0132_1230_1320_2301
Counting sequence:
1, 1, 2, 6, 20, 64, 189, 517, 1329, 3257, 7700, 17720, 39962, 88750, 194791, 423607, 914447, 1962207, 4189462, 8906914, 18867056, 39837036, 83875457, 176148121, 369083805, 771734389, 1610592264, 3355419452, 6979294454, 14495483162, 30064735115, 62276984875, 128848972507, 266287919995, 549755754986, 1133871300102, 2336462135212, 4810363289272, 9895604558597, 20340965012589, 41781441743561, 85761906843121, 175921860308412, 360639813761536, 738871813702290, 1512927999641414, 3096224743622639, 6333186975777887, 12947848928459879, 26458647810551767, 54043195528175230, 110338190870284330, 225179981368208552, 459367161991449540, 936748722492695881, 1909526242004695297, 3891110078047684277, 7926335344171617837, 16140901064495370032, 32858262881294616964, 66869447267196566766, 136044737543607347506, 276701161105642638867, 562625694248140647251, 1143698132569991479475, 2324289753287402737139, 4722366482869644399314, 9592306918328965975822, 19479761741837285589604, 39549819294033277693072, 80280230208783967604045, 162921643659002758784085, 330565653800875163808097, 670576040567489619129385, 1360041547066457820261476, 2757862025995872803445192, 5591281915717659931589642, 11333679558887148511367902, 22969590572677954317835767, 46543644055163223224524039, 94296213929941075625332671, 191010279499111409601738191, 386856262276681335904046822, 783383931110279705207577266, 1586110675334393477212379360, 3210906976896455088017377564, 6499185206248246443218070289, 13153112917407165420800753257, 26615710844635675910328615629, 53850391708914041958109231077, 108938723457113464191120137560, 220353326992797688932041192140, 445658414142736898963681671142, 901220348599756840126559251530, 1822247737828079764651507535771, 3684109556913291698099790225787, 7447447276340847733793127719307, 15053350877710224142773346799467, 30423614405477505635920873007802, 61481054111069125972590101377814, 124229758822366481346676909877276
Generating function in Maple syntax:
-(6*x^6-28*x^5+49*x^4-48*x^3+27*x^2-8*x+1)/(2*x-1)^2/(x-1)^5
Generating function in latex syntax:
-\frac{6 x^{6}-28 x^{5}+49 x^{4}-48 x^{3}+27 x^{2}-8 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-6*x**6 + 28*x**5 - 49*x**4 + 48*x**3 - 27*x**2 + 8*x - 1)/((x - 1)**5*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^5*F(x)+6*x^6-28*x^5+49*x^4-48*x^3+27*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{5} F \! \left(x \right)+6 x^{6}-28 x^{5}+49 x^{4}-48 x^{3}+27 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
3+1/24*n^2+1/12*n-1/24*n^4-1/12*n^3+2^n*n-2^(n+1)
Explicit closed form in latex syntax:
3+\frac{n^{2}}{24}+\frac{n}{12}-\frac{n^{4}}{24}-\frac{n^{3}}{12}+2^{n} n -2^{n +1}
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(n+2) = -1/24*n^4+1/4*n^3+13/24*n^2-4*a(n)+4*a(n+1)-3/4*n+2, n >= 7
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(n +2\right) = -\frac{n^{4}}{24}+\frac{n^{3}}{4}+\frac{13 n^{2}}{24}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{3 n}{4}+2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15621/
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[47,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[29,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[25,x]
F[20,x] = 0
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[24,x]
F[23,x] = F[12,x]
F[24,x] = F[19,x]
F[25,x] = F[12,x]*F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[1,x]+F[26,x]
F[29,x] = F[20,x]+F[30,x]+F[45,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[10,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[10,x]*F[12,x]
F[35,x] = F[29,x]+F[36,x]
F[36,x] = 2*F[20,x]+F[37,x]+F[41,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[33,x]
F[40,x] = F[36,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[20,x]+F[43,x]+F[45,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[10,x]+F[42,x]
F[45,x] = F[12,x]*F[46,x]
F[46,x] = F[26,x]+F[42,x]
F[47,x] = F[14,x]+F[48,x]
F[48,x] = F[154,x]+F[20,x]+F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[51,x]+F[97,x]
F[51,x] = F[52,x]+F[6,x]
F[52,x] = F[20,x]+F[34,x]+F[53,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[58,x]
F[55,x] = F[12,x]+F[56,x]
F[56,x] = F[20,x]+F[34,x]+F[57,x]
F[57,x] = F[12,x]*F[55,x]
F[58,x] = F[59,x]+F[77,x]
F[59,x] = F[20,x]+F[60,x]+F[61,x]
F[60,x] = x^2
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[63,x]+F[66,x]
F[63,x] = F[12,x]+F[64,x]
F[64,x] = F[20,x]+F[60,x]+F[65,x]
F[65,x] = x^2
F[66,x] = F[59,x]+F[67,x]
F[67,x] = F[20,x]+F[68,x]+F[69,x]+F[73,x]
F[68,x] = 0
F[69,x] = F[12,x]*F[70,x]
F[70,x] = F[71,x]+F[72,x]
F[71,x] = F[64,x]
F[72,x] = F[67,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[75,x]
F[75,x] = F[12,x]*F[76,x]
F[76,x] = F[12,x]+F[74,x]
F[77,x] = F[20,x]+F[78,x]+F[79,x]+F[95,x]
F[78,x] = F[12,x]*F[33,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[81,x]+F[84,x]
F[81,x] = F[56,x]+F[82,x]
F[82,x] = 2*F[20,x]+F[78,x]+F[83,x]
F[83,x] = F[12,x]*F[56,x]
F[84,x] = F[77,x]+F[85,x]
F[85,x] = 2*F[20,x]+F[86,x]+F[87,x]+F[91,x]
F[86,x] = 0
F[87,x] = F[12,x]*F[88,x]
F[88,x] = F[89,x]+F[90,x]
F[89,x] = F[82,x]
F[90,x] = F[85,x]
F[91,x] = F[12,x]*F[92,x]
F[92,x] = 2*F[20,x]+F[93,x]+F[95,x]
F[93,x] = F[12,x]*F[94,x]
F[94,x] = F[56,x]+F[92,x]
F[95,x] = F[12,x]*F[96,x]
F[96,x] = F[74,x]+F[92,x]
F[97,x] = F[48,x]+F[98,x]
F[98,x] = F[103,x]+F[20,x]+F[41,x]+F[99,x]
F[99,x] = F[100,x]*F[12,x]
F[100,x] = F[101,x]+F[102,x]
F[101,x] = F[52,x]
F[102,x] = F[98,x]
F[103,x] = F[104,x]*F[12,x]
F[104,x] = F[105,x]+F[109,x]
F[105,x] = F[106,x]+F[107,x]
F[106,x] = F[25,x]
F[107,x] = 2*F[20,x]+F[108,x]+F[41,x]
F[108,x] = F[105,x]*F[12,x]
F[109,x] = F[110,x]+F[131,x]
F[110,x] = 2*F[20,x]+F[111,x]+F[115,x]
F[111,x] = F[112,x]*F[12,x]
F[112,x] = F[113,x]+F[20,x]+F[25,x]
F[113,x] = F[114,x]*F[12,x]
F[114,x] = F[112,x]+F[12,x]
F[115,x] = F[116,x]*F[12,x]
F[116,x] = F[117,x]+F[120,x]
F[117,x] = F[106,x]+F[118,x]
F[118,x] = 2*F[20,x]+F[111,x]+F[119,x]
F[119,x] = F[106,x]*F[12,x]
F[120,x] = F[110,x]+F[121,x]
F[121,x] = 2*F[20,x]+F[122,x]+F[123,x]+F[127,x]
F[122,x] = 0
F[123,x] = F[12,x]*F[124,x]
F[124,x] = F[125,x]+F[126,x]
F[125,x] = F[118,x]
F[126,x] = F[121,x]
F[127,x] = F[12,x]*F[128,x]
F[128,x] = F[129,x]
F[129,x] = F[12,x]*F[130,x]
F[130,x] = F[106,x]+F[128,x]
F[131,x] = 2*F[20,x]+F[132,x]+F[136,x]+F[152,x]
F[132,x] = F[12,x]*F[133,x]
F[133,x] = 2*F[20,x]+F[134,x]+F[41,x]
F[134,x] = F[12,x]*F[135,x]
F[135,x] = F[133,x]+F[33,x]
F[136,x] = F[12,x]*F[137,x]
F[137,x] = F[138,x]+F[141,x]
F[138,x] = F[107,x]+F[139,x]
F[139,x] = 3*F[20,x]+F[132,x]+F[140,x]
F[140,x] = F[107,x]*F[12,x]
F[141,x] = F[131,x]+F[142,x]
F[142,x] = 3*F[20,x]+F[143,x]+F[144,x]+F[148,x]
F[143,x] = 0
F[144,x] = F[12,x]*F[145,x]
F[145,x] = F[146,x]+F[147,x]
F[146,x] = F[139,x]
F[147,x] = F[142,x]
F[148,x] = F[12,x]*F[149,x]
F[149,x] = 3*F[20,x]+F[150,x]+F[152,x]
F[150,x] = F[12,x]*F[151,x]
F[151,x] = F[107,x]+F[149,x]
F[152,x] = F[12,x]*F[153,x]
F[153,x] = F[128,x]+F[149,x]
F[154,x] = F[12,x]*F[155,x]
F[155,x] = F[156,x]+F[46,x]
F[156,x] = F[157,x]+F[172,x]
F[157,x] = F[113,x]+F[158,x]+F[20,x]
F[158,x] = F[12,x]*F[159,x]
F[159,x] = F[160,x]+F[161,x]
F[160,x] = F[112,x]+F[26,x]
F[161,x] = F[157,x]+F[162,x]
F[162,x] = F[163,x]+F[164,x]+F[168,x]+F[20,x]
F[163,x] = 0
F[164,x] = F[12,x]*F[165,x]
F[165,x] = F[166,x]+F[167,x]
F[166,x] = F[112,x]
F[167,x] = F[162,x]
F[168,x] = F[12,x]*F[169,x]
F[169,x] = F[170,x]
F[170,x] = F[12,x]*F[171,x]
F[171,x] = F[169,x]+F[26,x]
F[172,x] = F[173,x]+F[176,x]+F[190,x]+F[20,x]
F[173,x] = F[12,x]*F[174,x]
F[174,x] = F[175,x]+F[56,x]
F[175,x] = F[108,x]+F[173,x]+F[20,x]+F[41,x]
F[176,x] = F[12,x]*F[177,x]
F[177,x] = F[178,x]+F[179,x]
F[178,x] = F[133,x]+F[42,x]
F[179,x] = F[172,x]+F[180,x]
F[180,x] = 2*F[20,x]+F[181,x]+F[182,x]+F[186,x]
F[181,x] = 0
F[182,x] = F[12,x]*F[183,x]
F[183,x] = F[184,x]+F[185,x]
F[184,x] = F[133,x]
F[185,x] = F[180,x]
F[186,x] = F[12,x]*F[187,x]
F[187,x] = 2*F[20,x]+F[188,x]+F[190,x]
F[188,x] = F[12,x]*F[189,x]
F[189,x] = F[187,x]+F[42,x]
F[190,x] = F[12,x]*F[191,x]
F[191,x] = F[169,x]+F[187,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_{47}\! \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_{29}\! \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_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \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_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{20}\! \left(x \right)+F_{30}\! \left(x \right)+F_{45}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{10}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{35}\! \left(x \right) = F_{29}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{37}\! \left(x \right)+F_{41}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{33}\! \left(x \right)
F_{40}\! \left(x \right) = F_{36}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{20}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{26}\! \left(x \right)+F_{42}\! \left(x \right)
F_{47}\! \left(x \right) = F_{14}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{154}\! \left(x \right)+F_{20}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{97}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{6}\! \left(x \right)
F_{52}\! \left(x \right) = F_{20}\! \left(x \right)+F_{34}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{58}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{20}\! \left(x \right)+F_{34}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{77}\! \left(x \right)
F_{59}\! \left(x \right) = F_{20}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)
F_{60}\! \left(x \right) = x^{2}
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{66}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{20}\! \left(x \right)+F_{60}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = x^{2}
F_{66}\! \left(x \right) = F_{59}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{20}\! \left(x \right)+F_{68}\! \left(x \right)+F_{69}\! \left(x \right)+F_{73}\! \left(x \right)
F_{68}\! \left(x \right) = 0
F_{69}\! \left(x \right) = F_{12}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)+F_{72}\! \left(x \right)
F_{71}\! \left(x \right) = F_{64}\! \left(x \right)
F_{72}\! \left(x \right) = F_{67}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{12}\! \left(x \right)+F_{74}\! \left(x \right)
F_{77}\! \left(x \right) = F_{20}\! \left(x \right)+F_{78}\! \left(x \right)+F_{79}\! \left(x \right)+F_{95}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{84}\! \left(x \right)
F_{81}\! \left(x \right) = F_{56}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{78}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{84}\! \left(x \right) = F_{77}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{86}\! \left(x \right)+F_{87}\! \left(x \right)+F_{91}\! \left(x \right)
F_{86}\! \left(x \right) = 0
F_{87}\! \left(x \right) = F_{12}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)+F_{90}\! \left(x \right)
F_{89}\! \left(x \right) = F_{82}\! \left(x \right)
F_{90}\! \left(x \right) = F_{85}\! \left(x \right)
F_{91}\! \left(x \right) = F_{12}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{93}\! \left(x \right)+F_{95}\! \left(x \right)
F_{93}\! \left(x \right) = F_{12}\! \left(x \right) F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{56}\! \left(x \right)+F_{92}\! \left(x \right)
F_{95}\! \left(x \right) = F_{12}\! \left(x \right) F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{74}\! \left(x \right)+F_{92}\! \left(x \right)
F_{97}\! \left(x \right) = F_{48}\! \left(x \right)+F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{103}\! \left(x \right)+F_{20}\! \left(x \right)+F_{41}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right) F_{12}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{102}\! \left(x \right)
F_{101}\! \left(x \right) = F_{52}\! \left(x \right)
F_{102}\! \left(x \right) = F_{98}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{12}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{109}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{107}\! \left(x \right)
F_{106}\! \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_{41}\! \left(x \right)
F_{108}\! \left(x \right) = F_{105}\! \left(x \right) F_{12}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right)+F_{131}\! \left(x \right)
F_{110}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{111}\! \left(x \right)+F_{115}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right) F_{12}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)+F_{20}\! \left(x \right)+F_{25}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right) F_{12}\! \left(x \right)
F_{114}\! \left(x \right) = F_{112}\! \left(x \right)+F_{12}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right) F_{12}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right)+F_{120}\! \left(x \right)
F_{117}\! \left(x \right) = F_{106}\! \left(x \right)+F_{118}\! \left(x \right)
F_{118}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{111}\! \left(x \right)+F_{119}\! \left(x \right)
F_{119}\! \left(x \right) = F_{106}\! \left(x \right) F_{12}\! \left(x \right)
F_{120}\! \left(x \right) = F_{110}\! \left(x \right)+F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{122}\! \left(x \right)+F_{123}\! \left(x \right)+F_{127}\! \left(x \right)
F_{122}\! \left(x \right) = 0
F_{123}\! \left(x \right) = F_{12}\! \left(x \right) F_{124}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right)+F_{126}\! \left(x \right)
F_{125}\! \left(x \right) = F_{118}\! \left(x \right)
F_{126}\! \left(x \right) = F_{121}\! \left(x \right)
F_{127}\! \left(x \right) = F_{12}\! \left(x \right) F_{128}\! \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_{106}\! \left(x \right)+F_{128}\! \left(x \right)
F_{131}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{132}\! \left(x \right)+F_{136}\! \left(x \right)+F_{152}\! \left(x \right)
F_{132}\! \left(x \right) = F_{12}\! \left(x \right) F_{133}\! \left(x \right)
F_{133}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{134}\! \left(x \right)+F_{41}\! \left(x \right)
F_{134}\! \left(x \right) = F_{12}\! \left(x \right) F_{135}\! \left(x \right)
F_{135}\! \left(x \right) = F_{133}\! \left(x \right)+F_{33}\! \left(x \right)
F_{136}\! \left(x \right) = F_{12}\! \left(x \right) F_{137}\! \left(x \right)
F_{137}\! \left(x \right) = F_{138}\! \left(x \right)+F_{141}\! \left(x \right)
F_{138}\! \left(x \right) = F_{107}\! \left(x \right)+F_{139}\! \left(x \right)
F_{139}\! \left(x \right) = 3 F_{20}\! \left(x \right)+F_{132}\! \left(x \right)+F_{140}\! \left(x \right)
F_{140}\! \left(x \right) = F_{107}\! \left(x \right) F_{12}\! \left(x \right)
F_{141}\! \left(x \right) = F_{131}\! \left(x \right)+F_{142}\! \left(x \right)
F_{142}\! \left(x \right) = 3 F_{20}\! \left(x \right)+F_{143}\! \left(x \right)+F_{144}\! \left(x \right)+F_{148}\! \left(x \right)
F_{143}\! \left(x \right) = 0
F_{144}\! \left(x \right) = F_{12}\! \left(x \right) F_{145}\! \left(x \right)
F_{145}\! \left(x \right) = F_{146}\! \left(x \right)+F_{147}\! \left(x \right)
F_{146}\! \left(x \right) = F_{139}\! \left(x \right)
F_{147}\! \left(x \right) = F_{142}\! \left(x \right)
F_{148}\! \left(x \right) = F_{12}\! \left(x \right) F_{149}\! \left(x \right)
F_{149}\! \left(x \right) = 3 F_{20}\! \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_{107}\! \left(x \right)+F_{149}\! \left(x \right)
F_{152}\! \left(x \right) = F_{12}\! \left(x \right) F_{153}\! \left(x \right)
F_{153}\! \left(x \right) = F_{128}\! \left(x \right)+F_{149}\! \left(x \right)
F_{154}\! \left(x \right) = F_{12}\! \left(x \right) F_{155}\! \left(x \right)
F_{155}\! \left(x \right) = F_{156}\! \left(x \right)+F_{46}\! \left(x \right)
F_{156}\! \left(x \right) = F_{157}\! \left(x \right)+F_{172}\! \left(x \right)
F_{157}\! \left(x \right) = F_{113}\! \left(x \right)+F_{158}\! \left(x \right)+F_{20}\! \left(x \right)
F_{158}\! \left(x \right) = F_{12}\! \left(x \right) F_{159}\! \left(x \right)
F_{159}\! \left(x \right) = F_{160}\! \left(x \right)+F_{161}\! \left(x \right)
F_{160}\! \left(x \right) = F_{112}\! \left(x \right)+F_{26}\! \left(x \right)
F_{161}\! \left(x \right) = F_{157}\! \left(x \right)+F_{162}\! \left(x \right)
F_{162}\! \left(x \right) = F_{163}\! \left(x \right)+F_{164}\! \left(x \right)+F_{168}\! \left(x \right)+F_{20}\! \left(x \right)
F_{163}\! \left(x \right) = 0
F_{164}\! \left(x \right) = F_{12}\! \left(x \right) F_{165}\! \left(x \right)
F_{165}\! \left(x \right) = F_{166}\! \left(x \right)+F_{167}\! \left(x \right)
F_{166}\! \left(x \right) = F_{112}\! \left(x \right)
F_{167}\! \left(x \right) = F_{162}\! \left(x \right)
F_{168}\! \left(x \right) = F_{12}\! \left(x \right) F_{169}\! \left(x \right)
F_{169}\! \left(x \right) = F_{170}\! \left(x \right)
F_{170}\! \left(x \right) = F_{12}\! \left(x \right) F_{171}\! \left(x \right)
F_{171}\! \left(x \right) = F_{169}\! \left(x \right)+F_{26}\! \left(x \right)
F_{172}\! \left(x \right) = F_{173}\! \left(x \right)+F_{176}\! \left(x \right)+F_{190}\! \left(x \right)+F_{20}\! \left(x \right)
F_{173}\! \left(x \right) = F_{12}\! \left(x \right) F_{174}\! \left(x \right)
F_{174}\! \left(x \right) = F_{175}\! \left(x \right)+F_{56}\! \left(x \right)
F_{175}\! \left(x \right) = F_{108}\! \left(x \right)+F_{173}\! \left(x \right)+F_{20}\! \left(x \right)+F_{41}\! \left(x \right)
F_{176}\! \left(x \right) = F_{12}\! \left(x \right) F_{177}\! \left(x \right)
F_{177}\! \left(x \right) = F_{178}\! \left(x \right)+F_{179}\! \left(x \right)
F_{178}\! \left(x \right) = F_{133}\! \left(x \right)+F_{42}\! \left(x \right)
F_{179}\! \left(x \right) = F_{172}\! \left(x \right)+F_{180}\! \left(x \right)
F_{180}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{181}\! \left(x \right)+F_{182}\! \left(x \right)+F_{186}\! \left(x \right)
F_{181}\! \left(x \right) = 0
F_{182}\! \left(x \right) = F_{12}\! \left(x \right) F_{183}\! \left(x \right)
F_{183}\! \left(x \right) = F_{184}\! \left(x \right)+F_{185}\! \left(x \right)
F_{184}\! \left(x \right) = F_{133}\! \left(x \right)
F_{185}\! \left(x \right) = F_{180}\! \left(x \right)
F_{186}\! \left(x \right) = F_{12}\! \left(x \right) F_{187}\! \left(x \right)
F_{187}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{188}\! \left(x \right)+F_{190}\! \left(x \right)
F_{188}\! \left(x \right) = F_{12}\! \left(x \right) F_{189}\! \left(x \right)
F_{189}\! \left(x \right) = F_{187}\! \left(x \right)+F_{42}\! \left(x \right)
F_{190}\! \left(x \right) = F_{12}\! \left(x \right) F_{191}\! \left(x \right)
F_{191}\! \left(x \right) = F_{169}\! \left(x \right)+F_{187}\! \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_47(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_29(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_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_25(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_24(x))
Eq(F_23(x), F_12(x))
Eq(F_24(x), F_19(x))
Eq(F_25(x), F_12(x)*F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_1(x) + F_26(x))
Eq(F_29(x), F_20(x) + F_30(x) + F_45(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_10(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_10(x)*F_12(x))
Eq(F_35(x), F_29(x) + F_36(x))
Eq(F_36(x), 2*F_20(x) + F_37(x) + F_41(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_33(x))
Eq(F_40(x), F_36(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_20(x) + F_43(x) + F_45(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_10(x) + F_42(x))
Eq(F_45(x), F_12(x)*F_46(x))
Eq(F_46(x), F_26(x) + F_42(x))
Eq(F_47(x), F_14(x) + F_48(x))
Eq(F_48(x), F_154(x) + F_20(x) + F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_97(x))
Eq(F_51(x), F_52(x) + F_6(x))
Eq(F_52(x), F_20(x) + F_34(x) + F_53(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_58(x))
Eq(F_55(x), F_12(x) + F_56(x))
Eq(F_56(x), F_20(x) + F_34(x) + F_57(x))
Eq(F_57(x), F_12(x)*F_55(x))
Eq(F_58(x), F_59(x) + F_77(x))
Eq(F_59(x), F_20(x) + F_60(x) + F_61(x))
Eq(F_60(x), x**2)
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_66(x))
Eq(F_63(x), F_12(x) + F_64(x))
Eq(F_64(x), F_20(x) + F_60(x) + F_65(x))
Eq(F_65(x), x**2)
Eq(F_66(x), F_59(x) + F_67(x))
Eq(F_67(x), F_20(x) + F_68(x) + F_69(x) + F_73(x))
Eq(F_68(x), 0)
Eq(F_69(x), F_12(x)*F_70(x))
Eq(F_70(x), F_71(x) + F_72(x))
Eq(F_71(x), F_64(x))
Eq(F_72(x), F_67(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_75(x))
Eq(F_75(x), F_12(x)*F_76(x))
Eq(F_76(x), F_12(x) + F_74(x))
Eq(F_77(x), F_20(x) + F_78(x) + F_79(x) + F_95(x))
Eq(F_78(x), F_12(x)*F_33(x))
Eq(F_79(x), F_12(x)*F_80(x))
Eq(F_80(x), F_81(x) + F_84(x))
Eq(F_81(x), F_56(x) + F_82(x))
Eq(F_82(x), 2*F_20(x) + F_78(x) + F_83(x))
Eq(F_83(x), F_12(x)*F_56(x))
Eq(F_84(x), F_77(x) + F_85(x))
Eq(F_85(x), 2*F_20(x) + F_86(x) + F_87(x) + F_91(x))
Eq(F_86(x), 0)
Eq(F_87(x), F_12(x)*F_88(x))
Eq(F_88(x), F_89(x) + F_90(x))
Eq(F_89(x), F_82(x))
Eq(F_90(x), F_85(x))
Eq(F_91(x), F_12(x)*F_92(x))
Eq(F_92(x), 2*F_20(x) + F_93(x) + F_95(x))
Eq(F_93(x), F_12(x)*F_94(x))
Eq(F_94(x), F_56(x) + F_92(x))
Eq(F_95(x), F_12(x)*F_96(x))
Eq(F_96(x), F_74(x) + F_92(x))
Eq(F_97(x), F_48(x) + F_98(x))
Eq(F_98(x), F_103(x) + F_20(x) + F_41(x) + F_99(x))
Eq(F_99(x), F_100(x)*F_12(x))
Eq(F_100(x), F_101(x) + F_102(x))
Eq(F_101(x), F_52(x))
Eq(F_102(x), F_98(x))
Eq(F_103(x), F_104(x)*F_12(x))
Eq(F_104(x), F_105(x) + F_109(x))
Eq(F_105(x), F_106(x) + F_107(x))
Eq(F_106(x), F_25(x))
Eq(F_107(x), F_108(x) + 2*F_20(x) + F_41(x))
Eq(F_108(x), F_105(x)*F_12(x))
Eq(F_109(x), F_110(x) + F_131(x))
Eq(F_110(x), F_111(x) + F_115(x) + 2*F_20(x))
Eq(F_111(x), F_112(x)*F_12(x))
Eq(F_112(x), F_113(x) + F_20(x) + F_25(x))
Eq(F_113(x), F_114(x)*F_12(x))
Eq(F_114(x), F_112(x) + F_12(x))
Eq(F_115(x), F_116(x)*F_12(x))
Eq(F_116(x), F_117(x) + F_120(x))
Eq(F_117(x), F_106(x) + F_118(x))
Eq(F_118(x), F_111(x) + F_119(x) + 2*F_20(x))
Eq(F_119(x), F_106(x)*F_12(x))
Eq(F_120(x), F_110(x) + F_121(x))
Eq(F_121(x), F_122(x) + F_123(x) + F_127(x) + 2*F_20(x))
Eq(F_122(x), 0)
Eq(F_123(x), F_12(x)*F_124(x))
Eq(F_124(x), F_125(x) + F_126(x))
Eq(F_125(x), F_118(x))
Eq(F_126(x), F_121(x))
Eq(F_127(x), F_12(x)*F_128(x))
Eq(F_128(x), F_129(x))
Eq(F_129(x), F_12(x)*F_130(x))
Eq(F_130(x), F_106(x) + F_128(x))
Eq(F_131(x), F_132(x) + F_136(x) + F_152(x) + 2*F_20(x))
Eq(F_132(x), F_12(x)*F_133(x))
Eq(F_133(x), F_134(x) + 2*F_20(x) + F_41(x))
Eq(F_134(x), F_12(x)*F_135(x))
Eq(F_135(x), F_133(x) + F_33(x))
Eq(F_136(x), F_12(x)*F_137(x))
Eq(F_137(x), F_138(x) + F_141(x))
Eq(F_138(x), F_107(x) + F_139(x))
Eq(F_139(x), F_132(x) + F_140(x) + 3*F_20(x))
Eq(F_140(x), F_107(x)*F_12(x))
Eq(F_141(x), F_131(x) + F_142(x))
Eq(F_142(x), F_143(x) + F_144(x) + F_148(x) + 3*F_20(x))
Eq(F_143(x), 0)
Eq(F_144(x), F_12(x)*F_145(x))
Eq(F_145(x), F_146(x) + F_147(x))
Eq(F_146(x), F_139(x))
Eq(F_147(x), F_142(x))
Eq(F_148(x), F_12(x)*F_149(x))
Eq(F_149(x), F_150(x) + F_152(x) + 3*F_20(x))
Eq(F_150(x), F_12(x)*F_151(x))
Eq(F_151(x), F_107(x) + F_149(x))
Eq(F_152(x), F_12(x)*F_153(x))
Eq(F_153(x), F_128(x) + F_149(x))
Eq(F_154(x), F_12(x)*F_155(x))
Eq(F_155(x), F_156(x) + F_46(x))
Eq(F_156(x), F_157(x) + F_172(x))
Eq(F_157(x), F_113(x) + F_158(x) + F_20(x))
Eq(F_158(x), F_12(x)*F_159(x))
Eq(F_159(x), F_160(x) + F_161(x))
Eq(F_160(x), F_112(x) + F_26(x))
Eq(F_161(x), F_157(x) + F_162(x))
Eq(F_162(x), F_163(x) + F_164(x) + F_168(x) + F_20(x))
Eq(F_163(x), 0)
Eq(F_164(x), F_12(x)*F_165(x))
Eq(F_165(x), F_166(x) + F_167(x))
Eq(F_166(x), F_112(x))
Eq(F_167(x), F_162(x))
Eq(F_168(x), F_12(x)*F_169(x))
Eq(F_169(x), F_170(x))
Eq(F_170(x), F_12(x)*F_171(x))
Eq(F_171(x), F_169(x) + F_26(x))
Eq(F_172(x), F_173(x) + F_176(x) + F_190(x) + F_20(x))
Eq(F_173(x), F_12(x)*F_174(x))
Eq(F_174(x), F_175(x) + F_56(x))
Eq(F_175(x), F_108(x) + F_173(x) + F_20(x) + F_41(x))
Eq(F_176(x), F_12(x)*F_177(x))
Eq(F_177(x), F_178(x) + F_179(x))
Eq(F_178(x), F_133(x) + F_42(x))
Eq(F_179(x), F_172(x) + F_180(x))
Eq(F_180(x), F_181(x) + F_182(x) + F_186(x) + 2*F_20(x))
Eq(F_181(x), 0)
Eq(F_182(x), F_12(x)*F_183(x))
Eq(F_183(x), F_184(x) + F_185(x))
Eq(F_184(x), F_133(x))
Eq(F_185(x), F_180(x))
Eq(F_186(x), F_12(x)*F_187(x))
Eq(F_187(x), F_188(x) + F_190(x) + 2*F_20(x))
Eq(F_188(x), F_12(x)*F_189(x))
Eq(F_189(x), F_187(x) + F_42(x))
Eq(F_190(x), F_12(x)*F_191(x))
Eq(F_191(x), F_169(x) + F_187(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:
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"pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], 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