123,x] = F[120,x]+F[124,x]
F[124,x] = F[125,x]
F[125,x] = F[126,x]*F[4,x]
F[126,x] = F[127,x]+F[128,x]
F[127,x] = F[96,x]
F[128,x] = F[124,x]
F[129,x] = F[130,x]
F[130,x] = F[131,x]*F[4,x]
F[131,x] = F[111,x]+F[132,x]
F[132,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_{25}\! \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_{24}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \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_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)
F_{24}\! \left(x \right) = F_{20}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{62}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{26}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{60}\! \left(x \right)
F_{34}\! \left(x \right) = 0
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{30}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{33}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{42}\! \left(x \right)+F_{46}\! \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_{38}\! \left(x \right)
F_{45}\! \left(x \right) = F_{41}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{34}\! \left(x \right)+F_{49}\! \left(x \right)+F_{59}\! \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_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{39}\! \left(x \right)
F_{53}\! \left(x \right) = F_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{46}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{52}\! \left(x \right)
F_{58}\! \left(x \right) = F_{54}\! \left(x \right)
F_{59}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{26}\! \left(x \right)+F_{48}\! \left(x \right)
F_{62}\! \left(x \right) = F_{112}\! \left(x \right)+F_{34}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{73}\! \left(x \right)
F_{65}\! \left(x \right) = F_{30}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{34}\! \left(x \right)+F_{67}\! \left(x \right)+F_{71}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \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_{70}\! \left(x \right) = x^{2}
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{30}\! \left(x \right)+F_{52}\! \left(x \right)
F_{73}\! \left(x \right) = F_{101}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{34}\! \left(x \right)+F_{63}\! \left(x \right)+F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{26}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{34}\! \left(x \right)+F_{59}\! \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_{83}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{34}\! \left(x \right)+F_{67}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{83}\! \left(x \right) = F_{77}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{34}\! \left(x \right)+F_{85}\! \left(x \right)+F_{92}\! \left(x \right)+F_{97}\! \left(x \right)
F_{85}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)+F_{88}\! \left(x \right)
F_{87}\! \left(x \right) = F_{52}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{85}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{4}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{77}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{77}\! \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_{59}\! \left(x \right)
F_{97}\! \left(x \right) = F_{4}\! \left(x \right) F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right) F_{4}\! \left(x \right)
F_{100}\! \left(x \right) = F_{26}\! \left(x \right)+F_{96}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{107}\! \left(x \right)+F_{34}\! \left(x \right)+F_{92}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right) F_{4}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{105}\! \left(x \right)
F_{104}\! \left(x \right) = F_{38}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)
F_{106}\! \left(x \right) = 2 F_{34}\! \left(x \right)+F_{102}\! \left(x \right)+F_{90}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{4}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{98}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right) F_{4}\! \left(x \right)
F_{111}\! \left(x \right) = F_{77}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{4}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)+F_{119}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{26}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{34}\! \left(x \right)+F_{78}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right) F_{4}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)+F_{26}\! \left(x \right)
F_{118}\! \left(x \right) = F_{116}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)+F_{129}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right) F_{4}\! \left(x \right)
F_{122}\! \left(x \right) = F_{100}\! \left(x \right)+F_{123}\! \left(x \right)
F_{123}\! \left(x \right) = F_{120}\! \left(x \right)+F_{124}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right) F_{4}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)+F_{128}\! \left(x \right)
F_{127}\! \left(x \right) = F_{96}\! \left(x \right)
F_{128}\! \left(x \right) = F_{124}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right) F_{4}\! \left(x \right)
F_{131}\! \left(x \right) = F_{111}\! \left(x \right)+F_{132}\! \left(x \right)
F_{132}\! \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_25(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_24(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_4(x))
Eq(F_18(x), F_14(x) + F_19(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_23(x))
Eq(F_22(x), F_4(x))
Eq(F_23(x), F_19(x))
Eq(F_24(x), F_20(x))
Eq(F_25(x), F_26(x) + F_62(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_1(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_17(x)*F_4(x))
Eq(F_32(x), F_26(x) + F_33(x))
Eq(F_33(x), F_34(x) + F_35(x) + F_60(x))
Eq(F_34(x), 0)
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_30(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_22(x)*F_4(x))
Eq(F_40(x), F_33(x) + F_41(x))
Eq(F_41(x), 2*F_34(x) + F_42(x) + F_46(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_38(x))
Eq(F_45(x), F_41(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_34(x) + F_49(x) + F_59(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_53(x))
Eq(F_51(x), F_4(x) + F_52(x))
Eq(F_52(x), F_39(x))
Eq(F_53(x), F_48(x) + F_54(x))
Eq(F_54(x), 2*F_34(x) + F_46(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_52(x))
Eq(F_58(x), F_54(x))
Eq(F_59(x), F_26(x)*F_4(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_26(x) + F_48(x))
Eq(F_62(x), F_112(x) + F_34(x) + F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_73(x))
Eq(F_65(x), F_30(x) + F_66(x))
Eq(F_66(x), F_34(x) + F_67(x) + F_71(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_4(x) + F_69(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), x**2)
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_30(x) + F_52(x))
Eq(F_73(x), F_101(x) + F_74(x))
Eq(F_74(x), F_34(x) + F_63(x) + F_75(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_26(x) + F_77(x))
Eq(F_77(x), F_34(x) + F_59(x) + F_78(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_80(x) + F_83(x))
Eq(F_80(x), F_4(x) + F_81(x))
Eq(F_81(x), F_34(x) + F_67(x) + F_82(x))
Eq(F_82(x), F_30(x)*F_4(x))
Eq(F_83(x), F_77(x) + F_84(x))
Eq(F_84(x), F_34(x) + F_85(x) + F_92(x) + F_97(x))
Eq(F_85(x), F_4(x)*F_86(x))
Eq(F_86(x), F_87(x) + F_88(x))
Eq(F_87(x), F_52(x))
Eq(F_88(x), F_89(x))
Eq(F_89(x), 2*F_34(x) + F_85(x) + F_90(x))
Eq(F_90(x), F_4(x)*F_91(x))
Eq(F_91(x), F_77(x))
Eq(F_92(x), F_4(x)*F_93(x))
Eq(F_93(x), F_77(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_59(x))
Eq(F_97(x), F_4(x)*F_98(x))
Eq(F_98(x), F_99(x))
Eq(F_99(x), F_100(x)*F_4(x))
Eq(F_100(x), F_26(x) + F_96(x))
Eq(F_101(x), F_102(x) + F_107(x) + F_34(x) + F_92(x))
Eq(F_102(x), F_103(x)*F_4(x))
Eq(F_103(x), F_104(x) + F_105(x))
Eq(F_104(x), F_38(x))
Eq(F_105(x), F_106(x))
Eq(F_106(x), F_102(x) + 2*F_34(x) + F_90(x))
Eq(F_107(x), F_108(x)*F_4(x))
Eq(F_108(x), F_109(x) + F_98(x))
Eq(F_109(x), F_110(x))
Eq(F_110(x), F_111(x)*F_4(x))
Eq(F_111(x), F_77(x))
Eq(F_112(x), F_113(x)*F_4(x))
Eq(F_113(x), F_114(x) + F_119(x))
Eq(F_114(x), F_115(x) + F_26(x))
Eq(F_115(x), F_116(x) + F_34(x) + F_78(x))
Eq(F_116(x), F_117(x)*F_4(x))
Eq(F_117(x), F_118(x) + F_26(x))
Eq(F_118(x), F_116(x))
Eq(F_119(x), F_120(x) + F_129(x))
Eq(F_120(x), F_121(x))
Eq(F_121(x), F_122(x)*F_4(x))
Eq(F_122(x), F_100(x) + F_123(x))
Eq(F_123(x), F_120(x) + F_124(x))
Eq(F_124(x), F_125(x))
Eq(F_125(x), F_126(x)*F_4(x))
Eq(F_126(x), F_127(x) + F_128(x))
Eq(F_127(x), F_96(x))
Eq(F_128(x), F_124(x))
Eq(F_129(x), F_130(x))
Eq(F_130(x), F_131(x)*F_4(x))
Eq(F_131(x), F_111(x) + F_132(x))
Eq(F_132(x), F_130(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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], "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, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 1], [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, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[1, 2], [1, 2], [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, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 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