0132_0231_0321_3120
Counting sequence:
1, 1, 2, 6, 20, 64, 190, 528, 1395, 3550, 8784, 21272, 50645, 118918, 276010, 634304, 1445191, 3267774, 7339044, 16382840, 36371129, 80345574, 176683262, 386922496, 844101355, 1835005374, 3976197240, 8589931288, 18505265565, 39761997830, 85228252754, 182267669632, 388962970255, 828391811198, 1760936584780, 3736621540344, 7915624718721, 16741782511334, 35356170771494, 74560632248640, 157024004331059, 330265805181694, 693791837114272, 1455753395162136, 3051144767064165, 6388162557363334, 13361265300717690, 27918799252470848, 58282912365131735, 121562005566894910, 253327479039569524, 527484106355747192, 1097470934194824265, 2281636161216552678, 4740038607807420750, 9840365185804505984, 20414817110870429051, 42324829298027890494, 87694091944158265544, 181585136975578364440, 375780352907794150189, 777213209293090679814, 1606596116669641221282, 3319261011763112402176, 6854118344887705228575, 14146346861526012349694, 29182749124608510608540, 60173279168440557321208, 124017460407549315461905, 255487405420877289826790, 526101140982196412031094, 1082897664103045518005952, 2228071536198934482324675, 4582466375814631974175358, 9421121133324942201255920, 19361702579765545376083224, 39776492885211021634888181, 81687495420679122906437254, 167700678539666840828693194, 344166069271539743174082240, 706088236518669352351740519, 1448142016170873922651071934, 2969121812973529253078276292, 6085732575940043265470794616, 12470069829324899937094106585, 25544602568457114461541396454, 52312638066364233647885541790, 107101156211299227845569434368, 219212101019081478991121368075, 448559836109812008982979425470, 917623054120288128768975709464, 1876717099556636496747072197912, 3837304636774857507118560183485, 7844207058931812111898300511238, 16031323508745674559943658562290, 32755893439491162073830826824064, 66913135003453374618847462408111, 136658676816791699306664121183614, 279041588375238997004461792926444, 569650488477560587297581002754040, 1162673284896829154185257471014049
Generating function in Maple syntax:
(x^8-2*x^7-14*x^6+49*x^5-77*x^4+68*x^3-34*x^2+9*x-1)/(2*x-1)^3/(x-1)^4
Generating function in latex syntax:
\frac{x^{8}-2 x^{7}-14 x^{6}+49 x^{5}-77 x^{4}+68 x^{3}-34 x^{2}+9 x -1}{\left(2 x -1\right)^{3} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(x**8 - 2*x**7 - 14*x**6 + 49*x**5 - 77*x**4 + 68*x**3 - 34*x**2 + 9*x - 1)/((x - 1)**4*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^3*(x-1)^4*F(x)+x^8-2*x^7-14*x^6+49*x^5-77*x^4+68*x^3-34*x^2+9*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{3} \left(x -1\right)^{4} F \! \left(x \right)+x^{8}-2 x^{7}-14 x^{6}+49 x^{5}-77 x^{4}+68 x^{3}-34 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/96*(9*n^2-21*n+150)*2^n-1/6*n^3-5/6*n-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(9 n^{2}-21 n +150\right) 2^{n}}{96}-\frac{n^{3}}{6}-\frac{5 n}{6}-1 & \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) = 190
a(7) = 528
a(8) = 1395
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)+1/6*n*(n-2)*(n-7), 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) = 190
a \! \left(7\right) = 528
a \! \left(8\right) = 1395
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)+\frac{n \left(n -2\right) \left(n -7\right)}{6}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21217/
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[21,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[21,x]*F[7,x]
F[7,x] = F[140,x]+F[8,x]
F[8,x] = F[87,x]+F[9,x]
F[9,x] = F[10,x]+F[47,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[21,x]
F[13,x] = F[14,x]+F[27,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[21,x]
F[17,x] = F[18,x]+F[22,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[21,x]
F[21,x] = x
F[22,x] = F[21,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[26,x]
F[24,x] = 0
F[25,x] = F[19,x]*F[21,x]
F[26,x] = F[21,x]*F[22,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[29,x]
F[29,x] = F[21,x]*F[30,x]
F[30,x] = F[1,x]+F[28,x]
F[31,x] = F[24,x]+F[32,x]+F[34,x]
F[32,x] = F[21,x]*F[33,x]
F[33,x] = F[15,x]+F[31,x]
F[34,x] = F[21,x]*F[35,x]
F[35,x] = F[36,x]+F[41,x]
F[36,x] = F[28,x]+F[37,x]
F[37,x] = F[24,x]+F[38,x]+F[40,x]
F[38,x] = F[21,x]*F[39,x]
F[39,x] = F[19,x]+F[37,x]
F[40,x] = F[21,x]*F[36,x]
F[41,x] = F[42,x]+F[44,x]
F[42,x] = F[43,x]
F[43,x] = F[21,x]*F[28,x]
F[44,x] = 2*F[24,x]+F[45,x]+F[46,x]
F[45,x] = F[21,x]*F[37,x]
F[46,x] = F[21,x]*F[41,x]
F[47,x] = F[28,x]+F[48,x]
F[48,x] = F[24,x]+F[38,x]+F[49,x]
F[49,x] = F[21,x]*F[50,x]
F[50,x] = F[51,x]+F[66,x]
F[51,x] = F[28,x]+F[52,x]
F[52,x] = F[24,x]+F[38,x]+F[53,x]
F[53,x] = F[21,x]*F[54,x]
F[54,x] = F[36,x]+F[55,x]
F[55,x] = F[56,x]+F[60,x]
F[56,x] = F[24,x]+F[43,x]+F[57,x]
F[57,x] = F[21,x]*F[58,x]
F[58,x] = F[21,x]+F[59,x]
F[59,x] = F[57,x]
F[60,x] = F[24,x]+F[45,x]+F[61,x]+F[65,x]
F[61,x] = F[21,x]*F[62,x]
F[62,x] = F[63,x]+F[64,x]
F[63,x] = F[25,x]
F[64,x] = F[61,x]
F[65,x] = F[21,x]*F[55,x]
F[66,x] = F[67,x]+F[70,x]
F[67,x] = F[24,x]+F[57,x]+F[68,x]
F[68,x] = F[21,x]*F[69,x]
F[69,x] = F[28,x]+F[67,x]
F[70,x] = F[24,x]+F[61,x]+F[71,x]+F[73,x]
F[71,x] = F[21,x]*F[72,x]
F[72,x] = F[52,x]+F[70,x]
F[73,x] = F[21,x]*F[74,x]
F[74,x] = F[75,x]+F[80,x]
F[75,x] = F[67,x]+F[76,x]
F[76,x] = F[24,x]+F[61,x]+F[77,x]+F[79,x]
F[77,x] = F[21,x]*F[78,x]
F[78,x] = F[37,x]+F[76,x]
F[79,x] = F[21,x]*F[75,x]
F[80,x] = F[81,x]+F[83,x]
F[81,x] = F[82,x]
F[82,x] = F[21,x]*F[67,x]
F[83,x] = 2*F[24,x]+F[84,x]+F[85,x]+F[86,x]
F[84,x] = 0
F[85,x] = F[21,x]*F[76,x]
F[86,x] = F[21,x]*F[80,x]
F[87,x] = F[113,x]*F[88,x]
F[88,x] = F[89,x]
F[89,x] = F[21,x]*F[90,x]
F[90,x] = F[91,x]+F[97,x]
F[91,x] = F[1,x]+F[92,x]
F[92,x] = F[93,x]
F[93,x] = F[21,x]*F[94,x]
F[94,x] = F[18,x]+F[95,x]
F[95,x] = F[21,x]+F[96,x]
F[96,x] = F[25,x]
F[97,x] = F[28,x]+F[98,x]
F[98,x] = F[101,x]+F[24,x]+F[99,x]
F[99,x] = F[100,x]*F[21,x]
F[100,x] = F[92,x]+F[98,x]
F[101,x] = F[102,x]*F[21,x]
F[102,x] = F[103,x]+F[108,x]
F[103,x] = F[104,x]+F[28,x]
F[104,x] = F[105,x]+F[107,x]+F[24,x]
F[105,x] = F[106,x]*F[21,x]
F[106,x] = F[104,x]+F[19,x]
F[107,x] = F[103,x]*F[21,x]
F[108,x] = F[109,x]+F[111,x]
F[109,x] = F[110,x]
F[110,x] = F[21,x]*F[28,x]
F[111,x] = F[112,x]
F[112,x] = F[104,x]*F[21,x]
F[113,x] = F[10,x]+F[114,x]
F[114,x] = F[115,x]+F[28,x]
F[115,x] = F[116,x]+F[24,x]+F[38,x]
F[116,x] = F[117,x]*F[21,x]
F[117,x] = F[118,x]+F[120,x]
F[118,x] = F[119,x]+F[28,x]
F[119,x] = F[24,x]+F[34,x]+F[38,x]
F[120,x] = F[121,x]+F[124,x]
F[121,x] = F[122,x]
F[122,x] = F[123,x]*F[21,x]
F[123,x] = F[121,x]+F[28,x]
F[124,x] = 2*F[24,x]+F[125,x]+F[127,x]
F[125,x] = F[126,x]*F[21,x]
F[126,x] = F[119,x]+F[124,x]
F[127,x] = F[128,x]*F[21,x]
F[128,x] = F[129,x]+F[134,x]
F[129,x] = F[121,x]+F[130,x]
F[130,x] = 2*F[24,x]+F[131,x]+F[133,x]
F[131,x] = F[132,x]*F[21,x]
F[132,x] = F[130,x]+F[37,x]
F[133,x] = F[129,x]*F[21,x]
F[134,x] = F[135,x]+F[137,x]
F[135,x] = F[136,x]
F[136,x] = F[121,x]*F[21,x]
F[137,x] = 3*F[24,x]+F[138,x]+F[139,x]
F[138,x] = F[130,x]*F[21,x]
F[139,x] = F[134,x]*F[21,x]
F[140,x] = F[141,x]
F[141,x] = F[142,x]*F[144,x]
F[142,x] = F[143,x]+F[28,x]
F[143,x] = F[1,x]+F[88,x]
F[144,x] = F[145,x]+F[146,x]
F[145,x] = F[19,x]*F[30,x]
F[146,x] = F[147,x]
F[147,x] = F[144,x]*F[18,x]*F[21,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_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{21}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{140}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{87}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{47}\! \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_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{27}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{21}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = x
F_{22}\! \left(x \right) = F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{19}\! \left(x \right) F_{21}\! \left(x \right)
F_{26}\! \left(x \right) = F_{21}\! \left(x \right) F_{22}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{24}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{32}\! \left(x \right) = F_{21}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{21}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)
F_{36}\! \left(x \right) = F_{28}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{24}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)
F_{38}\! \left(x \right) = F_{21}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{19}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{21}\! \left(x \right) F_{36}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{21}\! \left(x \right) F_{28}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{21}\! \left(x \right) F_{37}\! \left(x \right)
F_{46}\! \left(x \right) = F_{21}\! \left(x \right) F_{41}\! \left(x \right)
F_{47}\! \left(x \right) = F_{28}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{24}\! \left(x \right)+F_{38}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{21}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{66}\! \left(x \right)
F_{51}\! \left(x \right) = F_{28}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{24}\! \left(x \right)+F_{38}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{21}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{36}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{60}\! \left(x \right)
F_{56}\! \left(x \right) = F_{24}\! \left(x \right)+F_{43}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{21}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{24}\! \left(x \right)+F_{45}\! \left(x \right)+F_{61}\! \left(x \right)+F_{65}\! \left(x \right)
F_{61}\! \left(x \right) = F_{21}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{64}\! \left(x \right)
F_{63}\! \left(x \right) = F_{25}\! \left(x \right)
F_{64}\! \left(x \right) = F_{61}\! \left(x \right)
F_{65}\! \left(x \right) = F_{21}\! \left(x \right) F_{55}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{70}\! \left(x \right)
F_{67}\! \left(x \right) = F_{24}\! \left(x \right)+F_{57}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{21}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{28}\! \left(x \right)+F_{67}\! \left(x \right)
F_{70}\! \left(x \right) = F_{24}\! \left(x \right)+F_{61}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)
F_{71}\! \left(x \right) = F_{21}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{52}\! \left(x \right)+F_{70}\! \left(x \right)
F_{73}\! \left(x \right) = F_{21}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)+F_{80}\! \left(x \right)
F_{75}\! \left(x \right) = F_{67}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{24}\! \left(x \right)+F_{61}\! \left(x \right)+F_{77}\! \left(x \right)+F_{79}\! \left(x \right)
F_{77}\! \left(x \right) = F_{21}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{37}\! \left(x \right)+F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{21}\! \left(x \right) F_{75}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{83}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{21}\! \left(x \right) F_{67}\! \left(x \right)
F_{83}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{84}\! \left(x \right)+F_{85}\! \left(x \right)+F_{86}\! \left(x \right)
F_{84}\! \left(x \right) = 0
F_{85}\! \left(x \right) = F_{21}\! \left(x \right) F_{76}\! \left(x \right)
F_{86}\! \left(x \right) = F_{21}\! \left(x \right) F_{80}\! \left(x \right)
F_{87}\! \left(x \right) = F_{113}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{21}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)+F_{97}\! \left(x \right)
F_{91}\! \left(x \right) = F_{1}\! \left(x \right)+F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{21}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{18}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{21}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{25}\! \left(x \right)
F_{97}\! \left(x \right) = F_{28}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{101}\! \left(x \right)+F_{24}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right) F_{21}\! \left(x \right)
F_{100}\! \left(x \right) = F_{92}\! \left(x \right)+F_{98}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{21}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{108}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{28}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{107}\! \left(x \right)+F_{24}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{21}\! \left(x \right)
F_{106}\! \left(x \right) = F_{104}\! \left(x \right)+F_{19}\! \left(x \right)
F_{107}\! \left(x \right) = F_{103}\! \left(x \right) F_{21}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{111}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)
F_{110}\! \left(x \right) = F_{21}\! \left(x \right) F_{28}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{104}\! \left(x \right) F_{21}\! \left(x \right)
F_{113}\! \left(x \right) = F_{10}\! \left(x \right)+F_{114}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{28}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{24}\! \left(x \right)+F_{38}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right) F_{21}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)+F_{120}\! \left(x \right)
F_{118}\! \left(x \right) = F_{119}\! \left(x \right)+F_{28}\! \left(x \right)
F_{119}\! \left(x \right) = F_{24}\! \left(x \right)+F_{34}\! \left(x \right)+F_{38}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)+F_{124}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right) F_{21}\! \left(x \right)
F_{123}\! \left(x \right) = F_{121}\! \left(x \right)+F_{28}\! \left(x \right)
F_{124}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{125}\! \left(x \right)+F_{127}\! \left(x \right)
F_{125}\! \left(x \right) = F_{126}\! \left(x \right) F_{21}\! \left(x \right)
F_{126}\! \left(x \right) = F_{119}\! \left(x \right)+F_{124}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right) F_{21}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right)+F_{134}\! \left(x \right)
F_{129}\! \left(x \right) = F_{121}\! \left(x \right)+F_{130}\! \left(x \right)
F_{130}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{131}\! \left(x \right)+F_{133}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right) F_{21}\! \left(x \right)
F_{132}\! \left(x \right) = F_{130}\! \left(x \right)+F_{37}\! \left(x \right)
F_{133}\! \left(x \right) = F_{129}\! \left(x \right) F_{21}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)+F_{137}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right)
F_{136}\! \left(x \right) = F_{121}\! \left(x \right) F_{21}\! \left(x \right)
F_{137}\! \left(x \right) = 3 F_{24}\! \left(x \right)+F_{138}\! \left(x \right)+F_{139}\! \left(x \right)
F_{138}\! \left(x \right) = F_{130}\! \left(x \right) F_{21}\! \left(x \right)
F_{139}\! \left(x \right) = F_{134}\! \left(x \right) F_{21}\! \left(x \right)
F_{140}\! \left(x \right) = F_{141}\! \left(x \right)
F_{141}\! \left(x \right) = F_{142}\! \left(x \right) F_{144}\! \left(x \right)
F_{142}\! \left(x \right) = F_{143}\! \left(x \right)+F_{28}\! \left(x \right)
F_{143}\! \left(x \right) = F_{1}\! \left(x \right)+F_{88}\! \left(x \right)
F_{144}\! \left(x \right) = F_{145}\! \left(x \right)+F_{146}\! \left(x \right)
F_{145}\! \left(x \right) = F_{19}\! \left(x \right) F_{30}\! \left(x \right)
F_{146}\! \left(x \right) = F_{147}\! \left(x \right)
F_{147}\! \left(x \right) = F_{144}\! \left(x \right) F_{18}\! \left(x \right) F_{21}\! \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_21(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_21(x)*F_7(x))
Eq(F_7(x), F_140(x) + F_8(x))
Eq(F_8(x), F_87(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_47(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_21(x))
Eq(F_13(x), F_14(x) + F_27(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_21(x))
Eq(F_17(x), F_18(x) + F_22(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_21(x))
Eq(F_21(x), x)
Eq(F_22(x), F_21(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_26(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_19(x)*F_21(x))
Eq(F_26(x), F_21(x)*F_22(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_21(x)*F_30(x))
Eq(F_30(x), F_1(x) + F_28(x))
Eq(F_31(x), F_24(x) + F_32(x) + F_34(x))
Eq(F_32(x), F_21(x)*F_33(x))
Eq(F_33(x), F_15(x) + F_31(x))
Eq(F_34(x), F_21(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_41(x))
Eq(F_36(x), F_28(x) + F_37(x))
Eq(F_37(x), F_24(x) + F_38(x) + F_40(x))
Eq(F_38(x), F_21(x)*F_39(x))
Eq(F_39(x), F_19(x) + F_37(x))
Eq(F_40(x), F_21(x)*F_36(x))
Eq(F_41(x), F_42(x) + F_44(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_21(x)*F_28(x))
Eq(F_44(x), 2*F_24(x) + F_45(x) + F_46(x))
Eq(F_45(x), F_21(x)*F_37(x))
Eq(F_46(x), F_21(x)*F_41(x))
Eq(F_47(x), F_28(x) + F_48(x))
Eq(F_48(x), F_24(x) + F_38(x) + F_49(x))
Eq(F_49(x), F_21(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_66(x))
Eq(F_51(x), F_28(x) + F_52(x))
Eq(F_52(x), F_24(x) + F_38(x) + F_53(x))
Eq(F_53(x), F_21(x)*F_54(x))
Eq(F_54(x), F_36(x) + F_55(x))
Eq(F_55(x), F_56(x) + F_60(x))
Eq(F_56(x), F_24(x) + F_43(x) + F_57(x))
Eq(F_57(x), F_21(x)*F_58(x))
Eq(F_58(x), F_21(x) + F_59(x))
Eq(F_59(x), F_57(x))
Eq(F_60(x), F_24(x) + F_45(x) + F_61(x) + F_65(x))
Eq(F_61(x), F_21(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_64(x))
Eq(F_63(x), F_25(x))
Eq(F_64(x), F_61(x))
Eq(F_65(x), F_21(x)*F_55(x))
Eq(F_66(x), F_67(x) + F_70(x))
Eq(F_67(x), F_24(x) + F_57(x) + F_68(x))
Eq(F_68(x), F_21(x)*F_69(x))
Eq(F_69(x), F_28(x) + F_67(x))
Eq(F_70(x), F_24(x) + F_61(x) + F_71(x) + F_73(x))
Eq(F_71(x), F_21(x)*F_72(x))
Eq(F_72(x), F_52(x) + F_70(x))
Eq(F_73(x), F_21(x)*F_74(x))
Eq(F_74(x), F_75(x) + F_80(x))
Eq(F_75(x), F_67(x) + F_76(x))
Eq(F_76(x), F_24(x) + F_61(x) + F_77(x) + F_79(x))
Eq(F_77(x), F_21(x)*F_78(x))
Eq(F_78(x), F_37(x) + F_76(x))
Eq(F_79(x), F_21(x)*F_75(x))
Eq(F_80(x), F_81(x) + F_83(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_21(x)*F_67(x))
Eq(F_83(x), 2*F_24(x) + F_84(x) + F_85(x) + F_86(x))
Eq(F_84(x), 0)
Eq(F_85(x), F_21(x)*F_76(x))
Eq(F_86(x), F_21(x)*F_80(x))
Eq(F_87(x), F_113(x)*F_88(x))
Eq(F_88(x), F_89(x))
Eq(F_89(x), F_21(x)*F_90(x))
Eq(F_90(x), F_91(x) + F_97(x))
Eq(F_91(x), F_1(x) + F_92(x))
Eq(F_92(x), F_93(x))
Eq(F_93(x), F_21(x)*F_94(x))
Eq(F_94(x), F_18(x) + F_95(x))
Eq(F_95(x), F_21(x) + F_96(x))
Eq(F_96(x), F_25(x))
Eq(F_97(x), F_28(x) + F_98(x))
Eq(F_98(x), F_101(x) + F_24(x) + F_99(x))
Eq(F_99(x), F_100(x)*F_21(x))
Eq(F_100(x), F_92(x) + F_98(x))
Eq(F_101(x), F_102(x)*F_21(x))
Eq(F_102(x), F_103(x) + F_108(x))
Eq(F_103(x), F_104(x) + F_28(x))
Eq(F_104(x), F_105(x) + F_107(x) + F_24(x))
Eq(F_105(x), F_106(x)*F_21(x))
Eq(F_106(x), F_104(x) + F_19(x))
Eq(F_107(x), F_103(x)*F_21(x))
Eq(F_108(x), F_109(x) + F_111(x))
Eq(F_109(x), F_110(x))
Eq(F_110(x), F_21(x)*F_28(x))
Eq(F_111(x), F_112(x))
Eq(F_112(x), F_104(x)*F_21(x))
Eq(F_113(x), F_10(x) + F_114(x))
Eq(F_114(x), F_115(x) + F_28(x))
Eq(F_115(x), F_116(x) + F_24(x) + F_38(x))
Eq(F_116(x), F_117(x)*F_21(x))
Eq(F_117(x), F_118(x) + F_120(x))
Eq(F_118(x), F_119(x) + F_28(x))
Eq(F_119(x), F_24(x) + F_34(x) + F_38(x))
Eq(F_120(x), F_121(x) + F_124(x))
Eq(F_121(x), F_122(x))
Eq(F_122(x), F_123(x)*F_21(x))
Eq(F_123(x), F_121(x) + F_28(x))
Eq(F_124(x), F_125(x) + F_127(x) + 2*F_24(x))
Eq(F_125(x), F_126(x)*F_21(x))
Eq(F_126(x), F_119(x) + F_124(x))
Eq(F_127(x), F_128(x)*F_21(x))
Eq(F_128(x), F_129(x) + F_134(x))
Eq(F_129(x), F_121(x) + F_130(x))
Eq(F_130(x), F_131(x) + F_133(x) + 2*F_24(x))
Eq(F_131(x), F_132(x)*F_21(x))
Eq(F_132(x), F_130(x) + F_37(x))
Eq(F_133(x), F_129(x)*F_21(x))
Eq(F_134(x), F_135(x) + F_137(x))
Eq(F_135(x), F_136(x))
Eq(F_136(x), F_121(x)*F_21(x))
Eq(F_137(x), F_138(x) + F_139(x) + 3*F_24(x))
Eq(F_138(x), F_130(x)*F_21(x))
Eq(F_139(x), F_134(x)*F_21(x))
Eq(F_140(x), F_141(x))
Eq(F_141(x), F_142(x)*F_144(x))
Eq(F_142(x), F_143(x) + F_28(x))
Eq(F_143(x), F_1(x) + F_88(x))
Eq(F_144(x), F_145(x) + F_146(x))
Eq(F_145(x), F_19(x)*F_30(x))
Eq(F_146(x), F_147(x))
Eq(F_147(x), F_144(x)*F_18(x)*F_21(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "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": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 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