0132_0213_0231_2013
Counting sequence:
1, 1, 2, 6, 20, 69, 243, 869, 3145, 11491, 42312, 156807, 584288, 2187298, 8221257, 31009841, 117331070, 445174418, 1693270531, 6454992143, 24657428519, 94363587324, 361741068087, 1388892123038, 5340282880156, 20560742443041, 79259430563491, 305889059254747, 1181802118618960, 4570499385804947, 17692683248902634, 68550476534471406, 265822499525416836, 1031613845900279901, 4006527642399008655, 15571397416181852853, 60559148904455939048, 235673169608998140253, 917709643904587509054, 3575631631529015964270, 13939301242692758826397, 54369987660705801475529, 212176573414925582011127, 828411706354230917154696, 3235913176478977151503041, 12645648990004976329480208, 49439186215978952578377892, 193366037438897909408679069, 756590646274920839562980815, 2961469143350013194167003281, 11596124822151832749415155542, 45422628771057488294401579517, 177983589697271967936907236998, 697639176949023409691433421108, 2735395450217076856703528308697, 10728602286414391575491759432507, 42091613909571808460791246982307, 165186013541182498232135546901864, 648443718425109670486308943564247, 2546178644438042715834540679296670, 10000447827192502713762458044573194, 39288062307166909319004307113571753, 154386421877513842848005936000685219, 606822698201633308502550531894655301, 2385699752989551727477509296184000569, 9381411074701694566534276631574900035, 36899139656476796221870415653502834800, 145163490725157361504571055225793294679, 571201262637185112473543253238322229380, 2248066290964769144308475372892296644434, 8849423316489006943322176753942630083129, 34842116075313076956917367147647115147389, 137206805585348302866677606078552515877113, 540413666476914919145530680181857173569788, 2128896627143126483239581104678628291629077, 8388001886447295099186132354857805414600606, 33054937814021608991426643605267218083292730, 130282564875529065944128644225196594719456803, 513578341211559608190239803989079225946488181, 2024864283484980812281632296074317184891820863, 7984585779890462253596737214465622177367551764, 31490140609951876254570047661044744683813741839, 124211293741641528860213313322934086167658282202, 490016197082662145184111213964547551827507415878, 1933398094939533068861119952675293167644128720601, 7629433970202043369108613809586709797800586147355, 30110797416451867997338306256442615977951491146169, 118852922925756448680576536826414102866493591952226, 469195624172067912720083623783294781533523742243189, 1852479280822668149302022785341876816860281889687150, 7314875932632587662156899165109923595409942234649620, 28887747721547979640891042498207705273866193638818489, 114096536525248565801641184293713901159139910555108115, 450694522083775613385088177249833614573404821007104439, 1780500693326583257015432954955468867844374858674305824, 7034789445840325807921000037029076707973083821317427879, 27797658525519720422244666448741878809230374562150346088, 109853166131135355958843500379624249019231636445012534498, 434173499847714602861706828834905020465206319544257655217, 1716167064526079528550233986343384161486027649829531628281, 6784229016646046595383554276969278976966523656438855618695
Generating function in Maple syntax:
((x-1)*(-4*x+1)^(1/2)-2*x^2+5*x-1)/(2*x^3-8*x^2+10*x-2)
Generating function in latex syntax:
\frac{\left(x -1\right) \sqrt{-4 x +1}-2 x^{2}+5 x -1}{2 x^{3}-8 x^{2}+10 x -2}
Generating function in sympy syntax:
(-2*x**2 + 5*x + sqrt(1 - 4*x)*(x - 1) - 1)/(2*x**3 - 8*x**2 + 10*x - 2)
Implicit equation for the generating function in Maple syntax:
(x^3-4*x^2+5*x-1)*F(x)^2+(2*x^2-5*x+1)*F(x)+x = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-4 x^{2}+5 x -1\right) F \! \left(x \right)^{2}+\left(2 x^{2}-5 x +1\right) F \! \left(x \right)+x = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+5) = 2*(3+2*n)/(n+5)*a(n)-(41+21*n)/(n+5)*a(n+1)+(103+41*n)/(n+5)*a(n+2)-3*(37+11*n)/(n+5)*a(n+3)+2*(21+5*n)/(n+5)*a(n+4), n >= 5
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(n +5\right) = \frac{2 \left(3+2 n \right) a \! \left(n \right)}{n +5}-\frac{\left(41+21 n \right) a \! \left(n +1\right)}{n +5}+\frac{\left(103+41 n \right) a \! \left(n +2\right)}{n +5}-\frac{3 \left(37+11 n \right) a \! \left(n +3\right)}{n +5}+\frac{2 \left(21+5 n \right) a \! \left(n +4\right)}{n +5}, \quad n \geq 5
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21169/
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[9,x]
F[3,x] = F[1,x]+F[124,x]+F[4,x]
F[4,x] = F[5,x]
F[5,x] = F[27,x]*F[6,x]*F[9,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[10,x]*F[9,x]
F[9,x] = x
F[10,x] = F[11,x]+F[16,x]
F[11,x] = F[12,x,1]
F[12,x,k[0]] = F[1,x]+F[13,x,k[0]]
F[13,x,k[0]] = F[14,x,k[0]]
F[14,x,k[0]] = F[12,x,k[0]]*F[15,x,k[0]]
F[15,x,k[0]] = k[0]*x
F[16,x] = F[17,x]+F[20,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[9,x]
F[19,x] = F[1,x]+F[17,x]
F[20,x] = F[21,x]+F[22,x]+F[26,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[9,x]
F[23,x] = F[20,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[11,x]*F[9,x]
F[26,x] = F[16,x]*F[9,x]
F[27,x] = F[28,x,1]
F[28,x,k[0]] = F[1,x]+F[29,x,k[0]]+F[30,x,k[0]]
F[29,x,k[0]] = F[15,x,k[0]]*F[28,x,k[0]]
F[30,x,k[0]] = F[31,x,k[0]]*F[9,x]
F[31,x,k[0]] = F[1,x]+F[107,x,k[0]]+F[32,x,k[0]]+F[64,x,k[0]]
F[32,x,k[0]] = F[33,x,k[0]]
F[33,x,k[0]] = F[15,x,k[0]]*F[34,x,k[0]]
F[34,x,k[0]] = F[35,x,k[0]]+F[63,x,k[0]]
F[35,x,k[0]] = F[36,x]+F[43,x,k[0]]
F[36,x] = F[1,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[9,x]
F[39,x] = F[11,x]+F[40,x]
F[40,x] = F[17,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[40,x]*F[9,x]
F[43,x,k[0]] = F[44,x,k[0]]+F[47,x,k[0]]
F[44,x,k[0]] = F[45,x,k[0]]
F[45,x,k[0]] = F[15,x,k[0]]*F[46,x,k[0]]
F[46,x,k[0]] = F[1,x]+F[44,x,k[0]]
F[47,x,k[0]] = F[21,x]+F[48,x,k[0]]+F[50,x,k[0]]
F[48,x,k[0]] = F[15,x,k[0]]*F[49,x,k[0]]
F[49,x,k[0]] = F[37,x]+F[47,x,k[0]]
F[50,x,k[0]] = F[51,x,k[0]]*F[9,x]
F[51,x,k[0]] = F[52,x,k[0]]+F[55,x,k[0]]
F[52,x,k[0]] = F[44,x,k[0]]+F[53,x,k[0]]
F[53,x,k[0]] = F[54,x,k[0]]
F[54,x,k[0]] = F[52,x,k[0]]*F[9,x]
F[55,x,k[0]] = F[56,x,k[0]]+F[61,x,k[0]]
F[56,x,k[0]] = F[21,x]+F[57,x,k[0]]+F[59,x,k[0]]
F[57,x,k[0]] = F[15,x,k[0]]*F[58,x,k[0]]
F[58,x,k[0]] = F[17,x]+F[56,x,k[0]]
F[59,x,k[0]] = F[60,x,k[0]]*F[9,x]
F[60,x,k[0]] = F[44,x,k[0]]+F[56,x,k[0]]
F[61,x,k[0]] = F[62,x,k[0]]
F[62,x,k[0]] = F[55,x,k[0]]*F[9,x]
F[63,x,k[0]] = F[11,x]*F[46,x,k[0]]*F[9,x]
F[64,x,k[0]] = F[65,x,k[0]]
F[65,x,k[0]] = F[66,x,k[0]]*F[9,x]
F[66,x,k[0]] = F[105,x,k[0]]+F[11,x]+F[67,x,k[0]]+F[69,x,k[0]]
F[67,x,k[0]] = F[15,x,k[0]]*F[68,x,k[0]]
F[68,x,k[0]] = (F[34,x,k[0]]*k[0]-F[34,x,1])/(-1+k[0])
F[69,x,k[0]] = F[70,x,k[0]]
F[70,x,k[0]] = F[71,x,k[0]]*F[76,x]*F[9,x]
F[71,x,k[0]] = F[1,x]+F[72,x,k[0]]+F[74,x,k[0]]
F[72,x,k[0]] = F[73,x,k[0]]
F[73,x,k[0]] = F[15,x,k[0]]*F[19,x]*F[46,x,k[0]]
F[74,x,k[0]] = F[75,x,k[0]]
F[75,x,k[0]] = F[19,x]*F[71,x,k[0]]*F[9,x]
F[76,x] = F[77,x,1]
F[77,x,k[0]] = F[104,x,k[0]]+F[78,x]
F[78,x] = F[79,x,1]
F[79,x,k[0]] = F[80,x,k[0]]+F[87,x,k[0]]
F[80,x,k[0]] = F[1,x]+F[81,x,k[0]]
F[81,x,k[0]] = F[82,x,k[0]]
F[82,x,k[0]] = F[15,x,k[0]]*F[83,x,k[0]]
F[83,x,k[0]] = F[12,x,k[0]]+F[84,x,k[0]]
F[84,x,k[0]] = F[44,x,k[0]]+F[85,x,k[0]]
F[85,x,k[0]] = F[86,x,k[0]]
F[86,x,k[0]] = F[15,x,k[0]]*F[84,x,k[0]]
F[87,x,k[0]] = F[17,x]+F[88,x,k[0]]
F[88,x,k[0]] = F[21,x]+F[89,x,k[0]]+F[91,x,k[0]]
F[89,x,k[0]] = F[9,x]*F[90,x,k[0]]
F[90,x,k[0]] = F[81,x,k[0]]+F[88,x,k[0]]
F[91,x,k[0]] = F[15,x,k[0]]*F[92,x,k[0]]
F[92,x,k[0]] = F[93,x,k[0]]+F[96,x,k[0]]
F[93,x,k[0]] = F[17,x]+F[94,x,k[0]]
F[94,x,k[0]] = F[95,x,k[0]]
F[95,x,k[0]] = F[15,x,k[0]]*F[93,x,k[0]]
F[96,x,k[0]] = F[102,x,k[0]]+F[97,x,k[0]]
F[97,x,k[0]] = F[100,x,k[0]]+F[21,x]+F[98,x,k[0]]
F[98,x,k[0]] = F[9,x]*F[99,x,k[0]]
F[99,x,k[0]] = F[44,x,k[0]]+F[97,x,k[0]]
F[100,x,k[0]] = F[101,x,k[0]]*F[15,x,k[0]]
F[101,x,k[0]] = F[17,x]+F[97,x,k[0]]
F[102,x,k[0]] = F[103,x,k[0]]
F[103,x,k[0]] = F[15,x,k[0]]*F[96,x,k[0]]
F[104,x,k[0]] = F[12,x,k[0]]*F[15,x,k[0]]*F[19,x]
F[105,x,k[0]] = F[106,x,k[0]]
F[106,x,k[0]] = F[68,x,k[0]]*F[9,x]
F[107,x,k[0]] = F[108,x,k[0]]*F[9,x]
F[108,x,k[0]] = F[1,x]+F[109,x,k[0]]+F[116,x,k[0]]+F[119,x,k[0]]+F[121,x,k[0]]
F[109,x,k[0]] = F[110,x,k[0]]*F[15,x,k[0]]
F[110,x,k[0]] = (F[111,x,k[0]]*k[0]-F[111,x,1])/(-1+k[0])
F[111,x,k[0]] = F[1,x]+F[112,x,k[0]]+F[113,x,k[0]]+F[114,x,k[0]]
F[112,x,k[0]] = F[111,x,k[0]]*F[15,x,k[0]]
F[113,x,k[0]] = F[108,x,k[0]]*F[9,x]
F[114,x,k[0]] = F[115,x,k[0]]
F[115,x,k[0]] = F[66,x,k[0]]*F[9,x]
F[116,x,k[0]] = F[117,x,k[0]]
F[117,x,k[0]] = F[118,x]*F[71,x,k[0]]*F[9,x]
F[118,x] = F[111,x,1]
F[119,x,k[0]] = F[120,x,k[0]]*F[9,x]
F[120,x,k[0]] = (F[108,x,k[0]]*k[0]-F[108,x,1])/(-1+k[0])
F[121,x,k[0]] = F[122,x,k[0]]
F[122,x,k[0]] = F[123,x,k[0]]*F[9,x]
F[123,x,k[0]] = (F[66,x,k[0]]*k[0]-F[66,x,1])/(-1+k[0])
F[124,x] = F[30,x,1]
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_{9}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{124}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{27}\! \left(x \right) F_{6}\! \left(x \right) F_{9}\! \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_{10}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{16}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x , 1\right)
F_{12}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = F_{14}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{15}\! \left(x , y\right)
F_{15}\! \left(x , y\right) = y x
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{9}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{9}\! \left(x \right)
F_{23}\! \left(x \right) = F_{20}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right) F_{9}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{9}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x , 1\right)
F_{28}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{29}\! \left(x , y\right)+F_{30}\! \left(x , y\right)
F_{29}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{28}\! \left(x , y\right)
F_{30}\! \left(x , y\right) = F_{31}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{31}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{107}\! \left(x , y\right)+F_{32}\! \left(x , y\right)+F_{64}\! \left(x , y\right)
F_{32}\! \left(x , y\right) = F_{33}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{34}\! \left(x , y\right)
F_{34}\! \left(x , y\right) = F_{35}\! \left(x , y\right)+F_{63}\! \left(x , y\right)
F_{35}\! \left(x , y\right) = F_{36}\! \left(x \right)+F_{43}\! \left(x , y\right)
F_{36}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{9}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{17}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{40}\! \left(x \right) F_{9}\! \left(x \right)
F_{43}\! \left(x , y\right) = F_{44}\! \left(x , y\right)+F_{47}\! \left(x , y\right)
F_{44}\! \left(x , y\right) = F_{45}\! \left(x , y\right)
F_{45}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{46}\! \left(x , y\right)
F_{46}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{44}\! \left(x , y\right)
F_{47}\! \left(x , y\right) = F_{21}\! \left(x \right)+F_{48}\! \left(x , y\right)+F_{50}\! \left(x , y\right)
F_{48}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{49}\! \left(x , y\right)
F_{49}\! \left(x , y\right) = F_{37}\! \left(x \right)+F_{47}\! \left(x , y\right)
F_{50}\! \left(x , y\right) = F_{51}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{51}\! \left(x , y\right) = F_{52}\! \left(x , y\right)+F_{55}\! \left(x , y\right)
F_{52}\! \left(x , y\right) = F_{44}\! \left(x , y\right)+F_{53}\! \left(x , y\right)
F_{53}\! \left(x , y\right) = F_{54}\! \left(x , y\right)
F_{54}\! \left(x , y\right) = F_{52}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{55}\! \left(x , y\right) = F_{56}\! \left(x , y\right)+F_{61}\! \left(x , y\right)
F_{56}\! \left(x , y\right) = F_{21}\! \left(x \right)+F_{57}\! \left(x , y\right)+F_{59}\! \left(x , y\right)
F_{57}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{58}\! \left(x , y\right)
F_{58}\! \left(x , y\right) = F_{17}\! \left(x \right)+F_{56}\! \left(x , y\right)
F_{59}\! \left(x , y\right) = F_{60}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{60}\! \left(x , y\right) = F_{44}\! \left(x , y\right)+F_{56}\! \left(x , y\right)
F_{61}\! \left(x , y\right) = F_{62}\! \left(x , y\right)
F_{62}\! \left(x , y\right) = F_{55}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{63}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{46}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{64}\! \left(x , y\right) = F_{65}\! \left(x , y\right)
F_{65}\! \left(x , y\right) = F_{66}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{66}\! \left(x , y\right) = F_{105}\! \left(x , y\right)+F_{11}\! \left(x \right)+F_{67}\! \left(x , y\right)+F_{69}\! \left(x , y\right)
F_{67}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{68}\! \left(x , y\right)
F_{68}\! \left(x , y\right) = -\frac{-y F_{34}\! \left(x , y\right)+F_{34}\! \left(x , 1\right)}{-1+y}
F_{69}\! \left(x , y\right) = F_{70}\! \left(x , y\right)
F_{70}\! \left(x , y\right) = F_{71}\! \left(x , y\right) F_{76}\! \left(x \right) F_{9}\! \left(x \right)
F_{71}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{72}\! \left(x , y\right)+F_{74}\! \left(x , y\right)
F_{72}\! \left(x , y\right) = F_{73}\! \left(x , y\right)
F_{73}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{19}\! \left(x \right) F_{46}\! \left(x , y\right)
F_{74}\! \left(x , y\right) = F_{75}\! \left(x , y\right)
F_{75}\! \left(x , y\right) = F_{19}\! \left(x \right) F_{71}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{76}\! \left(x \right) = F_{77}\! \left(x , 1\right)
F_{77}\! \left(x , y\right) = F_{104}\! \left(x , y\right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{79}\! \left(x , 1\right)
F_{79}\! \left(x , y\right) = F_{80}\! \left(x , y\right)+F_{87}\! \left(x , y\right)
F_{80}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{81}\! \left(x , y\right)
F_{81}\! \left(x , y\right) = F_{82}\! \left(x , y\right)
F_{82}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{83}\! \left(x , y\right)
F_{83}\! \left(x , y\right) = F_{12}\! \left(x , y\right)+F_{84}\! \left(x , y\right)
F_{84}\! \left(x , y\right) = F_{44}\! \left(x , y\right)+F_{85}\! \left(x , y\right)
F_{85}\! \left(x , y\right) = F_{86}\! \left(x , y\right)
F_{86}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{84}\! \left(x , y\right)
F_{87}\! \left(x , y\right) = F_{17}\! \left(x \right)+F_{88}\! \left(x , y\right)
F_{88}\! \left(x , y\right) = F_{21}\! \left(x \right)+F_{89}\! \left(x , y\right)+F_{91}\! \left(x , y\right)
F_{89}\! \left(x , y\right) = F_{9}\! \left(x \right) F_{90}\! \left(x , y\right)
F_{90}\! \left(x , y\right) = F_{81}\! \left(x , y\right)+F_{88}\! \left(x , y\right)
F_{91}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{92}\! \left(x , y\right)
F_{92}\! \left(x , y\right) = F_{93}\! \left(x , y\right)+F_{96}\! \left(x , y\right)
F_{93}\! \left(x , y\right) = F_{17}\! \left(x \right)+F_{94}\! \left(x , y\right)
F_{94}\! \left(x , y\right) = F_{95}\! \left(x , y\right)
F_{95}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{93}\! \left(x , y\right)
F_{96}\! \left(x , y\right) = F_{102}\! \left(x , y\right)+F_{97}\! \left(x , y\right)
F_{97}\! \left(x , y\right) = F_{100}\! \left(x , y\right)+F_{21}\! \left(x \right)+F_{98}\! \left(x , y\right)
F_{98}\! \left(x , y\right) = F_{9}\! \left(x \right) F_{99}\! \left(x , y\right)
F_{99}\! \left(x , y\right) = F_{44}\! \left(x , y\right)+F_{97}\! \left(x , y\right)
F_{100}\! \left(x , y\right) = F_{101}\! \left(x , y\right) F_{15}\! \left(x , y\right)
F_{101}\! \left(x , y\right) = F_{17}\! \left(x \right)+F_{97}\! \left(x , y\right)
F_{102}\! \left(x , y\right) = F_{103}\! \left(x , y\right)
F_{103}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{96}\! \left(x , y\right)
F_{104}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{15}\! \left(x , y\right) F_{19}\! \left(x \right)
F_{105}\! \left(x , y\right) = F_{106}\! \left(x , y\right)
F_{106}\! \left(x , y\right) = F_{68}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{107}\! \left(x , y\right) = F_{108}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{108}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{109}\! \left(x , y\right)+F_{116}\! \left(x , y\right)+F_{119}\! \left(x , y\right)+F_{121}\! \left(x , y\right)
F_{109}\! \left(x , y\right) = F_{110}\! \left(x , y\right) F_{15}\! \left(x , y\right)
F_{110}\! \left(x , y\right) = -\frac{-y F_{111}\! \left(x , y\right)+F_{111}\! \left(x , 1\right)}{-1+y}
F_{111}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{112}\! \left(x , y\right)+F_{113}\! \left(x , y\right)+F_{114}\! \left(x , y\right)
F_{112}\! \left(x , y\right) = F_{111}\! \left(x , y\right) F_{15}\! \left(x , y\right)
F_{113}\! \left(x , y\right) = F_{108}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{114}\! \left(x , y\right) = F_{115}\! \left(x , y\right)
F_{115}\! \left(x , y\right) = F_{66}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{116}\! \left(x , y\right) = F_{117}\! \left(x , y\right)
F_{117}\! \left(x , y\right) = F_{118}\! \left(x \right) F_{71}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{118}\! \left(x \right) = F_{111}\! \left(x , 1\right)
F_{119}\! \left(x , y\right) = F_{120}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{120}\! \left(x , y\right) = -\frac{-y F_{108}\! \left(x , y\right)+F_{108}\! \left(x , 1\right)}{-1+y}
F_{121}\! \left(x , y\right) = F_{122}\! \left(x , y\right)
F_{122}\! \left(x , y\right) = F_{123}\! \left(x , y\right) F_{9}\! \left(x \right)
F_{123}\! \left(x , y\right) = -\frac{-y F_{66}\! \left(x , y\right)+F_{66}\! \left(x , 1\right)}{-1+y}
F_{124}\! \left(x \right) = F_{30}\! \left(x , 1\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)*F_9(x))
Eq(F_3(x), F_1(x) + F_124(x) + F_4(x))
Eq(F_4(x), F_5(x))
Eq(F_5(x), F_27(x)*F_6(x)*F_9(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_10(x)*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x) + F_16(x))
Eq(F_11(x), F_12(x, 1))
Eq(F_12(x, k_0), F_1(x) + F_13(x, k_0))
Eq(F_13(x, k_0), F_14(x, k_0))
Eq(F_14(x, k_0), F_12(x, k_0)*F_15(x, k_0))
Eq(F_15(x, k_0), k_0*x)
Eq(F_16(x), F_17(x) + F_20(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_9(x))
Eq(F_19(x), F_1(x) + F_17(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_26(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_9(x))
Eq(F_23(x), F_20(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_11(x)*F_9(x))
Eq(F_26(x), F_16(x)*F_9(x))
Eq(F_27(x), F_28(x, 1))
Eq(F_28(x, k_0), F_1(x) + F_29(x, k_0) + F_30(x, k_0))
Eq(F_29(x, k_0), F_15(x, k_0)*F_28(x, k_0))
Eq(F_30(x, k_0), F_31(x, k_0)*F_9(x))
Eq(F_31(x, k_0), F_1(x) + F_107(x, k_0) + F_32(x, k_0) + F_64(x, k_0))
Eq(F_32(x, k_0), F_33(x, k_0))
Eq(F_33(x, k_0), F_15(x, k_0)*F_34(x, k_0))
Eq(F_34(x, k_0), F_35(x, k_0) + F_63(x, k_0))
Eq(F_35(x, k_0), F_36(x) + F_43(x, k_0))
Eq(F_36(x), F_1(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_9(x))
Eq(F_39(x), F_11(x) + F_40(x))
Eq(F_40(x), F_17(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_40(x)*F_9(x))
Eq(F_43(x, k_0), F_44(x, k_0) + F_47(x, k_0))
Eq(F_44(x, k_0), F_45(x, k_0))
Eq(F_45(x, k_0), F_15(x, k_0)*F_46(x, k_0))
Eq(F_46(x, k_0), F_1(x) + F_44(x, k_0))
Eq(F_47(x, k_0), F_21(x) + F_48(x, k_0) + F_50(x, k_0))
Eq(F_48(x, k_0), F_15(x, k_0)*F_49(x, k_0))
Eq(F_49(x, k_0), F_37(x) + F_47(x, k_0))
Eq(F_50(x, k_0), F_51(x, k_0)*F_9(x))
Eq(F_51(x, k_0), F_52(x, k_0) + F_55(x, k_0))
Eq(F_52(x, k_0), F_44(x, k_0) + F_53(x, k_0))
Eq(F_53(x, k_0), F_54(x, k_0))
Eq(F_54(x, k_0), F_52(x, k_0)*F_9(x))
Eq(F_55(x, k_0), F_56(x, k_0) + F_61(x, k_0))
Eq(F_56(x, k_0), F_21(x) + F_57(x, k_0) + F_59(x, k_0))
Eq(F_57(x, k_0), F_15(x, k_0)*F_58(x, k_0))
Eq(F_58(x, k_0), F_17(x) + F_56(x, k_0))
Eq(F_59(x, k_0), F_60(x, k_0)*F_9(x))
Eq(F_60(x, k_0), F_44(x, k_0) + F_56(x, k_0))
Eq(F_61(x, k_0), F_62(x, k_0))
Eq(F_62(x, k_0), F_55(x, k_0)*F_9(x))
Eq(F_63(x, k_0), F_11(x)*F_46(x, k_0)*F_9(x))
Eq(F_64(x, k_0), F_65(x, k_0))
Eq(F_65(x, k_0), F_66(x, k_0)*F_9(x))
Eq(F_66(x, k_0), F_105(x, k_0) + F_11(x) + F_67(x, k_0) + F_69(x, k_0))
Eq(F_67(x, k_0), F_15(x, k_0)*F_68(x, k_0))
Eq(F_68(x, k_0), (-k_0*F_34(x, k_0) + F_34(x, 1))/(1 - k_0))
Eq(F_69(x, k_0), F_70(x, k_0))
Eq(F_70(x, k_0), F_71(x, k_0)*F_76(x)*F_9(x))
Eq(F_71(x, k_0), F_1(x) + F_72(x, k_0) + F_74(x, k_0))
Eq(F_72(x, k_0), F_73(x, k_0))
Eq(F_73(x, k_0), F_15(x, k_0)*F_19(x)*F_46(x, k_0))
Eq(F_74(x, k_0), F_75(x, k_0))
Eq(F_75(x, k_0), F_19(x)*F_71(x, k_0)*F_9(x))
Eq(F_76(x), F_77(x, 1))
Eq(F_77(x, k_0), F_104(x, k_0) + F_78(x))
Eq(F_78(x), F_79(x, 1))
Eq(F_79(x, k_0), F_80(x, k_0) + F_87(x, k_0))
Eq(F_80(x, k_0), F_1(x) + F_81(x, k_0))
Eq(F_81(x, k_0), F_82(x, k_0))
Eq(F_82(x, k_0), F_15(x, k_0)*F_83(x, k_0))
Eq(F_83(x, k_0), F_12(x, k_0) + F_84(x, k_0))
Eq(F_84(x, k_0), F_44(x, k_0) + F_85(x, k_0))
Eq(F_85(x, k_0), F_86(x, k_0))
Eq(F_86(x, k_0), F_15(x, k_0)*F_84(x, k_0))
Eq(F_87(x, k_0), F_17(x) + F_88(x, k_0))
Eq(F_88(x, k_0), F_21(x) + F_89(x, k_0) + F_91(x, k_0))
Eq(F_89(x, k_0), F_9(x)*F_90(x, k_0))
Eq(F_90(x, k_0), F_81(x, k_0) + F_88(x, k_0))
Eq(F_91(x, k_0), F_15(x, k_0)*F_92(x, k_0))
Eq(F_92(x, k_0), F_93(x, k_0) + F_96(x, k_0))
Eq(F_93(x, k_0), F_17(x) + F_94(x, k_0))
Eq(F_94(x, k_0), F_95(x, k_0))
Eq(F_95(x, k_0), F_15(x, k_0)*F_93(x, k_0))
Eq(F_96(x, k_0), F_102(x, k_0) + F_97(x, k_0))
Eq(F_97(x, k_0), F_100(x, k_0) + F_21(x) + F_98(x, k_0))
Eq(F_98(x, k_0), F_9(x)*F_99(x, k_0))
Eq(F_99(x, k_0), F_44(x, k_0) + F_97(x, k_0))
Eq(F_100(x, k_0), F_101(x, k_0)*F_15(x, k_0))
Eq(F_101(x, k_0), F_17(x) + F_97(x, k_0))
Eq(F_102(x, k_0), F_103(x, k_0))
Eq(F_103(x, k_0), F_15(x, k_0)*F_96(x, k_0))
Eq(F_104(x, k_0), F_12(x, k_0)*F_15(x, k_0)*F_19(x))
Eq(F_105(x, k_0), F_106(x, k_0))
Eq(F_106(x, k_0), F_68(x, k_0)*F_9(x))
Eq(F_107(x, k_0), F_108(x, k_0)*F_9(x))
Eq(F_108(x, k_0), F_1(x) + F_109(x, k_0) + F_116(x, k_0) + F_119(x, k_0) + F_121(x, k_0))
Eq(F_109(x, k_0), F_110(x, k_0)*F_15(x, k_0))
Eq(F_110(x, k_0), (-k_0*F_111(x, k_0) + F_111(x, 1))/(1 - k_0))
Eq(F_111(x, k_0), F_1(x) + F_112(x, k_0) + F_113(x, k_0) + F_114(x, k_0))
Eq(F_112(x, k_0), F_111(x, k_0)*F_15(x, k_0))
Eq(F_113(x, k_0), F_108(x, k_0)*F_9(x))
Eq(F_114(x, k_0), F_115(x, k_0))
Eq(F_115(x, k_0), F_66(x, k_0)*F_9(x))
Eq(F_116(x, k_0), F_117(x, k_0))
Eq(F_117(x, k_0), F_118(x)*F_71(x, k_0)*F_9(x))
Eq(F_118(x), F_111(x, 1))
Eq(F_119(x, k_0), F_120(x, k_0)*F_9(x))
Eq(F_120(x, k_0), (-k_0*F_108(x, k_0) + F_108(x, 1))/(1 - k_0))
Eq(F_121(x, k_0), F_122(x, k_0))
Eq(F_122(x, k_0), F_123(x, k_0)*F_9(x))
Eq(F_123(x, k_0), (-k_0*F_66(x, k_0) + F_66(x, 1))/(1 - k_0))
Eq(F_124(x), F_30(x, 1))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "row_placements_tracked_fusion", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, 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