0132_1302_2310
Counting sequence:
1, 1, 2, 6, 21, 73, 239, 734, 2133, 5924, 15859, 41202, 104433, 259312, 632815, 1521646, 3612653, 8482796, 19726315, 45481962, 104071145, 236519400, 534249447, 1200095206, 2682257381, 5967445988, 13220446179, 29175578594, 64156073953, 140609847264, 307224379359, 669343809502, 1454383300573, 3152237559772, 6816113098715, 14705968021466, 31662498906073, 68036576935896, 145925808848855, 312433100980182, 667815874920405, 1425173228027860, 3036851115917267, 6461829836439506, 13730701207666641, 29138157647691728, 61757369108922319, 130736330589077454, 276443611501952973, 583902247000211404, 1232015973062541259, 2596888135132512202, 5468495847534624713, 11504726723094642632, 24182078199165878215, 50784841198043398086, 106564174382840676293, 223428081113477676996, 468086130870379872195, 979911221321782001602, 2049894435190973923265, 4285193059841536425920, 8951859022519863345087, 18688281128924686057406, 38989499442794351034301, 81293648211333387124668, 169396450828874812489659, 352775533665621465104314, 734254201109934992523193, 1527408856047224583356344, 3175643885777246729797559, 6599064437952560156901302, 13706078420718786527100853, 28453143503005150470995892, 59039025768836304461627315, 122446240534327030745923506, 253836643187209169526587313, 525981901228503953546674096, 1089431058132095672218812335, 2255515569014131425546338222, 4667813705259535072187383725, 9656219426308124618642161580, 19967827762574830078621908907, 41275145333282669282818260906, 85287298722172859017171304361, 176165879360063377967358672808, 363751272012206543118991884199, 750820305233140053759716491174, 1549243273539425745902663368613, 3195645497349640009318112690084, 6589554829116079203288038244259, 13583606566068656411399365328802, 27992300168321018776018996952993, 57667270335040690690694628507552, 118765491488960812813272179802015, 244525344199884024179724411469726, 503306805892178487605187137503133, 1035665588626306279501706922295196, 2130524518171082302227990368485275, 4381634299688874926273354629382042, 9008875903171969299086695504805785
Generating function in Maple syntax:
(x^7+2*x^6-27*x^5+59*x^4-61*x^3+33*x^2-9*x+1)/(x-1)^2/(2*x-1)^4
Generating function in latex syntax:
\frac{x^{7}+2 x^{6}-27 x^{5}+59 x^{4}-61 x^{3}+33 x^{2}-9 x +1}{\left(x -1\right)^{2} \left(2 x -1\right)^{4}}
Generating function in sympy syntax:
(x**7 + 2*x**6 - 27*x**5 + 59*x**4 - 61*x**3 + 33*x**2 - 9*x + 1)/((x - 1)**2*(2*x - 1)**4)
Implicit equation for the generating function in Maple syntax:
-(x-1)^2*(2*x-1)^4*F(x)+x^7+2*x^6-27*x^5+59*x^4-61*x^3+33*x^2-9*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{2} \left(2 x -1\right)^{4} F \! \left(x \right)+x^{7}+2 x^{6}-27 x^{5}+59 x^{4}-61 x^{3}+33 x^{2}-9 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/192*(n^3+39*n^2-262*n+696)*2^n-n-3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(n^{3}+39 n^{2}-262 n +696\right) 2^{n}}{192}-n -3 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 73
a(6) = 239
a(7) = 734
a(n+4) = -16*a(n)+32*a(n+1)-24*a(n+2)+8*a(n+3)+1-n, n >= 8
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 21
a \! \left(5\right) = 73
a \! \left(6\right) = 239
a \! \left(7\right) = 734
a \! \left(n +4\right) = -16 a \! \left(n \right)+32 a \! \left(n +1\right)-24 a \! \left(n +2\right)+8 a \! \left(n +3\right)+1-n, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14733/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[12,x]*F[4,x]
F[4,x] = F[38,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[26,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]+F[23,x]+F[25,x]
F[22,x] = 0
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[10,x]+F[21,x]
F[25,x] = F[12,x]*F[17,x]
F[26,x] = F[22,x]+F[27,x]+F[37,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[10,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[29,x]
F[32,x] = F[21,x]+F[33,x]
F[33,x] = 2*F[22,x]+F[34,x]+F[36,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[30,x]+F[33,x]
F[36,x] = F[12,x]*F[32,x]
F[37,x] = F[12,x]*F[13,x]
F[38,x] = F[39,x]+F[57,x]
F[39,x] = F[40,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[1,x]+F[14,x]
F[43,x] = F[18,x]+F[44,x]
F[44,x] = F[22,x]+F[45,x]+F[47,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[14,x]+F[44,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[17,x]+F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[18,x]+F[50,x]
F[53,x] = 2*F[22,x]+F[54,x]+F[56,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[21,x]+F[53,x]
F[56,x] = F[12,x]*F[49,x]
F[57,x] = F[105,x]+F[22,x]+F[58,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[60,x]+F[72,x]
F[60,x] = F[6,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[29,x]+F[64,x]
F[64,x] = F[65,x]+F[68,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[67,x]
F[67,x] = F[10,x]+F[65,x]
F[68,x] = 2*F[22,x]+F[69,x]+F[71,x]
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[30,x]+F[68,x]
F[71,x] = F[12,x]*F[64,x]
F[72,x] = F[102,x]+F[73,x]
F[73,x] = F[22,x]+F[74,x]+F[76,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[6,x]+F[73,x]
F[76,x] = F[12,x]*F[77,x]
F[77,x] = F[17,x]+F[78,x]
F[78,x] = F[44,x]+F[79,x]
F[79,x] = F[101,x]+F[22,x]+F[80,x]+F[94,x]
F[80,x] = F[12,x]*F[81,x]
F[81,x] = F[82,x]+F[83,x]
F[82,x] = F[62,x]
F[83,x] = 2*F[22,x]+F[80,x]+F[84,x]
F[84,x] = F[12,x]*F[85,x]
F[85,x] = F[32,x]+F[86,x]
F[86,x] = F[87,x]+F[90,x]
F[87,x] = F[88,x]
F[88,x] = F[12,x]*F[89,x]
F[89,x] = F[21,x]+F[87,x]
F[90,x] = 3*F[22,x]+F[91,x]+F[93,x]
F[91,x] = F[12,x]*F[92,x]
F[92,x] = F[33,x]+F[90,x]
F[93,x] = F[12,x]*F[86,x]
F[94,x] = F[12,x]*F[95,x]
F[95,x] = F[32,x]+F[96,x]
F[96,x] = F[53,x]+F[97,x]
F[97,x] = 3*F[22,x]+F[100,x]+F[98,x]
F[98,x] = F[12,x]*F[99,x]
F[99,x] = F[33,x]+F[97,x]
F[100,x] = F[12,x]*F[96,x]
F[101,x] = F[12,x]*F[78,x]
F[102,x] = 2*F[22,x]+F[103,x]+F[84,x]
F[103,x] = F[104,x]*F[12,x]
F[104,x] = F[102,x]+F[61,x]
F[105,x] = F[106,x]*F[12,x]
F[106,x] = F[107,x]+F[114,x]
F[107,x] = F[108,x]+F[39,x]
F[108,x] = F[109,x]+F[113,x]+F[22,x]
F[109,x] = F[110,x]*F[12,x]
F[110,x] = F[111,x]+F[112,x]
F[111,x] = F[10,x]+F[82,x]
F[112,x] = F[21,x]+F[83,x]
F[113,x] = F[107,x]*F[12,x]
F[114,x] = F[115,x]+F[124,x]
F[115,x] = F[116,x]+F[22,x]+F[47,x]
F[116,x] = F[117,x]*F[12,x]
F[117,x] = F[118,x]+F[120,x]
F[118,x] = F[119,x]+F[14,x]
F[119,x] = F[62,x]
F[120,x] = F[121,x]+F[44,x]
F[121,x] = 2*F[22,x]+F[122,x]+F[84,x]
F[122,x] = F[12,x]*F[123,x]
F[123,x] = F[119,x]+F[121,x]
F[124,x] = F[125,x]+F[161,x]+F[22,x]+F[94,x]
F[125,x] = F[12,x]*F[126,x]
F[126,x] = F[127,x]+F[142,x]
F[127,x] = F[128,x]+F[82,x]
F[128,x] = F[129,x]
F[129,x] = F[12,x]*F[130,x]
F[130,x] = F[131,x]+F[134,x]
F[131,x] = F[132,x]+F[30,x]
F[132,x] = F[133,x]
F[133,x] = F[12,x]*F[131,x]
F[134,x] = F[135,x]+F[138,x]
F[135,x] = F[136,x]
F[136,x] = F[12,x]*F[137,x]
F[137,x] = F[135,x]+F[30,x]
F[138,x] = 3*F[22,x]+F[139,x]+F[141,x]
F[139,x] = F[12,x]*F[140,x]
F[140,x] = F[132,x]+F[138,x]
F[141,x] = F[12,x]*F[134,x]
F[142,x] = F[143,x]+F[83,x]
F[143,x] = 3*F[22,x]+F[144,x]+F[146,x]
F[144,x] = F[12,x]*F[145,x]
F[145,x] = F[128,x]+F[143,x]
F[146,x] = F[12,x]*F[147,x]
F[147,x] = F[148,x]+F[153,x]
F[148,x] = F[149,x]+F[33,x]
F[149,x] = 3*F[22,x]+F[150,x]+F[152,x]
F[150,x] = F[12,x]*F[151,x]
F[151,x] = F[132,x]+F[149,x]
F[152,x] = F[12,x]*F[148,x]
F[153,x] = F[154,x]+F[157,x]
F[154,x] = F[155,x]
F[155,x] = F[12,x]*F[156,x]
F[156,x] = F[154,x]+F[33,x]
F[157,x] = 4*F[22,x]+F[158,x]+F[160,x]
F[158,x] = F[12,x]*F[159,x]
F[159,x] = F[149,x]+F[157,x]
F[160,x] = F[12,x]*F[153,x]
F[161,x] = F[114,x]*F[12,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{38}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{26}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{37}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{21}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{57}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{43}\! \left(x \right) = F_{18}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{22}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{17}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{18}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{54}\! \left(x \right)+F_{56}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{21}\! \left(x \right)+F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{57}\! \left(x \right) = F_{105}\! \left(x \right)+F_{22}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{72}\! \left(x \right)
F_{60}\! \left(x \right) = F_{6}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{29}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{68}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{10}\! \left(x \right)+F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{69}\! \left(x \right)+F_{71}\! \left(x \right)
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{30}\! \left(x \right)+F_{68}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{72}\! \left(x \right) = F_{102}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{22}\! \left(x \right)+F_{74}\! \left(x \right)+F_{76}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{6}\! \left(x \right)+F_{73}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{17}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{44}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{101}\! \left(x \right)+F_{22}\! \left(x \right)+F_{80}\! \left(x \right)+F_{94}\! \left(x \right)
F_{80}\! \left(x \right) = F_{12}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)+F_{83}\! \left(x \right)
F_{82}\! \left(x \right) = F_{62}\! \left(x \right)
F_{83}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{80}\! \left(x \right)+F_{84}\! \left(x \right)
F_{84}\! \left(x \right) = F_{12}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{32}\! \left(x \right)+F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)+F_{90}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{12}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{21}\! \left(x \right)+F_{87}\! \left(x \right)
F_{90}\! \left(x \right) = 3 F_{22}\! \left(x \right)+F_{91}\! \left(x \right)+F_{93}\! \left(x \right)
F_{91}\! \left(x \right) = F_{12}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{33}\! \left(x \right)+F_{90}\! \left(x \right)
F_{93}\! \left(x \right) = F_{12}\! \left(x \right) F_{86}\! \left(x \right)
F_{94}\! \left(x \right) = F_{12}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{32}\! \left(x \right)+F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{53}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = 3 F_{22}\! \left(x \right)+F_{100}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{12}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{33}\! \left(x \right)+F_{97}\! \left(x \right)
F_{100}\! \left(x \right) = F_{12}\! \left(x \right) F_{96}\! \left(x \right)
F_{101}\! \left(x \right) = F_{12}\! \left(x \right) F_{78}\! \left(x \right)
F_{102}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{103}\! \left(x \right)+F_{84}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{12}\! \left(x \right)
F_{104}\! \left(x \right) = F_{102}\! \left(x \right)+F_{61}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{12}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{114}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)+F_{39}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{113}\! \left(x \right)+F_{22}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right) F_{12}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{112}\! \left(x \right)
F_{111}\! \left(x \right) = F_{10}\! \left(x \right)+F_{82}\! \left(x \right)
F_{112}\! \left(x \right) = F_{21}\! \left(x \right)+F_{83}\! \left(x \right)
F_{113}\! \left(x \right) = F_{107}\! \left(x \right) F_{12}\! \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_{22}\! \left(x \right)+F_{47}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right) F_{12}\! \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_{14}\! \left(x \right)
F_{119}\! \left(x \right) = F_{62}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)+F_{44}\! \left(x \right)
F_{121}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{122}\! \left(x \right)+F_{84}\! \left(x \right)
F_{122}\! \left(x \right) = F_{12}\! \left(x \right) F_{123}\! \left(x \right)
F_{123}\! \left(x \right) = F_{119}\! \left(x \right)+F_{121}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)+F_{161}\! \left(x \right)+F_{22}\! \left(x \right)+F_{94}\! \left(x \right)
F_{125}\! \left(x \right) = F_{12}\! \left(x \right) F_{126}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right)+F_{142}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right)+F_{82}\! \left(x \right)
F_{128}\! \left(x \right) = F_{129}\! \left(x \right)
F_{129}\! \left(x \right) = F_{12}\! \left(x \right) F_{130}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right)+F_{134}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)+F_{30}\! \left(x \right)
F_{132}\! \left(x \right) = F_{133}\! \left(x \right)
F_{133}\! \left(x \right) = F_{12}\! \left(x \right) F_{131}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)+F_{138}\! \left(x \right)
F_{135}\! \left(x \right) = F_{136}\! \left(x \right)
F_{136}\! \left(x \right) = F_{12}\! \left(x \right) F_{137}\! \left(x \right)
F_{137}\! \left(x \right) = F_{135}\! \left(x \right)+F_{30}\! \left(x \right)
F_{138}\! \left(x \right) = 3 F_{22}\! \left(x \right)+F_{139}\! \left(x \right)+F_{141}\! \left(x \right)
F_{139}\! \left(x \right) = F_{12}\! \left(x \right) F_{140}\! \left(x \right)
F_{140}\! \left(x \right) = F_{132}\! \left(x \right)+F_{138}\! \left(x \right)
F_{141}\! \left(x \right) = F_{12}\! \left(x \right) F_{134}\! \left(x \right)
F_{142}\! \left(x \right) = F_{143}\! \left(x \right)+F_{83}\! \left(x \right)
F_{143}\! \left(x \right) = 3 F_{22}\! \left(x \right)+F_{144}\! \left(x \right)+F_{146}\! \left(x \right)
F_{144}\! \left(x \right) = F_{12}\! \left(x \right) F_{145}\! \left(x \right)
F_{145}\! \left(x \right) = F_{128}\! \left(x \right)+F_{143}\! \left(x \right)
F_{146}\! \left(x \right) = F_{12}\! \left(x \right) F_{147}\! \left(x \right)
F_{147}\! \left(x \right) = F_{148}\! \left(x \right)+F_{153}\! \left(x \right)
F_{148}\! \left(x \right) = F_{149}\! \left(x \right)+F_{33}\! \left(x \right)
F_{149}\! \left(x \right) = 3 F_{22}\! \left(x \right)+F_{150}\! \left(x \right)+F_{152}\! \left(x \right)
F_{150}\! \left(x \right) = F_{12}\! \left(x \right) F_{151}\! \left(x \right)
F_{151}\! \left(x \right) = F_{132}\! \left(x \right)+F_{149}\! \left(x \right)
F_{152}\! \left(x \right) = F_{12}\! \left(x \right) F_{148}\! \left(x \right)
F_{153}\! \left(x \right) = F_{154}\! \left(x \right)+F_{157}\! \left(x \right)
F_{154}\! \left(x \right) = F_{155}\! \left(x \right)
F_{155}\! \left(x \right) = F_{12}\! \left(x \right) F_{156}\! \left(x \right)
F_{156}\! \left(x \right) = F_{154}\! \left(x \right)+F_{33}\! \left(x \right)
F_{157}\! \left(x \right) = 4 F_{22}\! \left(x \right)+F_{158}\! \left(x \right)+F_{160}\! \left(x \right)
F_{158}\! \left(x \right) = F_{12}\! \left(x \right) F_{159}\! \left(x \right)
F_{159}\! \left(x \right) = F_{149}\! \left(x \right)+F_{157}\! \left(x \right)
F_{160}\! \left(x \right) = F_{12}\! \left(x \right) F_{153}\! \left(x \right)
F_{161}\! \left(x \right) = F_{114}\! \left(x \right) F_{12}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_12(x)*F_4(x))
Eq(F_4(x), F_38(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_26(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_25(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_10(x) + F_21(x))
Eq(F_25(x), F_12(x)*F_17(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_37(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_10(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_29(x))
Eq(F_32(x), F_21(x) + F_33(x))
Eq(F_33(x), 2*F_22(x) + F_34(x) + F_36(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_30(x) + F_33(x))
Eq(F_36(x), F_12(x)*F_32(x))
Eq(F_37(x), F_12(x)*F_13(x))
Eq(F_38(x), F_39(x) + F_57(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_1(x) + F_14(x))
Eq(F_43(x), F_18(x) + F_44(x))
Eq(F_44(x), F_22(x) + F_45(x) + F_47(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_14(x) + F_44(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_17(x) + F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_18(x) + F_50(x))
Eq(F_53(x), 2*F_22(x) + F_54(x) + F_56(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_21(x) + F_53(x))
Eq(F_56(x), F_12(x)*F_49(x))
Eq(F_57(x), F_105(x) + F_22(x) + F_58(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_72(x))
Eq(F_60(x), F_6(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_29(x) + F_64(x))
Eq(F_64(x), F_65(x) + F_68(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_67(x))
Eq(F_67(x), F_10(x) + F_65(x))
Eq(F_68(x), 2*F_22(x) + F_69(x) + F_71(x))
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_30(x) + F_68(x))
Eq(F_71(x), F_12(x)*F_64(x))
Eq(F_72(x), F_102(x) + F_73(x))
Eq(F_73(x), F_22(x) + F_74(x) + F_76(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_6(x) + F_73(x))
Eq(F_76(x), F_12(x)*F_77(x))
Eq(F_77(x), F_17(x) + F_78(x))
Eq(F_78(x), F_44(x) + F_79(x))
Eq(F_79(x), F_101(x) + F_22(x) + F_80(x) + F_94(x))
Eq(F_80(x), F_12(x)*F_81(x))
Eq(F_81(x), F_82(x) + F_83(x))
Eq(F_82(x), F_62(x))
Eq(F_83(x), 2*F_22(x) + F_80(x) + F_84(x))
Eq(F_84(x), F_12(x)*F_85(x))
Eq(F_85(x), F_32(x) + F_86(x))
Eq(F_86(x), F_87(x) + F_90(x))
Eq(F_87(x), F_88(x))
Eq(F_88(x), F_12(x)*F_89(x))
Eq(F_89(x), F_21(x) + F_87(x))
Eq(F_90(x), 3*F_22(x) + F_91(x) + F_93(x))
Eq(F_91(x), F_12(x)*F_92(x))
Eq(F_92(x), F_33(x) + F_90(x))
Eq(F_93(x), F_12(x)*F_86(x))
Eq(F_94(x), F_12(x)*F_95(x))
Eq(F_95(x), F_32(x) + F_96(x))
Eq(F_96(x), F_53(x) + F_97(x))
Eq(F_97(x), F_100(x) + 3*F_22(x) + F_98(x))
Eq(F_98(x), F_12(x)*F_99(x))
Eq(F_99(x), F_33(x) + F_97(x))
Eq(F_100(x), F_12(x)*F_96(x))
Eq(F_101(x), F_12(x)*F_78(x))
Eq(F_102(x), F_103(x) + 2*F_22(x) + F_84(x))
Eq(F_103(x), F_104(x)*F_12(x))
Eq(F_104(x), F_102(x) + F_61(x))
Eq(F_105(x), F_106(x)*F_12(x))
Eq(F_106(x), F_107(x) + F_114(x))
Eq(F_107(x), F_108(x) + F_39(x))
Eq(F_108(x), F_109(x) + F_113(x) + F_22(x))
Eq(F_109(x), F_110(x)*F_12(x))
Eq(F_110(x), F_111(x) + F_112(x))
Eq(F_111(x), F_10(x) + F_82(x))
Eq(F_112(x), F_21(x) + F_83(x))
Eq(F_113(x), F_107(x)*F_12(x))
Eq(F_114(x), F_115(x) + F_124(x))
Eq(F_115(x), F_116(x) + F_22(x) + F_47(x))
Eq(F_116(x), F_117(x)*F_12(x))
Eq(F_117(x), F_118(x) + F_120(x))
Eq(F_118(x), F_119(x) + F_14(x))
Eq(F_119(x), F_62(x))
Eq(F_120(x), F_121(x) + F_44(x))
Eq(F_121(x), F_122(x) + 2*F_22(x) + F_84(x))
Eq(F_122(x), F_12(x)*F_123(x))
Eq(F_123(x), F_119(x) + F_121(x))
Eq(F_124(x), F_125(x) + F_161(x) + F_22(x) + F_94(x))
Eq(F_125(x), F_12(x)*F_126(x))
Eq(F_126(x), F_127(x) + F_142(x))
Eq(F_127(x), F_128(x) + F_82(x))
Eq(F_128(x), F_129(x))
Eq(F_129(x), F_12(x)*F_130(x))
Eq(F_130(x), F_131(x) + F_134(x))
Eq(F_131(x), F_132(x) + F_30(x))
Eq(F_132(x), F_133(x))
Eq(F_133(x), F_12(x)*F_131(x))
Eq(F_134(x), F_135(x) + F_138(x))
Eq(F_135(x), F_136(x))
Eq(F_136(x), F_12(x)*F_137(x))
Eq(F_137(x), F_135(x) + F_30(x))
Eq(F_138(x), F_139(x) + F_141(x) + 3*F_22(x))
Eq(F_139(x), F_12(x)*F_140(x))
Eq(F_140(x), F_132(x) + F_138(x))
Eq(F_141(x), F_12(x)*F_134(x))
Eq(F_142(x), F_143(x) + F_83(x))
Eq(F_143(x), F_144(x) + F_146(x) + 3*F_22(x))
Eq(F_144(x), F_12(x)*F_145(x))
Eq(F_145(x), F_128(x) + F_143(x))
Eq(F_146(x), F_12(x)*F_147(x))
Eq(F_147(x), F_148(x) + F_153(x))
Eq(F_148(x), F_149(x) + F_33(x))
Eq(F_149(x), F_150(x) + F_152(x) + 3*F_22(x))
Eq(F_150(x), F_12(x)*F_151(x))
Eq(F_151(x), F_132(x) + F_149(x))
Eq(F_152(x), F_12(x)*F_148(x))
Eq(F_153(x), F_154(x) + F_157(x))
Eq(F_154(x), F_155(x))
Eq(F_155(x), F_12(x)*F_156(x))
Eq(F_156(x), F_154(x) + F_33(x))
Eq(F_157(x), F_158(x) + F_160(x) + 4*F_22(x))
Eq(F_158(x), F_12(x)*F_159(x))
Eq(F_159(x), F_149(x) + F_157(x))
Eq(F_160(x), F_12(x)*F_153(x))
Eq(F_161(x), F_114(x)*F_12(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 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": 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