0132_0213_1320_3201
Counting sequence:
1, 1, 2, 6, 20, 62, 171, 429, 1007, 2261, 4932, 10562, 22358, 46984, 98273, 204907, 426289, 885311, 1835938, 3802472, 7866272, 16255578, 33557927, 69210521, 142612035, 293608377, 603988496, 1241524574, 2550149562, 5234506556, 10737436157, 22011728407, 45097181093, 92341825219, 188978593678, 386547094052, 790274025124, 1614907751726, 3298534938083, 6734508781797, 13743895416407, 28037546585693, 57174604730652, 116548232640186, 237494511705950, 483785116338992, 985162418616921, 2005509209206019, 4081387162461017, 8303511813135751, 16888498602826682, 34339947158904352, 69805794224465128, 141863388262412354, 288230376151973983, 585467951558448497, 1188950301626118059, 2413929400270917441, 4899916394579457128, 9943947977234440022, 20176126330620235842, 40928713413543511908, 83010348331693458677, 168326539672600168751, 341264765363627250765, 691752902764108768907, 1401952549601926545446, 2840798587351271612764, 5755384150997380811372, 11658342254584437373846, 23611832414348226868507, 47813960639055158638397, 96808512898827727782463, 195978209039090277324389, 396678784561050198963188, 802802302087839687400242, 1624494070107157954645094, 3286767072077273069930456, 6649092007880460462149073, 13449299743212749569941147, 27200830941329156432296577, 55006124792465627450614223, 111221175404545884074530002, 224860202448321026496992216, 454556108175100569691250480, 918783622907118172778510122, 1856910058928070412350594071, 3752505744083808958289972809, 7582382740622954183759236627, 15319507986156580901878864841, 30948500982134506872480413632, 62515971983911703882408190542, 126269884007108788039713201098, 255015648092788336629222237324, 514983056342718194358038445293, 1039869632999719430915267241287, 2099546306628004946228917706229, 4238706694513142061254604498771, 8556641551540548460102749929918, 17271739428109625595392584609172, 34860391506276308541159341730612
Generating function in Maple syntax:
(2*x^8-6*x^7-6*x^6+30*x^5-49*x^4+48*x^3-27*x^2+8*x-1)/(2*x-1)^2/(-1+x)^5
Generating function in latex syntax:
\frac{2 x^{8}-6 x^{7}-6 x^{6}+30 x^{5}-49 x^{4}+48 x^{3}-27 x^{2}+8 x -1}{\left(2 x -1\right)^{2} \left(-1+x \right)^{5}}
Generating function in sympy syntax:
(2*x**8 - 6*x**7 - 6*x**6 + 30*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*(-1+x)^5*F(x)+2*x^8-6*x^7-6*x^6+30*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(-1+x \right)^{5} F \! \left(x \right)+2 x^{8}-6 x^{7}-6 x^{6}+30 x^{5}-49 x^{4}+48 x^{3}-27 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,-3-1/24*n^2-17/12*n-7/12*n^3+1/24*n^4+1/4*2^n*n+5*2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -3-\frac{n^{2}}{24}-\frac{17 n}{12}-\frac{7 n^{3}}{12}+\frac{n^{4}}{24}+\frac{2^{n} n}{4}+5 \,2^{-1+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) = 62
a(6) = 171
a(7) = 429
a(8) = 1007
a(n+2) = -4*a(n)+4*a(n+1)+1/24*(-1+n)*(n^3-21*n^2+62*n+48), 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) = 62
a \! \left(6\right) = 171
a \! \left(7\right) = 429
a \! \left(8\right) = 1007
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{\left(-1+n \right) \left(n^{3}-21 n^{2}+62 n +48\right)}{24}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15294/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[25,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[23,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[17,x]*F[4,x]
F[23,x] = F[24,x]
F[24,x] = F[13,x]*F[4,x]
F[25,x] = F[26,x]+F[42,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[1,x]+F[14,x]
F[30,x] = F[18,x]+F[31,x]
F[31,x] = F[32,x]+F[33,x]+F[35,x]
F[32,x] = 0
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[14,x]+F[31,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]
F[37,x] = F[18,x]+F[38,x]
F[38,x] = F[32,x]+F[39,x]+F[41,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[11,x]+F[38,x]
F[41,x] = F[37,x]*F[4,x]
F[42,x] = F[32,x]+F[43,x]+F[97,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[63,x]
F[45,x] = F[46,x]+F[7,x]
F[46,x] = F[32,x]+F[47,x]+F[55,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[52,x]
F[49,x] = F[11,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[49,x]
F[52,x] = F[21,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[52,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[13,x]+F[57,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[49,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[57,x]
F[63,x] = F[64,x]+F[77,x]
F[64,x] = F[32,x]+F[65,x]+F[67,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[64,x]+F[7,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[37,x]+F[69,x]
F[69,x] = F[70,x]+F[73,x]
F[70,x] = F[71,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[14,x]+F[70,x]
F[73,x] = 2*F[32,x]+F[74,x]+F[76,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[23,x]+F[73,x]
F[76,x] = F[4,x]*F[69,x]
F[77,x] = F[32,x]+F[78,x]+F[80,x]+F[87,x]
F[78,x] = F[4,x]*F[79,x]
F[79,x] = F[46,x]+F[77,x]
F[80,x] = F[4,x]*F[81,x]
F[81,x] = F[82,x]
F[82,x] = F[38,x]+F[83,x]
F[83,x] = 2*F[32,x]+F[84,x]+F[86,x]
F[84,x] = F[4,x]*F[85,x]
F[85,x] = F[50,x]+F[83,x]
F[86,x] = F[4,x]*F[82,x]
F[87,x] = F[4,x]*F[88,x]
F[88,x] = F[69,x]+F[89,x]
F[89,x] = F[90,x]+F[93,x]
F[90,x] = F[91,x]
F[91,x] = F[4,x]*F[92,x]
F[92,x] = F[58,x]+F[90,x]
F[93,x] = 3*F[32,x]+F[94,x]+F[96,x]
F[94,x] = F[4,x]*F[95,x]
F[95,x] = F[61,x]+F[93,x]
F[96,x] = F[4,x]*F[89,x]
F[97,x] = F[4,x]*F[98,x]
F[98,x] = F[110,x]+F[99,x]
F[99,x] = F[100,x]+F[26,x]
F[100,x] = F[101,x]+F[109,x]+F[32,x]
F[101,x] = F[102,x]*F[4,x]
F[102,x] = F[103,x]+F[105,x]
F[103,x] = F[104,x]+F[11,x]
F[104,x] = F[24,x]+F[32,x]+F[47,x]
F[105,x] = F[106,x]+F[38,x]
F[106,x] = F[107,x]+F[32,x]+F[76,x]+F[80,x]
F[107,x] = F[108,x]*F[4,x]
F[108,x] = F[104,x]+F[106,x]
F[109,x] = F[4,x]*F[99,x]
F[110,x] = F[111,x]+F[129,x]
F[111,x] = F[112,x]
F[112,x] = F[113,x]*F[4,x]
F[113,x] = F[114,x]+F[115,x]
F[114,x] = F[14,x]+F[58,x]
F[115,x] = F[116,x]+F[70,x]
F[116,x] = 2*F[32,x]+F[117,x]+F[119,x]
F[117,x] = F[118,x]*F[4,x]
F[118,x] = F[116,x]+F[58,x]
F[119,x] = F[120,x]*F[4,x]
F[120,x] = F[121,x]
F[121,x] = F[122,x]+F[125,x]
F[122,x] = F[123,x]
F[123,x] = F[124,x]*F[4,x]
F[124,x] = F[11,x]+F[122,x]
F[125,x] = 2*F[32,x]+F[126,x]+F[128,x]
F[126,x] = F[127,x]*F[4,x]
F[127,x] = F[125,x]+F[50,x]
F[128,x] = F[121,x]*F[4,x]
F[129,x] = 2*F[32,x]+F[130,x]+F[139,x]
F[130,x] = F[131,x]*F[4,x]
F[131,x] = F[132,x]+F[134,x]
F[132,x] = F[133,x]+F[23,x]
F[133,x] = F[62,x]
F[134,x] = F[135,x]+F[73,x]
F[135,x] = 2*F[32,x]+F[136,x]+F[138,x]+F[96,x]
F[136,x] = F[137,x]*F[4,x]
F[137,x] = F[133,x]+F[135,x]
F[138,x] = 0
F[139,x] = F[110,x]*F[4,x]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{25}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{23}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \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_{20}\! \left(x \right) F_{4}\! \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_{22}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{42}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = 0
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right)+F_{31}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)+F_{41}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{11}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{32}\! \left(x \right)+F_{43}\! \left(x \right)+F_{97}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{63}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{7}\! \left(x \right)
F_{46}\! \left(x \right) = F_{32}\! \left(x \right)+F_{47}\! \left(x \right)+F_{55}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{52}\! \left(x \right)
F_{49}\! \left(x \right) = F_{11}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{21}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{13}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{61}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{49}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{77}\! \left(x \right)
F_{64}\! \left(x \right) = F_{32}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{64}\! \left(x \right)+F_{7}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{37}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{73}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{14}\! \left(x \right)+F_{70}\! \left(x \right)
F_{73}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{74}\! \left(x \right)+F_{76}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{23}\! \left(x \right)+F_{73}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{77}\! \left(x \right) = F_{32}\! \left(x \right)+F_{78}\! \left(x \right)+F_{80}\! \left(x \right)+F_{87}\! \left(x \right)
F_{78}\! \left(x \right) = F_{4}\! \left(x \right) F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{46}\! \left(x \right)+F_{77}\! \left(x \right)
F_{80}\! \left(x \right) = F_{4}\! \left(x \right) F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{38}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{84}\! \left(x \right)+F_{86}\! \left(x \right)
F_{84}\! \left(x \right) = F_{4}\! \left(x \right) F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{50}\! \left(x \right)+F_{83}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{87}\! \left(x \right) = F_{4}\! \left(x \right) F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{69}\! \left(x \right)+F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{90}\! \left(x \right)+F_{93}\! \left(x \right)
F_{90}\! \left(x \right) = F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{4}\! \left(x \right) F_{92}\! \left(x \right)
F_{92}\! \left(x \right) = F_{58}\! \left(x \right)+F_{90}\! \left(x \right)
F_{93}\! \left(x \right) = 3 F_{32}\! \left(x \right)+F_{94}\! \left(x \right)+F_{96}\! \left(x \right)
F_{94}\! \left(x \right) = F_{4}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{61}\! \left(x \right)+F_{93}\! \left(x \right)
F_{96}\! \left(x \right) = F_{4}\! \left(x \right) F_{89}\! \left(x \right)
F_{97}\! \left(x \right) = F_{4}\! \left(x \right) F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{110}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)+F_{26}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right)+F_{109}\! \left(x \right)+F_{32}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{4}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right)+F_{105}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{11}\! \left(x \right)
F_{104}\! \left(x \right) = F_{24}\! \left(x \right)+F_{32}\! \left(x \right)+F_{47}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{38}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{32}\! \left(x \right)+F_{76}\! \left(x \right)+F_{80}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right) F_{4}\! \left(x \right)
F_{108}\! \left(x \right) = F_{104}\! \left(x \right)+F_{106}\! \left(x \right)
F_{109}\! \left(x \right) = F_{4}\! \left(x \right) F_{99}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{129}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right) F_{4}\! \left(x \right)
F_{113}\! \left(x \right) = F_{114}\! \left(x \right)+F_{115}\! \left(x \right)
F_{114}\! \left(x \right) = F_{14}\! \left(x \right)+F_{58}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{70}\! \left(x \right)
F_{116}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{117}\! \left(x \right)+F_{119}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right) F_{4}\! \left(x \right)
F_{118}\! \left(x \right) = F_{116}\! \left(x \right)+F_{58}\! \left(x \right)
F_{119}\! \left(x \right) = F_{120}\! \left(x \right) F_{4}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right)
F_{121}\! \left(x \right) = F_{122}\! \left(x \right)+F_{125}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right)
F_{123}\! \left(x \right) = F_{124}\! \left(x \right) F_{4}\! \left(x \right)
F_{124}\! \left(x \right) = F_{11}\! \left(x \right)+F_{122}\! \left(x \right)
F_{125}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{126}\! \left(x \right)+F_{128}\! \left(x \right)
F_{126}\! \left(x \right) = F_{127}\! \left(x \right) F_{4}\! \left(x \right)
F_{127}\! \left(x \right) = F_{125}\! \left(x \right)+F_{50}\! \left(x \right)
F_{128}\! \left(x \right) = F_{121}\! \left(x \right) F_{4}\! \left(x \right)
F_{129}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{130}\! \left(x \right)+F_{139}\! \left(x \right)
F_{130}\! \left(x \right) = F_{131}\! \left(x \right) F_{4}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right)+F_{134}\! \left(x \right)
F_{132}\! \left(x \right) = F_{133}\! \left(x \right)+F_{23}\! \left(x \right)
F_{133}\! \left(x \right) = F_{62}\! \left(x \right)
F_{134}\! \left(x \right) = F_{135}\! \left(x \right)+F_{73}\! \left(x \right)
F_{135}\! \left(x \right) = 2 F_{32}\! \left(x \right)+F_{136}\! \left(x \right)+F_{138}\! \left(x \right)+F_{96}\! \left(x \right)
F_{136}\! \left(x \right) = F_{137}\! \left(x \right) F_{4}\! \left(x \right)
F_{137}\! \left(x \right) = F_{133}\! \left(x \right)+F_{135}\! \left(x \right)
F_{138}\! \left(x \right) = 0
F_{139}\! \left(x \right) = F_{110}\! \left(x \right) F_{4}\! \left(x \right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_25(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_23(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_17(x)*F_4(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_13(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_42(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_1(x) + F_14(x))
Eq(F_30(x), F_18(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_33(x) + F_35(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_14(x) + F_31(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_18(x) + F_38(x))
Eq(F_38(x), F_32(x) + F_39(x) + F_41(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_11(x) + F_38(x))
Eq(F_41(x), F_37(x)*F_4(x))
Eq(F_42(x), F_32(x) + F_43(x) + F_97(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_63(x))
Eq(F_45(x), F_46(x) + F_7(x))
Eq(F_46(x), F_32(x) + F_47(x) + F_55(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_52(x))
Eq(F_49(x), F_11(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_49(x))
Eq(F_52(x), F_21(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_52(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_13(x) + F_57(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_49(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_57(x))
Eq(F_63(x), F_64(x) + F_77(x))
Eq(F_64(x), F_32(x) + F_65(x) + F_67(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_64(x) + F_7(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_37(x) + F_69(x))
Eq(F_69(x), F_70(x) + F_73(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_14(x) + F_70(x))
Eq(F_73(x), 2*F_32(x) + F_74(x) + F_76(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_23(x) + F_73(x))
Eq(F_76(x), F_4(x)*F_69(x))
Eq(F_77(x), F_32(x) + F_78(x) + F_80(x) + F_87(x))
Eq(F_78(x), F_4(x)*F_79(x))
Eq(F_79(x), F_46(x) + F_77(x))
Eq(F_80(x), F_4(x)*F_81(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_38(x) + F_83(x))
Eq(F_83(x), 2*F_32(x) + F_84(x) + F_86(x))
Eq(F_84(x), F_4(x)*F_85(x))
Eq(F_85(x), F_50(x) + F_83(x))
Eq(F_86(x), F_4(x)*F_82(x))
Eq(F_87(x), F_4(x)*F_88(x))
Eq(F_88(x), F_69(x) + F_89(x))
Eq(F_89(x), F_90(x) + F_93(x))
Eq(F_90(x), F_91(x))
Eq(F_91(x), F_4(x)*F_92(x))
Eq(F_92(x), F_58(x) + F_90(x))
Eq(F_93(x), 3*F_32(x) + F_94(x) + F_96(x))
Eq(F_94(x), F_4(x)*F_95(x))
Eq(F_95(x), F_61(x) + F_93(x))
Eq(F_96(x), F_4(x)*F_89(x))
Eq(F_97(x), F_4(x)*F_98(x))
Eq(F_98(x), F_110(x) + F_99(x))
Eq(F_99(x), F_100(x) + F_26(x))
Eq(F_100(x), F_101(x) + F_109(x) + F_32(x))
Eq(F_101(x), F_102(x)*F_4(x))
Eq(F_102(x), F_103(x) + F_105(x))
Eq(F_103(x), F_104(x) + F_11(x))
Eq(F_104(x), F_24(x) + F_32(x) + F_47(x))
Eq(F_105(x), F_106(x) + F_38(x))
Eq(F_106(x), F_107(x) + F_32(x) + F_76(x) + F_80(x))
Eq(F_107(x), F_108(x)*F_4(x))
Eq(F_108(x), F_104(x) + F_106(x))
Eq(F_109(x), F_4(x)*F_99(x))
Eq(F_110(x), F_111(x) + F_129(x))
Eq(F_111(x), F_112(x))
Eq(F_112(x), F_113(x)*F_4(x))
Eq(F_113(x), F_114(x) + F_115(x))
Eq(F_114(x), F_14(x) + F_58(x))
Eq(F_115(x), F_116(x) + F_70(x))
Eq(F_116(x), F_117(x) + F_119(x) + 2*F_32(x))
Eq(F_117(x), F_118(x)*F_4(x))
Eq(F_118(x), F_116(x) + F_58(x))
Eq(F_119(x), F_120(x)*F_4(x))
Eq(F_120(x), F_121(x))
Eq(F_121(x), F_122(x) + F_125(x))
Eq(F_122(x), F_123(x))
Eq(F_123(x), F_124(x)*F_4(x))
Eq(F_124(x), F_11(x) + F_122(x))
Eq(F_125(x), F_126(x) + F_128(x) + 2*F_32(x))
Eq(F_126(x), F_127(x)*F_4(x))
Eq(F_127(x), F_125(x) + F_50(x))
Eq(F_128(x), F_121(x)*F_4(x))
Eq(F_129(x), F_130(x) + F_139(x) + 2*F_32(x))
Eq(F_130(x), F_131(x)*F_4(x))
Eq(F_131(x), F_132(x) + F_134(x))
Eq(F_132(x), F_133(x) + F_23(x))
Eq(F_133(x), F_62(x))
Eq(F_134(x), F_135(x) + F_73(x))
Eq(F_135(x), F_136(x) + F_138(x) + 2*F_32(x) + F_96(x))
Eq(F_136(x), F_137(x)*F_4(x))
Eq(F_137(x), F_133(x) + F_135(x))
Eq(F_138(x), 0)
Eq(F_139(x), F_110(x)*F_4(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": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 2, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 2, 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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 2, 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], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [3, 2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 2], [1, 0], [1, 2]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [3, 2, 0, 1], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": 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