0231_1032_3102
Counting sequence:
1, 1, 2, 6, 21, 73, 244, 794, 2553, 8179, 26192, 83906, 268883, 861815, 2762484, 8855204, 28385839, 90992639, 291683062, 935010104, 2997239539, 9607859097, 30798658994, 98727239296, 316477019741, 1014489059421, 3252015115894, 10424560242258, 33416651637465, 107119396954821, 343378664280584, 1100724149268710, 3528447684198397, 11310684033232975, 36257182973850532, 116224917373414990, 372567042183052663, 1194289521196760019, 3828377980196090888, 12272131421335344400, 39339169330091125267, 126104438621892578459, 404236533484166392362, 1295808274387948412964, 4153803392038147631383, 13315305173411764834837, 42683135220343584405542, 136823753455988158708524, 438598041431182762042273, 1405956473845371036872921, 4506891093032528484901802, 14447152313969776214566622, 46311349814002665826011101, 148454247244355009567695233, 475880396779635491914663772, 1525467652443703055903495410, 4889992473738486874385610305, 15675210388705054948262237419, 50247975237130678844225895224, 161073373359671957080836253978, 516331881693265253287648951515, 1655137695897145282249294559087, 5305658801071557466556275149148, 17007657660850831370186909636540, 54519227480341835714037800373303, 174765168980039588845105735005559, 560221883181208419708011710959006, 1795830142967132319480469611247440, 5756658208487361014826689019544443, 18453367574391712516602204662824145, 59153550984415564757146078622297306, 189620814735285280228821716549893144, 607842687083084297910494851044678949, 1948481936206086786979306553863329685, 6245994130389984970438355678203466014, 20021967846839654534978611918152735658, 64181808066292725029741796058792732449, 205739241925145983285612119579708374141, 659511424549039459359782473159603683568, 2114109661534346313918594712975230725950, 6776925303529039287216944731077761338565, 21723904585100930076093690557200606342599, 69637484446938760625624218520359625243916, 223227791353103449744333037625704927172646, 715572183977349644646670420714809684523135, 2293816318202779117525517714275552433379659, 7352987468584861874078227776876914083923696, 23570511851414254126299196567314942124050088, 75556912249788353983050459255538521766299803, 242202928163381438527470831413995670298839699, 776398302474045731261197728550468467626296658, 2488798664225712002545894659876526733104800956, 7978016916463765648010972525185337073335283231, 25574087142635027147278390920992872105447992269, 81979511954831624210498725568173107946312178638, 262791017441567993613588781494656783286766076228, 842394852094558707183566392180043649231437095657, 2700355186201144077464705392710313194935071655057, 8656176036110081411492917344269887563202362906322, 27747973285520633343945914541097420048392061324470, 88948054919637173149137972866439140079284052893861
Generating function in Maple syntax:
-(x^2-x+1)*(2*x-1)^3/(4*x^3-7*x^2+5*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{\left(x^{2}-x +1\right) \left(2 x -1\right)^{3}}{\left(4 x^{3}-7 x^{2}+5 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(2*x - 1)**3*(-x**2 + x - 1)/((x - 1)**3*(4*x**3 - 7*x**2 + 5*x - 1))
Implicit equation for the generating function in Maple syntax:
(4*x^3-7*x^2+5*x-1)*(x-1)^3*F(x)+(x^2-x+1)*(2*x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(4 x^{3}-7 x^{2}+5 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(x^{2}-x +1\right) \left(2 x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
1/42834*(((165*I-55*3^(1/2))*59^(1/2)-4543*I*3^(1/2)+4543)*(71+6*59^(1/2)*3^(1/2))^(1/3)+28556+((-579*I-193*3^(1/2))*59^(1/2)+2183*I*3^(1/2)+2183)*(71+6*59^(1/2)*3^(1/2))^(2/3))*(1/2904*((71*I-6*59^(1/2))*3^(1/2)-18*I*59^(1/2)+71)*(71+6*59^(1/2)*3^(1/2))^(2/3)-1/24*I*3^(1/2)*(71+6*59^(1/2)*3^(1/2))^(1/3)+1/24*(71+6*59^(1/2)*3^(1/2))^(1/3)+7/12)^(-n)+1/42834*(((-165*I-55*3^(1/2))*59^(1/2)+4543*I*3^(1/2)+4543)*(71+6*59^(1/2)*3^(1/2))^(1/3)+28556+((579*I-193*3^(1/2))*59^(1/2)-2183*I*3^(1/2)+2183)*(71+6*59^(1/2)*3^(1/2))^(2/3))*(1/2904*((-71*I-6*59^(1/2))*3^(1/2)+18*I*59^(1/2)+71)*(71+6*59^(1/2)*3^(1/2))^(2/3)+1/24*I*3^(1/2)*(71+6*59^(1/2)*3^(1/2))^(1/3)+1/24*(71+6*59^(1/2)*3^(1/2))^(1/3)+7/12)^(-n)+1/42834*((110*59^(1/2)*3^(1/2)-9086)*(71+6*59^(1/2)*3^(1/2))^(1/3)+386*(71+6*59^(1/2)*3^(1/2))^(2/3)*59^(1/2)*3^(1/2)-4366*(71+6*59^(1/2)*3^(1/2))^(2/3)+28556)*(-1/12*(71+6*59^(1/2)*3^(1/2))^(1/3)+7/12-71/1452*(71+6*59^(1/2)*3^(1/2))^(2/3)+1/242*(71+6*59^(1/2)*3^(1/2))^(2/3)*59^(1/2)*3^(1/2))^(-n)+1/2*n^2-5/2*n-1
Explicit closed form in latex syntax:
\frac{\left(\left(\left(165 \,\mathrm{I}-55 \sqrt{3}\right) \sqrt{59}-4543 \,\mathrm{I} \sqrt{3}+4543\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+28556+\left(\left(-579 \,\mathrm{I}-193 \sqrt{3}\right) \sqrt{59}+2183 \,\mathrm{I} \sqrt{3}+2183\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(71 \,\mathrm{I}-6 \sqrt{59}\right) \sqrt{3}-18 \,\mathrm{I} \sqrt{59}+71\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2904}-\frac{\mathrm{I} \sqrt{3}\, \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{\left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{7}{12}\right)^{-n}}{42834}+\frac{\left(\left(\left(-165 \,\mathrm{I}-55 \sqrt{3}\right) \sqrt{59}+4543 \,\mathrm{I} \sqrt{3}+4543\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+28556+\left(\left(579 \,\mathrm{I}-193 \sqrt{3}\right) \sqrt{59}-2183 \,\mathrm{I} \sqrt{3}+2183\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(-71 \,\mathrm{I}-6 \sqrt{59}\right) \sqrt{3}+18 \,\mathrm{I} \sqrt{59}+71\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{2904}+\frac{\mathrm{I} \sqrt{3}\, \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{\left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{24}+\frac{7}{12}\right)^{-n}}{42834}+\frac{\left(\left(110 \sqrt{59}\, \sqrt{3}-9086\right) \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+386 \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{59}\, \sqrt{3}-4366 \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+28556\right) \left(-\frac{\left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{7}{12}-\frac{71 \left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{1452}+\frac{\left(71+6 \sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{59}\, \sqrt{3}}{242}\right)^{-n}}{42834}+\frac{n^{2}}{2}-\frac{5 n}{2}-1
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 73
a(n+3) = -1/2*n^2+5*a(n+2)+4*a(n)-7*a(n+1)+5/2*n-1, n >= 6
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(n +3\right) = -\frac{n^{2}}{2}+5 a \! \left(n +2\right)+4 a \! \left(n \right)-7 a \! \left(n +1\right)+\frac{5 n}{2}-1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14763/
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[6,x]
F[10,x] = F[11,x]+F[16,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[15,x]
F[14,x] = F[1,x]+F[11,x]
F[15,x] = F[11,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[24,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[1,x]+F[20,x]
F[23,x] = F[17,x]
F[24,x] = F[23,x]
F[25,x] = F[26,x]+F[34,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[14,x]+F[29,x]
F[29,x] = F[26,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[29,x]+F[33,x]
F[33,x] = F[30,x]
F[34,x] = F[35,x]+F[36,x]+F[60,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[45,x]
F[38,x] = F[20,x]+F[39,x]
F[39,x] = F[35,x]+F[40,x]+F[43,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[38,x]+F[42,x]
F[42,x] = F[23,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[11,x]+F[23,x]
F[45,x] = F[46,x]+F[49,x]
F[46,x] = F[35,x]+F[36,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[26,x]+F[46,x]
F[49,x] = 2*F[35,x]+F[50,x]+F[58,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[45,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[46,x]+F[53,x]
F[57,x] = F[53,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[30,x]+F[53,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[25,x]+F[62,x]
F[62,x] = F[63,x]+F[84,x]
F[63,x] = F[35,x]+F[64,x]+F[81,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[66,x]+F[72,x]
F[66,x] = F[20,x]+F[67,x]
F[67,x] = F[35,x]+F[68,x]+F[70,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[66,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[11,x]
F[72,x] = F[73,x]+F[76,x]
F[73,x] = F[74,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[26,x]+F[73,x]
F[76,x] = 2*F[35,x]+F[77,x]+F[79,x]
F[77,x] = F[4,x]*F[78,x]
F[78,x] = F[72,x]
F[79,x] = F[4,x]*F[80,x]
F[80,x] = F[30,x]
F[81,x] = F[4,x]*F[82,x]
F[82,x] = F[29,x]+F[83,x]
F[83,x] = F[63,x]
F[84,x] = 2*F[35,x]+F[105,x]+F[85,x]
F[85,x] = F[4,x]*F[86,x]
F[86,x] = F[87,x]+F[96,x]
F[87,x] = F[88,x]+F[91,x]
F[88,x] = F[89,x]
F[89,x] = F[4,x]*F[90,x]
F[90,x] = F[20,x]+F[88,x]
F[91,x] = 2*F[35,x]+F[92,x]+F[94,x]
F[92,x] = F[4,x]*F[93,x]
F[93,x] = F[87,x]
F[94,x] = F[4,x]*F[95,x]
F[95,x] = F[23,x]
F[96,x] = F[100,x]+F[97,x]
F[97,x] = F[98,x]
F[98,x] = F[4,x]*F[99,x]
F[99,x] = F[46,x]+F[97,x]
F[100,x] = 3*F[35,x]+F[101,x]+F[103,x]
F[101,x] = F[102,x]*F[4,x]
F[102,x] = F[96,x]
F[103,x] = F[104,x]*F[4,x]
F[104,x] = F[53,x]
F[105,x] = F[106,x]*F[4,x]
F[106,x] = F[107,x]+F[56,x]
F[107,x] = F[108,x]
F[108,x] = 2*F[35,x]+F[105,x]+F[85,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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{16}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{15}\! \left(x \right) = F_{11}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{17}\! \left(x \right)
F_{24}\! \left(x \right) = F_{23}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{34}\! \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_{14}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{29}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{30}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{60}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{45}\! \left(x \right)
F_{38}\! \left(x \right) = F_{20}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{38}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{23}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right)+F_{23}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{46}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{26}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{50}\! \left(x \right)+F_{58}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{45}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \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_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{46}\! \left(x \right)+F_{53}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{30}\! \left(x \right)+F_{53}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{25}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{84}\! \left(x \right)
F_{63}\! \left(x \right) = F_{35}\! \left(x \right)+F_{64}\! \left(x \right)+F_{81}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)+F_{72}\! \left(x \right)
F_{66}\! \left(x \right) = F_{20}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{35}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{66}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{11}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)+F_{76}\! \left(x \right)
F_{73}\! \left(x \right) = F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{26}\! \left(x \right)+F_{73}\! \left(x \right)
F_{76}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{77}\! \left(x \right)+F_{79}\! \left(x \right)
F_{77}\! \left(x \right) = F_{4}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{72}\! \left(x \right)
F_{79}\! \left(x \right) = F_{4}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{30}\! \left(x \right)
F_{81}\! \left(x \right) = F_{4}\! \left(x \right) F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{29}\! \left(x \right)+F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{63}\! \left(x \right)
F_{84}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{105}\! \left(x \right)+F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{4}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{87}\! \left(x \right)+F_{96}\! \left(x \right)
F_{87}\! \left(x \right) = F_{88}\! \left(x \right)+F_{91}\! \left(x \right)
F_{88}\! \left(x \right) = F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{4}\! \left(x \right) F_{90}\! \left(x \right)
F_{90}\! \left(x \right) = F_{20}\! \left(x \right)+F_{88}\! \left(x \right)
F_{91}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{92}\! \left(x \right)+F_{94}\! \left(x \right)
F_{92}\! \left(x \right) = F_{4}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{87}\! \left(x \right)
F_{94}\! \left(x \right) = F_{4}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{23}\! \left(x \right)
F_{96}\! \left(x \right) = F_{100}\! \left(x \right)+F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{98}\! \left(x \right)
F_{98}\! \left(x \right) = F_{4}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{46}\! \left(x \right)+F_{97}\! \left(x \right)
F_{100}\! \left(x \right) = 3 F_{35}\! \left(x \right)+F_{101}\! \left(x \right)+F_{103}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right) F_{4}\! \left(x \right)
F_{102}\! \left(x \right) = F_{96}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right) F_{4}\! \left(x \right)
F_{104}\! \left(x \right) = F_{53}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right) F_{4}\! \left(x \right)
F_{106}\! \left(x \right) = F_{107}\! \left(x \right)+F_{56}\! \left(x \right)
F_{107}\! \left(x \right) = F_{108}\! \left(x \right)
F_{108}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{105}\! \left(x \right)+F_{85}\! \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_6(x))
Eq(F_10(x), F_11(x) + F_16(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_15(x))
Eq(F_14(x), F_1(x) + F_11(x))
Eq(F_15(x), F_11(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_24(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_1(x) + F_20(x))
Eq(F_23(x), F_17(x))
Eq(F_24(x), F_23(x))
Eq(F_25(x), F_26(x) + F_34(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_14(x) + F_29(x))
Eq(F_29(x), F_26(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_29(x) + F_33(x))
Eq(F_33(x), F_30(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_60(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_45(x))
Eq(F_38(x), F_20(x) + F_39(x))
Eq(F_39(x), F_35(x) + F_40(x) + F_43(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_38(x) + F_42(x))
Eq(F_42(x), F_23(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_11(x) + F_23(x))
Eq(F_45(x), F_46(x) + F_49(x))
Eq(F_46(x), F_35(x) + F_36(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_26(x) + F_46(x))
Eq(F_49(x), 2*F_35(x) + F_50(x) + F_58(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_45(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_46(x) + F_53(x))
Eq(F_57(x), F_53(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_30(x) + F_53(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_25(x) + F_62(x))
Eq(F_62(x), F_63(x) + F_84(x))
Eq(F_63(x), F_35(x) + F_64(x) + F_81(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_66(x) + F_72(x))
Eq(F_66(x), F_20(x) + F_67(x))
Eq(F_67(x), F_35(x) + F_68(x) + F_70(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_66(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_11(x))
Eq(F_72(x), F_73(x) + F_76(x))
Eq(F_73(x), F_74(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_26(x) + F_73(x))
Eq(F_76(x), 2*F_35(x) + F_77(x) + F_79(x))
Eq(F_77(x), F_4(x)*F_78(x))
Eq(F_78(x), F_72(x))
Eq(F_79(x), F_4(x)*F_80(x))
Eq(F_80(x), F_30(x))
Eq(F_81(x), F_4(x)*F_82(x))
Eq(F_82(x), F_29(x) + F_83(x))
Eq(F_83(x), F_63(x))
Eq(F_84(x), F_105(x) + 2*F_35(x) + F_85(x))
Eq(F_85(x), F_4(x)*F_86(x))
Eq(F_86(x), F_87(x) + F_96(x))
Eq(F_87(x), F_88(x) + F_91(x))
Eq(F_88(x), F_89(x))
Eq(F_89(x), F_4(x)*F_90(x))
Eq(F_90(x), F_20(x) + F_88(x))
Eq(F_91(x), 2*F_35(x) + F_92(x) + F_94(x))
Eq(F_92(x), F_4(x)*F_93(x))
Eq(F_93(x), F_87(x))
Eq(F_94(x), F_4(x)*F_95(x))
Eq(F_95(x), F_23(x))
Eq(F_96(x), F_100(x) + F_97(x))
Eq(F_97(x), F_98(x))
Eq(F_98(x), F_4(x)*F_99(x))
Eq(F_99(x), F_46(x) + F_97(x))
Eq(F_100(x), F_101(x) + F_103(x) + 3*F_35(x))
Eq(F_101(x), F_102(x)*F_4(x))
Eq(F_102(x), F_96(x))
Eq(F_103(x), F_104(x)*F_4(x))
Eq(F_104(x), F_53(x))
Eq(F_105(x), F_106(x)*F_4(x))
Eq(F_106(x), F_107(x) + F_56(x))
Eq(F_107(x), F_108(x))
Eq(F_108(x), F_105(x) + 2*F_35(x) + F_85(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "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, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "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, 0], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 0], [1, 0], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 2], [1, 0], [1, 2]]}, {"patt": [3, 1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 2, 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