0123_0132_1230_2301
Counting sequence:
1, 1, 2, 6, 20, 64, 189, 514, 1303, 3126, 7196, 16074, 35142, 75676, 161247, 341034, 717473, 1503602, 3141846, 6549778, 13628024, 28308840, 58718801, 121635250, 251661339, 520100398, 1073753248, 2214609978, 4563427802, 9395275380, 19327398675, 39728507098, 81604454661, 167503820010, 343597501946, 704374781378, 1443109187052, 2954937710656, 6047314204325, 12369506110242, 25288767789023, 51677046914854, 105553116742500, 215504279594794, 439804651744558, 897201488992332, 1829587349449607, 3729543442360842, 7599824372258601, 15481123720296034, 31525197392956254, 64176294691559922, 130604389195458080, 265712378016773144, 540431955286591545, 1098878309080770322, 2233785415178392931, 4539628424392366110, 9223372036857984104, 18734974449864798106, 38046409652029837122, 77245740808663013668, 156797324626535863099, 318206335271494878138, 645636042579839889677, 1309718829233384251866, 2656331146614182059266, 5386449269523196275106, 10920472491636062375700, 22136092888451470415088, 44862481587261638705933, 90905554795240680284546, 184172292831916174047399, 373066952146701983437078, 755578637259143246640044, 1530046740449765062633866, 3097872412762487274585110, 6271302689250888859251964, 12693721105953606351003183, 25689673666810869980283370, 51983810243429054531398321, 105176546306472738219795282, 212770944252174734770042598, 430377591782807986218627282, 870426590122533005813226696, 1760195993358900078398605000, 3559077612945468290363112225, 7195526478346272847881094770, 14545795461603218230096541099, 29401075933027781528888022542, 59421121885698253195193873712, 120080183810681886665253150970, 242636247699934533880268744618, 490224255557010588860095993812, 990352031428304219919344697443, 2000511103485174524237032698010, 4040636288227481217270792173589, 8160500738969226772135080470218, 16479457802966982219457198263562, 33275828255991021789288518876034, 67185481812096158279325332898620
Generating function in Maple syntax:
(4*x^8-9*x^7+34*x^6-77*x^5+97*x^4-75*x^3+35*x^2-9*x+1)/(2*x-1)^2/(x-1)^6
Generating function in latex syntax:
\frac{4 x^{8}-9 x^{7}+34 x^{6}-77 x^{5}+97 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:
(4*x**8 - 9*x**7 + 34*x**6 - 77*x**5 + 97*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)-4*x^8+9*x^7-34*x^6+77*x^5-97*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)-4 x^{8}+9 x^{7}-34 x^{6}+77 x^{5}-97 x^{4}+75 x^{3}-35 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-3+1/120*n^5-5/24*n^4+13/24*n^3-67/24*n^2-11/20*n+2^(-1+n)*n+3*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -3+\frac{n^{5}}{120}-\frac{5 n^{4}}{24}+\frac{13 n^{3}}{24}-\frac{67 n^{2}}{24}-\frac{11 n}{20}+2^{-1+n} n +3 \,2^{n} & \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) = 64
a(6) = 189
a(7) = 514
a(8) = 1303
a(n+2) = 1/120*n^5-7/24*n^4+53/24*n^3-137/24*n^2-4*a(n)+4*a(n+1)+467/60*n-2, n >= 9
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) = 64
a \! \left(6\right) = 189
a \! \left(7\right) = 514
a \! \left(8\right) = 1303
a \! \left(n +2\right) = \frac{n^{5}}{120}-\frac{7 n^{4}}{24}+\frac{53 n^{3}}{24}-\frac{137 n^{2}}{24}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{467 n}{60}-2, \quad n \geq 9
Specification 1
Strategy pack name: insertion_point_placements
Tree: http://permpal.com/tree/21140/
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[13,x]*F[4,x]
F[4,x] = F[29,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[5,x]+F[8,x]
F[8,x] = F[13,x]*F[9,x]
F[9,x] = F[10,x]+F[14,x]+F[8,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]
F[13,x] = x
F[14,x] = F[15,x]
F[15,x] = F[10,x]*F[13,x]*F[16,x]
F[16,x] = F[17,x]+F[24,x]
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[23,x]
F[20,x] = 0
F[21,x] = F[13,x]*F[22,x]
F[22,x] = F[11,x]+F[19,x]
F[23,x] = F[13,x]*F[18,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[20,x]+F[26,x]+F[28,x]
F[26,x] = F[13,x]*F[27,x]
F[27,x] = F[11,x]+F[25,x]
F[28,x] = F[13,x]*F[24,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[6,x]
F[31,x] = F[32,x]
F[32,x] = F[13,x]*F[33,x]
F[33,x] = F[34,x]+F[65,x]
F[34,x] = F[35,x]+F[39,x]
F[35,x] = F[36,x]+F[37,x]
F[36,x] = F[10,x]*F[11,x]
F[37,x] = F[38,x]
F[38,x] = F[10,x]^2*F[11,x]*F[13,x]
F[39,x] = F[36,x]+F[40,x]
F[40,x] = F[41,x]+F[63,x]
F[41,x] = F[20,x]+F[21,x]+F[42,x]
F[42,x] = F[13,x]*F[43,x]
F[43,x] = F[44,x]+F[49,x]
F[44,x] = F[11,x]+F[45,x]
F[45,x] = F[20,x]+F[46,x]+F[48,x]
F[46,x] = F[13,x]*F[47,x]
F[47,x] = F[13,x]+F[45,x]
F[48,x] = F[11,x]*F[13,x]
F[49,x] = F[41,x]+F[50,x]
F[50,x] = F[20,x]+F[51,x]+F[55,x]+F[59,x]
F[51,x] = F[13,x]*F[52,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[48,x]
F[54,x] = 2*F[20,x]+F[51,x]+F[55,x]
F[55,x] = F[13,x]*F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[13,x]*F[58,x]
F[58,x] = F[11,x]+F[56,x]
F[59,x] = F[13,x]*F[60,x]
F[60,x] = F[61,x]+F[62,x]
F[61,x] = F[45,x]
F[62,x] = F[50,x]
F[63,x] = F[64,x]
F[64,x] = F[11,x]^3
F[65,x] = F[66,x]+F[69,x]
F[66,x] = F[67,x]+F[68,x]
F[67,x] = F[11,x]*F[18,x]
F[68,x] = F[11,x]*F[37,x]
F[69,x] = F[115,x]+F[70,x]
F[70,x] = F[71,x]+F[87,x]
F[71,x] = F[72,x]
F[72,x] = F[13,x]*F[73,x]
F[73,x] = F[74,x]+F[76,x]
F[74,x] = F[10,x]*F[75,x]
F[75,x] = F[11,x]+F[36,x]
F[76,x] = F[67,x]+F[77,x]
F[77,x] = F[70,x]+F[78,x]
F[78,x] = F[11,x]*F[79,x]
F[79,x] = F[25,x]+F[80,x]
F[80,x] = F[81,x]
F[81,x] = F[13,x]*F[82,x]
F[82,x] = F[83,x]+F[86,x]
F[83,x] = F[11,x]*F[84,x]
F[84,x] = F[36,x]+F[85,x]
F[85,x] = F[10,x]*F[27,x]
F[86,x] = F[10,x]*F[11,x]*F[22,x]
F[87,x] = F[88,x]
F[88,x] = F[13,x]*F[89,x]
F[89,x] = F[105,x]+F[90,x]
F[90,x] = F[91,x]+F[94,x]
F[91,x] = F[92,x]+F[93,x]
F[92,x] = F[11,x]*F[75,x]
F[93,x] = F[11,x]^2*F[10,x]
F[94,x] = F[95,x]+F[97,x]
F[95,x] = F[96,x]
F[96,x] = F[11,x]^3
F[97,x] = F[103,x]+F[98,x]
F[98,x] = F[100,x]+F[99,x]
F[99,x] = F[64,x]
F[100,x] = F[101,x]+F[87,x]
F[101,x] = F[102,x]
F[102,x] = F[11,x]^2*F[19,x]
F[103,x] = F[104,x]
F[104,x] = F[11,x]^2*F[25,x]
F[105,x] = F[106,x]*F[11,x]
F[106,x] = F[107,x]+F[25,x]
F[107,x] = 2*F[20,x]+F[108,x]+F[113,x]
F[108,x] = F[109,x]*F[13,x]
F[109,x] = F[107,x]+F[110,x]
F[110,x] = F[111,x]
F[111,x] = F[112,x]*F[13,x]
F[112,x] = F[11,x]+F[110,x]
F[113,x] = F[114,x]*F[13,x]
F[114,x] = F[107,x]+F[25,x]
F[115,x] = F[116,x]+F[133,x]
F[116,x] = F[117,x]
F[117,x] = F[118,x]*F[13,x]
F[118,x] = F[119,x]+F[123,x]
F[119,x] = F[120,x]+F[121,x]
F[120,x] = F[10,x]*F[25,x]
F[121,x] = F[116,x]+F[122,x]
F[122,x] = F[104,x]
F[123,x] = F[124,x]+F[128,x]
F[124,x] = F[10,x]*F[125,x]
F[125,x] = F[126,x]+F[25,x]
F[126,x] = F[127,x]
F[127,x] = F[11,x]^3
F[128,x] = F[129,x]+F[130,x]
F[129,x] = F[10,x]*F[107,x]
F[130,x] = F[131,x]
F[131,x] = F[11,x]^2*F[132,x]
F[132,x] = F[19,x]+F[63,x]
F[133,x] = F[134,x]
F[134,x] = F[11,x]*F[80,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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{29}\! \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_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{5}\! \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_{10}\! \left(x \right)+F_{14}\! \left(x \right)+F_{8}\! \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_{13}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{24}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{13}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{23}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{20}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right) F_{24}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{6}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{65}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{39}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10} \left(x \right)^{2} F_{11}\! \left(x \right) F_{13}\! \left(x \right)
F_{39}\! \left(x \right) = F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{63}\! \left(x \right)
F_{41}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{13}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{49}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{20}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{13}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{13}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right)
F_{49}\! \left(x \right) = F_{41}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{20}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)+F_{59}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{48}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{13}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{13}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{11}\! \left(x \right)+F_{56}\! \left(x \right)
F_{59}\! \left(x \right) = F_{13}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{62}\! \left(x \right)
F_{61}\! \left(x \right) = F_{45}\! \left(x \right)
F_{62}\! \left(x \right) = F_{50}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{11} \left(x \right)^{3}
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{69}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{68}\! \left(x \right)
F_{67}\! \left(x \right) = F_{11}\! \left(x \right) F_{18}\! \left(x \right)
F_{68}\! \left(x \right) = F_{11}\! \left(x \right) F_{37}\! \left(x \right)
F_{69}\! \left(x \right) = F_{115}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{87}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{13}\! \left(x \right) F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)+F_{76}\! \left(x \right)
F_{74}\! \left(x \right) = F_{10}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{11}\! \left(x \right)+F_{36}\! \left(x \right)
F_{76}\! \left(x \right) = F_{67}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{70}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{11}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{25}\! \left(x \right)+F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{13}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{83}\! \left(x \right)+F_{86}\! \left(x \right)
F_{83}\! \left(x \right) = F_{11}\! \left(x \right) F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{36}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{10}\! \left(x \right) F_{27}\! \left(x \right)
F_{86}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{22}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{13}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{105}\! \left(x \right)+F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)+F_{94}\! \left(x \right)
F_{91}\! \left(x \right) = F_{92}\! \left(x \right)+F_{93}\! \left(x \right)
F_{92}\! \left(x \right) = F_{11}\! \left(x \right) F_{75}\! \left(x \right)
F_{93}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{10}\! \left(x \right)
F_{94}\! \left(x \right) = F_{95}\! \left(x \right)+F_{97}\! \left(x \right)
F_{95}\! \left(x \right) = F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{11} \left(x \right)^{3}
F_{97}\! \left(x \right) = F_{103}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{100}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{64}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{87}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)
F_{102}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{19}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)
F_{104}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{25}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{11}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{25}\! \left(x \right)
F_{107}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{108}\! \left(x \right)+F_{113}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right) F_{13}\! \left(x \right)
F_{109}\! \left(x \right) = F_{107}\! \left(x \right)+F_{110}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right) F_{13}\! \left(x \right)
F_{112}\! \left(x \right) = F_{11}\! \left(x \right)+F_{110}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right) F_{13}\! \left(x \right)
F_{114}\! \left(x \right) = F_{107}\! \left(x \right)+F_{25}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{133}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right) F_{13}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right)+F_{123}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right)+F_{121}\! \left(x \right)
F_{120}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right)
F_{121}\! \left(x \right) = F_{116}\! \left(x \right)+F_{122}\! \left(x \right)
F_{122}\! \left(x \right) = F_{104}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right)+F_{128}\! \left(x \right)
F_{124}\! \left(x \right) = F_{10}\! \left(x \right) F_{125}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right)+F_{25}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)
F_{127}\! \left(x \right) = F_{11} \left(x \right)^{3}
F_{128}\! \left(x \right) = F_{129}\! \left(x \right)+F_{130}\! \left(x \right)
F_{129}\! \left(x \right) = F_{10}\! \left(x \right) F_{107}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right)
F_{131}\! \left(x \right) = F_{11} \left(x \right)^{2} F_{132}\! \left(x \right)
F_{132}\! \left(x \right) = F_{19}\! \left(x \right)+F_{63}\! \left(x \right)
F_{133}\! \left(x \right) = F_{134}\! \left(x \right)
F_{134}\! \left(x \right) = F_{11}\! \left(x \right) F_{80}\! \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_13(x)*F_4(x))
Eq(F_4(x), F_29(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_13(x)*F_7(x))
Eq(F_7(x), F_5(x) + F_8(x))
Eq(F_8(x), F_13(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_14(x) + F_8(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_13(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_10(x)*F_13(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_24(x))
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_23(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_13(x)*F_22(x))
Eq(F_22(x), F_11(x) + F_19(x))
Eq(F_23(x), F_13(x)*F_18(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_20(x) + F_26(x) + F_28(x))
Eq(F_26(x), F_13(x)*F_27(x))
Eq(F_27(x), F_11(x) + F_25(x))
Eq(F_28(x), F_13(x)*F_24(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_6(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_13(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_65(x))
Eq(F_34(x), F_35(x) + F_39(x))
Eq(F_35(x), F_36(x) + F_37(x))
Eq(F_36(x), F_10(x)*F_11(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_10(x)**2*F_11(x)*F_13(x))
Eq(F_39(x), F_36(x) + F_40(x))
Eq(F_40(x), F_41(x) + F_63(x))
Eq(F_41(x), F_20(x) + F_21(x) + F_42(x))
Eq(F_42(x), F_13(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_49(x))
Eq(F_44(x), F_11(x) + F_45(x))
Eq(F_45(x), F_20(x) + F_46(x) + F_48(x))
Eq(F_46(x), F_13(x)*F_47(x))
Eq(F_47(x), F_13(x) + F_45(x))
Eq(F_48(x), F_11(x)*F_13(x))
Eq(F_49(x), F_41(x) + F_50(x))
Eq(F_50(x), F_20(x) + F_51(x) + F_55(x) + F_59(x))
Eq(F_51(x), F_13(x)*F_52(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_48(x))
Eq(F_54(x), 2*F_20(x) + F_51(x) + F_55(x))
Eq(F_55(x), F_13(x)*F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_13(x)*F_58(x))
Eq(F_58(x), F_11(x) + F_56(x))
Eq(F_59(x), F_13(x)*F_60(x))
Eq(F_60(x), F_61(x) + F_62(x))
Eq(F_61(x), F_45(x))
Eq(F_62(x), F_50(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_11(x)**3)
Eq(F_65(x), F_66(x) + F_69(x))
Eq(F_66(x), F_67(x) + F_68(x))
Eq(F_67(x), F_11(x)*F_18(x))
Eq(F_68(x), F_11(x)*F_37(x))
Eq(F_69(x), F_115(x) + F_70(x))
Eq(F_70(x), F_71(x) + F_87(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_13(x)*F_73(x))
Eq(F_73(x), F_74(x) + F_76(x))
Eq(F_74(x), F_10(x)*F_75(x))
Eq(F_75(x), F_11(x) + F_36(x))
Eq(F_76(x), F_67(x) + F_77(x))
Eq(F_77(x), F_70(x) + F_78(x))
Eq(F_78(x), F_11(x)*F_79(x))
Eq(F_79(x), F_25(x) + F_80(x))
Eq(F_80(x), F_81(x))
Eq(F_81(x), F_13(x)*F_82(x))
Eq(F_82(x), F_83(x) + F_86(x))
Eq(F_83(x), F_11(x)*F_84(x))
Eq(F_84(x), F_36(x) + F_85(x))
Eq(F_85(x), F_10(x)*F_27(x))
Eq(F_86(x), F_10(x)*F_11(x)*F_22(x))
Eq(F_87(x), F_88(x))
Eq(F_88(x), F_13(x)*F_89(x))
Eq(F_89(x), F_105(x) + F_90(x))
Eq(F_90(x), F_91(x) + F_94(x))
Eq(F_91(x), F_92(x) + F_93(x))
Eq(F_92(x), F_11(x)*F_75(x))
Eq(F_93(x), F_10(x)*F_11(x)**2)
Eq(F_94(x), F_95(x) + F_97(x))
Eq(F_95(x), F_96(x))
Eq(F_96(x), F_11(x)**3)
Eq(F_97(x), F_103(x) + F_98(x))
Eq(F_98(x), F_100(x) + F_99(x))
Eq(F_99(x), F_64(x))
Eq(F_100(x), F_101(x) + F_87(x))
Eq(F_101(x), F_102(x))
Eq(F_102(x), F_11(x)**2*F_19(x))
Eq(F_103(x), F_104(x))
Eq(F_104(x), F_11(x)**2*F_25(x))
Eq(F_105(x), F_106(x)*F_11(x))
Eq(F_106(x), F_107(x) + F_25(x))
Eq(F_107(x), F_108(x) + F_113(x) + 2*F_20(x))
Eq(F_108(x), F_109(x)*F_13(x))
Eq(F_109(x), F_107(x) + F_110(x))
Eq(F_110(x), F_111(x))
Eq(F_111(x), F_112(x)*F_13(x))
Eq(F_112(x), F_11(x) + F_110(x))
Eq(F_113(x), F_114(x)*F_13(x))
Eq(F_114(x), F_107(x) + F_25(x))
Eq(F_115(x), F_116(x) + F_133(x))
Eq(F_116(x), F_117(x))
Eq(F_117(x), F_118(x)*F_13(x))
Eq(F_118(x), F_119(x) + F_123(x))
Eq(F_119(x), F_120(x) + F_121(x))
Eq(F_120(x), F_10(x)*F_25(x))
Eq(F_121(x), F_116(x) + F_122(x))
Eq(F_122(x), F_104(x))
Eq(F_123(x), F_124(x) + F_128(x))
Eq(F_124(x), F_10(x)*F_125(x))
Eq(F_125(x), F_126(x) + F_25(x))
Eq(F_126(x), F_127(x))
Eq(F_127(x), F_11(x)**3)
Eq(F_128(x), F_129(x) + F_130(x))
Eq(F_129(x), F_10(x)*F_107(x))
Eq(F_130(x), F_131(x))
Eq(F_131(x), F_11(x)**2*F_132(x))
Eq(F_132(x), F_19(x) + F_63(x))
Eq(F_133(x), F_134(x))
Eq(F_134(x), F_11(x)*F_80(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "insertion_point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [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]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [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]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [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]]}], "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, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"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], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 2], [0, 2], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 2], [0, 2], [0, 2]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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]]}]]}, "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, 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", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], 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