0123_0132_0231_0321
Counting sequence:
1, 1, 2, 6, 20, 68, 232, 792, 2704, 9232, 31520, 107616, 367424, 1254464, 4283008, 14623104, 49926400, 170459392, 581984768, 1987020288, 6784111616, 23162405888, 79081400320, 270000789504, 921840357376, 3147359850496, 10745758687232, 36688315047936, 125261742817280, 427670341173248, 1460157879058432, 4985290833887232, 17020847577432064, 58112808641953792, 198409539412951040, 677412540367896576, 2312831082645684224, 7896499249846943744, 26960334834096406528, 92048340836691738624, 314272693678574141440, 1072994093040913088512, 3663430984806504071168, 12507735753144190107648, 42704081042963752288256, 145800852665566628937728, 497795248576339011174400, 1699579288974222786822144, 5802726658744213124939776, 19811748057028406926114816, 67641538910625201454579712, 230942659528443991966089216, 788487560292525564955197440, 2692064922113214275888611328, 9191284567867805973644050432, 31381008427244795342798979072, 107141464573243569423907815424, 365803841438484687010033303552, 1248932436607451609192317583360, 4264122063552837062749203726336, 14558623380996445032612179738624, 49706249396880106004950311501824, 169707750825527533954576886530048, 579418504508349923808406923116544, 1978258516382344627324473919406080, 6754197056512678661681081831391232, 23060271193286025392075379486752768, 78732690660118744244939354284228608, 268810220253902926195606658163408896, 917775499695374216292547924085178368, 3133481558273691012778978380013895680, 10698375233704015618530817671885225984, 36526537818268680448565313927513112576, 124709400805666690557199620366281998336, 425784527586129401331667853610101768192, 1453719308733184224212272173707843076096, 4963308179760478094185752987611168768000, 16945794101575543928318467603028988919808, 57856560046781219524902364436893618143232, 197534651983973790242972522541516494733312, 674425487842332721922085361292278742646784, 2302632647401383307202396400086081981120512, 7861679613920867784965414877759770439188480, 26841453160880704525456866710866917794512896, 91642453415681082531896637087948130299674624, 312886907340962921076672814930058685609672704, 1068262722532489519242897985544338481839341568, 3647277075448032234818246312317236556138020864, 12452582856727149900787189278180269260873400320, 42515777276012535133512264488086603931217559552, 145157943390595840732474679395985877203123437568, 495600219010358292662874188607770300950058631168, 1692084989260241489186547395639109449393987649536, 5777139519020249371420441205340897195675833335808, 19724388097560514507308670030085369883915358044160, 67343273352201559286393797709659685144309765505024, 229924317213685208130957850778468000809408345931776, 785010722150337713951043807694552632949013852717056, 2680194254173980439542259529221274530177238719004672, 9150755572395246330266950501495992854810927170584576, 31242633781233024441983282947541422358889231244328960
Generating function in Maple syntax:
-(3*x-1)/(2*x^2-4*x+1)
Generating function in latex syntax:
-\frac{3 x -1}{2 x^{2}-4 x +1}
Generating function in sympy syntax:
(1 - 3*x)/(2*x**2 - 4*x + 1)
Implicit equation for the generating function in Maple syntax:
(2*x^2-4*x+1)*F(x)+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{2}-4 x +1\right) F \! \left(x \right)+3 x -1 = 0
Explicit closed form in Maple syntax:
-1/4*(2+2^(1/2))*(2*(1-1/2*2^(1/2))^(-n)*2^(1/2)-(1+1/2*2^(1/2))^(-n)-3*(1-1/2*2^(1/2))^(-n))
Explicit closed form in latex syntax:
-\frac{\left(2+\sqrt{2}\right) \left(2 \left(1-\frac{\sqrt{2}}{2}\right)^{-n} \sqrt{2}-\left(1+\frac{\sqrt{2}}{2}\right)^{-n}-3 \left(1-\frac{\sqrt{2}}{2}\right)^{-n}\right)}{4}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(n+2) = -2*a(n)+4*a(n+1), n >= 2
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(n +2\right) = -2 a \! \left(n \right)+4 a \! \left(n +1\right), \quad n \geq 2
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14793/
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[17,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[131,x]+F[19,x]+F[20,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[31,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[19,x]+F[24,x]+F[27,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[16,x]+F[26,x]
F[26,x] = F[14,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[11,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[15,x]
F[31,x] = F[18,x]+F[32,x]
F[32,x] = F[129,x]+F[19,x]+F[33,x]+F[98,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[42,x]
F[35,x] = F[23,x]+F[36,x]
F[36,x] = F[19,x]+F[37,x]+F[38,x]+F[40,x]
F[37,x] = 0
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[26,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[29,x]
F[42,x] = F[32,x]+F[43,x]
F[43,x] = F[127,x]+F[19,x]+F[44,x]+F[48,x]+F[49,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[36,x]
F[47,x] = F[43,x]
F[48,x] = 0
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[52,x]
F[52,x] = 2*F[19,x]+F[53,x]+F[65,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[58,x]
F[55,x] = F[14,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[26,x]*F[4,x]
F[58,x] = F[52,x]+F[59,x]
F[59,x] = 2*F[19,x]+F[48,x]+F[60,x]+F[64,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[63,x]
F[62,x] = F[56,x]
F[63,x] = F[59,x]
F[64,x] = F[4,x]*F[51,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]
F[67,x] = F[126,x]+F[19,x]+F[68,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[73,x]
F[70,x] = F[4,x]+F[71,x]
F[71,x] = F[19,x]+F[24,x]+F[72,x]
F[72,x] = F[12,x]*F[4,x]
F[73,x] = F[67,x]+F[74,x]
F[74,x] = F[100,x]+F[19,x]+F[75,x]+F[98,x]
F[75,x] = F[4,x]*F[76,x]
F[76,x] = F[77,x]+F[80,x]
F[77,x] = F[71,x]+F[78,x]
F[78,x] = F[19,x]+F[37,x]+F[38,x]+F[79,x]
F[79,x] = F[30,x]*F[4,x]
F[80,x] = F[74,x]+F[81,x]
F[81,x] = F[19,x]+F[48,x]+F[49,x]+F[82,x]+F[86,x]
F[82,x] = F[4,x]*F[83,x]
F[83,x] = F[84,x]+F[85,x]
F[84,x] = F[78,x]
F[85,x] = F[81,x]
F[86,x] = F[4,x]*F[87,x]
F[87,x] = 2*F[19,x]+F[65,x]+F[88,x]
F[88,x] = F[4,x]*F[89,x]
F[89,x] = F[90,x]+F[92,x]
F[90,x] = F[30,x]+F[91,x]
F[91,x] = F[57,x]
F[92,x] = F[87,x]+F[93,x]
F[93,x] = 2*F[19,x]+F[48,x]+F[64,x]+F[94,x]
F[94,x] = F[4,x]*F[95,x]
F[95,x] = F[96,x]+F[97,x]
F[96,x] = F[91,x]
F[97,x] = F[93,x]
F[98,x] = F[4,x]*F[99,x]
F[99,x] = F[51,x]+F[66,x]
F[100,x] = F[101,x]*F[4,x]
F[101,x] = F[102,x]+F[124,x]+F[19,x]
F[102,x] = F[103,x]*F[4,x]
F[103,x] = F[104,x]+F[107,x]
F[104,x] = F[105,x]+F[12,x]
F[105,x] = F[106,x]+F[19,x]+F[24,x]
F[106,x] = F[11,x]*F[4,x]
F[107,x] = F[101,x]+F[108,x]
F[108,x] = F[109,x]+F[122,x]+F[19,x]+F[98,x]
F[109,x] = F[110,x]*F[4,x]
F[110,x] = F[111,x]+F[114,x]
F[111,x] = F[105,x]+F[112,x]
F[112,x] = F[113,x]+F[19,x]+F[37,x]+F[38,x]
F[113,x] = F[29,x]*F[4,x]
F[114,x] = F[108,x]+F[115,x]
F[115,x] = F[116,x]+F[120,x]+F[19,x]+F[48,x]+F[49,x]
F[116,x] = F[117,x]*F[4,x]
F[117,x] = F[118,x]+F[119,x]
F[118,x] = F[112,x]
F[119,x] = F[115,x]
F[120,x] = F[121,x]*F[4,x]
F[121,x] = F[87,x]
F[122,x] = F[123,x]*F[4,x]
F[123,x] = F[101,x]+F[52,x]
F[124,x] = F[125,x]*F[4,x]
F[125,x] = F[2,x]+F[67,x]
F[126,x] = F[2,x]*F[4,x]
F[127,x] = F[128,x]*F[4,x]
F[128,x] = F[121,x]
F[129,x] = F[130,x]*F[4,x]
F[130,x] = F[121,x]+F[123,x]
F[131,x] = F[132,x]*F[4,x]
F[132,x] = F[123,x]+F[125,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_{17}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \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_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{131}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{31}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)
F_{31}\! \left(x \right) = F_{18}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{129}\! \left(x \right)+F_{19}\! \left(x \right)+F_{33}\! \left(x \right)+F_{98}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{42}\! \left(x \right)
F_{35}\! \left(x \right) = F_{23}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = 0
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{26}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{29}\! \left(x \right)
F_{42}\! \left(x \right) = F_{32}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{127}\! \left(x \right)+F_{19}\! \left(x \right)+F_{44}\! \left(x \right)+F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{36}\! \left(x \right)
F_{47}\! \left(x \right) = F_{43}\! \left(x \right)
F_{48}\! \left(x \right) = 0
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{53}\! \left(x \right)+F_{65}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \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_{14}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{58}\! \left(x \right) = F_{52}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{60}\! \left(x \right)+F_{64}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{63}\! \left(x \right)
F_{62}\! \left(x \right) = F_{56}\! \left(x \right)
F_{63}\! \left(x \right) = F_{59}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{126}\! \left(x \right)+F_{19}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \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_{4}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{73}\! \left(x \right) = F_{67}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{100}\! \left(x \right)+F_{19}\! \left(x \right)+F_{75}\! \left(x \right)+F_{98}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x \right)+F_{80}\! \left(x \right)
F_{77}\! \left(x \right) = F_{71}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{80}\! \left(x \right) = F_{74}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{49}\! \left(x \right)+F_{82}\! \left(x \right)+F_{86}\! \left(x \right)
F_{82}\! \left(x \right) = F_{4}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{84}\! \left(x \right)+F_{85}\! \left(x \right)
F_{84}\! \left(x \right) = F_{78}\! \left(x \right)
F_{85}\! \left(x \right) = F_{81}\! \left(x \right)
F_{86}\! \left(x \right) = F_{4}\! \left(x \right) F_{87}\! \left(x \right)
F_{87}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{65}\! \left(x \right)+F_{88}\! \left(x \right)
F_{88}\! \left(x \right) = F_{4}\! \left(x \right) F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = F_{90}\! \left(x \right)+F_{92}\! \left(x \right)
F_{90}\! \left(x \right) = F_{30}\! \left(x \right)+F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{57}\! \left(x \right)
F_{92}\! \left(x \right) = F_{87}\! \left(x \right)+F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{64}\! \left(x \right)+F_{94}\! \left(x \right)
F_{94}\! \left(x \right) = F_{4}\! \left(x \right) F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{96}\! \left(x \right)+F_{97}\! \left(x \right)
F_{96}\! \left(x \right) = F_{91}\! \left(x \right)
F_{97}\! \left(x \right) = F_{93}\! \left(x \right)
F_{98}\! \left(x \right) = F_{4}\! \left(x \right) F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{51}\! \left(x \right)+F_{66}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{4}\! \left(x \right)
F_{101}\! \left(x \right) = F_{102}\! \left(x \right)+F_{124}\! \left(x \right)+F_{19}\! \left(x \right)
F_{102}\! \left(x \right) = F_{103}\! \left(x \right) F_{4}\! \left(x \right)
F_{103}\! \left(x \right) = F_{104}\! \left(x \right)+F_{107}\! \left(x \right)
F_{104}\! \left(x \right) = F_{105}\! \left(x \right)+F_{12}\! \left(x \right)
F_{105}\! \left(x \right) = F_{106}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)
F_{106}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{107}\! \left(x \right) = F_{101}\! \left(x \right)+F_{108}\! \left(x \right)
F_{108}\! \left(x \right) = F_{109}\! \left(x \right)+F_{122}\! \left(x \right)+F_{19}\! \left(x \right)+F_{98}\! \left(x \right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right) F_{4}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{114}\! \left(x \right)
F_{111}\! \left(x \right) = F_{105}\! \left(x \right)+F_{112}\! \left(x \right)
F_{112}\! \left(x \right) = F_{113}\! \left(x \right)+F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{113}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{114}\! \left(x \right) = F_{108}\! \left(x \right)+F_{115}\! \left(x \right)
F_{115}\! \left(x \right) = F_{116}\! \left(x \right)+F_{120}\! \left(x \right)+F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{116}\! \left(x \right) = F_{117}\! \left(x \right) F_{4}\! \left(x \right)
F_{117}\! \left(x \right) = F_{118}\! \left(x \right)+F_{119}\! \left(x \right)
F_{118}\! \left(x \right) = F_{112}\! \left(x \right)
F_{119}\! \left(x \right) = F_{115}\! \left(x \right)
F_{120}\! \left(x \right) = F_{121}\! \left(x \right) F_{4}\! \left(x \right)
F_{121}\! \left(x \right) = F_{87}\! \left(x \right)
F_{122}\! \left(x \right) = F_{123}\! \left(x \right) F_{4}\! \left(x \right)
F_{123}\! \left(x \right) = F_{101}\! \left(x \right)+F_{52}\! \left(x \right)
F_{124}\! \left(x \right) = F_{125}\! \left(x \right) F_{4}\! \left(x \right)
F_{125}\! \left(x \right) = F_{2}\! \left(x \right)+F_{67}\! \left(x \right)
F_{126}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{127}\! \left(x \right) = F_{128}\! \left(x \right) F_{4}\! \left(x \right)
F_{128}\! \left(x \right) = F_{121}\! \left(x \right)
F_{129}\! \left(x \right) = F_{130}\! \left(x \right) F_{4}\! \left(x \right)
F_{130}\! \left(x \right) = F_{121}\! \left(x \right)+F_{123}\! \left(x \right)
F_{131}\! \left(x \right) = F_{132}\! \left(x \right) F_{4}\! \left(x \right)
F_{132}\! \left(x \right) = F_{123}\! \left(x \right)+F_{125}\! \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_17(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_131(x) + F_19(x) + F_20(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_31(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_19(x) + F_24(x) + F_27(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_16(x) + F_26(x))
Eq(F_26(x), F_14(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_11(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_15(x))
Eq(F_31(x), F_18(x) + F_32(x))
Eq(F_32(x), F_129(x) + F_19(x) + F_33(x) + F_98(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_42(x))
Eq(F_35(x), F_23(x) + F_36(x))
Eq(F_36(x), F_19(x) + F_37(x) + F_38(x) + F_40(x))
Eq(F_37(x), 0)
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_26(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_29(x))
Eq(F_42(x), F_32(x) + F_43(x))
Eq(F_43(x), F_127(x) + F_19(x) + F_44(x) + F_48(x) + F_49(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_36(x))
Eq(F_47(x), F_43(x))
Eq(F_48(x), 0)
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), 2*F_19(x) + F_53(x) + F_65(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_58(x))
Eq(F_55(x), F_14(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_26(x)*F_4(x))
Eq(F_58(x), F_52(x) + F_59(x))
Eq(F_59(x), 2*F_19(x) + F_48(x) + F_60(x) + F_64(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_63(x))
Eq(F_62(x), F_56(x))
Eq(F_63(x), F_59(x))
Eq(F_64(x), F_4(x)*F_51(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_126(x) + F_19(x) + F_68(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_73(x))
Eq(F_70(x), F_4(x) + F_71(x))
Eq(F_71(x), F_19(x) + F_24(x) + F_72(x))
Eq(F_72(x), F_12(x)*F_4(x))
Eq(F_73(x), F_67(x) + F_74(x))
Eq(F_74(x), F_100(x) + F_19(x) + F_75(x) + F_98(x))
Eq(F_75(x), F_4(x)*F_76(x))
Eq(F_76(x), F_77(x) + F_80(x))
Eq(F_77(x), F_71(x) + F_78(x))
Eq(F_78(x), F_19(x) + F_37(x) + F_38(x) + F_79(x))
Eq(F_79(x), F_30(x)*F_4(x))
Eq(F_80(x), F_74(x) + F_81(x))
Eq(F_81(x), F_19(x) + F_48(x) + F_49(x) + F_82(x) + F_86(x))
Eq(F_82(x), F_4(x)*F_83(x))
Eq(F_83(x), F_84(x) + F_85(x))
Eq(F_84(x), F_78(x))
Eq(F_85(x), F_81(x))
Eq(F_86(x), F_4(x)*F_87(x))
Eq(F_87(x), 2*F_19(x) + F_65(x) + F_88(x))
Eq(F_88(x), F_4(x)*F_89(x))
Eq(F_89(x), F_90(x) + F_92(x))
Eq(F_90(x), F_30(x) + F_91(x))
Eq(F_91(x), F_57(x))
Eq(F_92(x), F_87(x) + F_93(x))
Eq(F_93(x), 2*F_19(x) + F_48(x) + F_64(x) + F_94(x))
Eq(F_94(x), F_4(x)*F_95(x))
Eq(F_95(x), F_96(x) + F_97(x))
Eq(F_96(x), F_91(x))
Eq(F_97(x), F_93(x))
Eq(F_98(x), F_4(x)*F_99(x))
Eq(F_99(x), F_51(x) + F_66(x))
Eq(F_100(x), F_101(x)*F_4(x))
Eq(F_101(x), F_102(x) + F_124(x) + F_19(x))
Eq(F_102(x), F_103(x)*F_4(x))
Eq(F_103(x), F_104(x) + F_107(x))
Eq(F_104(x), F_105(x) + F_12(x))
Eq(F_105(x), F_106(x) + F_19(x) + F_24(x))
Eq(F_106(x), F_11(x)*F_4(x))
Eq(F_107(x), F_101(x) + F_108(x))
Eq(F_108(x), F_109(x) + F_122(x) + F_19(x) + F_98(x))
Eq(F_109(x), F_110(x)*F_4(x))
Eq(F_110(x), F_111(x) + F_114(x))
Eq(F_111(x), F_105(x) + F_112(x))
Eq(F_112(x), F_113(x) + F_19(x) + F_37(x) + F_38(x))
Eq(F_113(x), F_29(x)*F_4(x))
Eq(F_114(x), F_108(x) + F_115(x))
Eq(F_115(x), F_116(x) + F_120(x) + F_19(x) + F_48(x) + F_49(x))
Eq(F_116(x), F_117(x)*F_4(x))
Eq(F_117(x), F_118(x) + F_119(x))
Eq(F_118(x), F_112(x))
Eq(F_119(x), F_115(x))
Eq(F_120(x), F_121(x)*F_4(x))
Eq(F_121(x), F_87(x))
Eq(F_122(x), F_123(x)*F_4(x))
Eq(F_123(x), F_101(x) + F_52(x))
Eq(F_124(x), F_125(x)*F_4(x))
Eq(F_125(x), F_2(x) + F_67(x))
Eq(F_126(x), F_2(x)*F_4(x))
Eq(F_127(x), F_128(x)*F_4(x))
Eq(F_128(x), F_121(x))
Eq(F_129(x), F_130(x)*F_4(x))
Eq(F_130(x), F_121(x) + F_123(x))
Eq(F_131(x), F_132(x)*F_4(x))
Eq(F_132(x), F_123(x) + F_125(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "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, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"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, 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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}], "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, 1, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"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, 3, 1], "pos": [[1, 0], [1, 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