0132_0213_1203_3012
Counting sequence:
1, 1, 2, 6, 20, 66, 216, 709, 2353, 7922, 27069, 93775, 328854, 1165527, 4168819, 15029189, 54555676, 199229653, 731424027, 2697916868, 9993461313, 37157645199, 138633670150, 518850928997, 1947388746469, 7328186076916, 27642927895535, 104504664888453, 395895544534346, 1502642426074165, 5713524472700813, 21760802938001885, 83008392287750894, 317105325637995887, 1213057760075477543, 4646448580620774690, 17819304726556705723, 68416557631799222775, 262969725145362629896, 1011811654507137653059, 3896908421928992391773, 15022638668074028178850, 57963977001907464529861, 223840094875066844369409, 865104191469971001987308, 3346062140422285919610435, 12951495043425430311644061, 50166330941038763117683984, 194445724226633558907319943, 754164300347289460079612497, 2926871682330083672610706894, 11365821653191689517921135545, 44161860281294399140674259949, 171685553390268522909149241372, 667807131699516393497433010743, 2598907301698603787399062128085, 10119171361242993678446153603138, 39419031519716449370090436643885, 153626663170081747082034909059101, 598990689817375182182768789070096, 2336468317583576971951576477168091, 9117586965985807804621426113454521, 35593793026615011602284461136895922, 139007230551372799194841272196730566, 543080193453804470276134776047173436, 2122505997156096938343465942487769350, 8298267775170214143613516424384087884, 32454498794282828545315534980928807182, 126971690568413328415114692069014003564, 496911691872912132517214020505001435444, 1945305490861607731422270889764244983606, 7617787107380625423358425104264312135398, 29839982987951633017709694562283198144702, 116921259460409636282276357298896779164598, 458258120759464590437522343379989111336538, 1796575034650253038548559422686848021292154, 7045242085773021540314467974461701593653210, 27634959847233159154074388676236094808195402, 108425445544538639154079008505345125641184210, 425510486567036697762437243211071987713771840, 1670295510058911608848149128721084570022569988, 6558096046614268789073827669438289741235996566, 25754976572289957232657117316075029947654800832, 101167513751684420729618997020181966831876003496, 397479941363369495990918577625425563200202084766, 1562001158037327752693574232689083988755181492780, 6139561172957221983199188638485417010384134891864, 24136877165659976532961907831937387142506906779582, 94909698924380112886657733956568985615572576932444, 373270734394040885943573082567348019844626855738624, 1468315034118775780759630067060458869822660522621866, 5776898002096821352121056722391156399441319058384288, 22732570858707486540009013753229958501277910708841752, 89470345396144723643592634056450763792688958674664038, 352196261772929215352517424433157445300247696871735388, 1386640405146502146848715928944547805930167413781303474, 5460279871325385747765103894756951910939921808031733160, 21504849632601463331630697670358532478583044172712791732, 84708487573528206861583502173158340723543697605501135890, 333722168000359019855033740239205727158168424379754974960, 1314950602664018723403787312486464700302549643524002020168
Generating function in Maple syntax:
1/2*((x-1)*(-4*x+1)^(1/2)-2*x^3-2*x^2-x+1)/x/(x^2+x-1)/(x-1)
Generating function in latex syntax:
\frac{\left(x -1\right) \sqrt{-4 x +1}-2 x^{3}-2 x^{2}-x +1}{2 x \left(x^{2}+x -1\right) \left(x -1\right)}
Generating function in sympy syntax:
(-x**3 - x**2 - x/2 + sqrt(1 - 4*x)*(x - 1)/2 + 1/2)/(x*(x - 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
x*(x-1)^2*(x^2+x-1)^2*F(x)^2+(x-1)*(x^2+x-1)*(2*x^3+2*x^2+x-1)*F(x)+x^5+2*x^4+2*x^3+x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{2} F \! \left(x \right)^{2}+\left(x -1\right) \left(x^{2}+x -1\right) \left(2 x^{3}+2 x^{2}+x -1\right) F \! \left(x \right)+x^{5}+2 x^{4}+2 x^{3}+x^{2}-3 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+3) = -2*(5+2*n)/(4+n)*a(n)-3*(n+2)/(4+n)*a(1+n)+(14+5*n)/(4+n)*a(n+2)+6*(n+2)/(4+n), 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 +3\right) = -\frac{2 \left(5+2 n \right) a \! \left(n \right)}{4+n}-\frac{3 \left(n +2\right) a \! \left(1+n \right)}{4+n}+\frac{\left(14+5 n \right) a \! \left(n +2\right)}{4+n}+\frac{6 n +12}{4+n}, \quad n \geq 5
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21190/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[20,x]*F[3,x]
F[3,x] = F[1,x]+F[109,x]+F[4,x]
F[4,x] = F[20,x]*F[5,x]
F[5,x] = F[6,x,1]
F[6,x,k[0]] = F[1,x]+F[106,x,k[0]]+F[108,x,k[0]]+F[7,x,k[0]]
F[7,x,k[0]] = F[20,x]*F[8,x,k[0]]
F[8,x,k[0]] = F[45,x,k[0]]+F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]+F[26,x,k[0]]
F[10,x,k[0]] = F[11,x,k[0]]+F[15,x,k[0]]
F[11,x,k[0]] = F[1,x]+F[12,x,k[0]]
F[12,x,k[0]] = F[13,x,k[0]]
F[13,x,k[0]] = F[11,x,k[0]]*F[14,x,k[0]]
F[14,x,k[0]] = k[0]*x
F[15,x,k[0]] = F[16,x]+F[24,x,k[0]]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[20,x]
F[18,x] = F[19,x]+F[21,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = x
F[21,x] = F[16,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[16,x]*F[20,x]
F[24,x,k[0]] = F[25,x,k[0]]
F[25,x,k[0]] = F[14,x,k[0]]*F[15,x,k[0]]
F[26,x,k[0]] = F[27,x,k[0]]+F[33,x,k[0]]
F[27,x,k[0]] = F[28,x]+F[31,x,k[0]]
F[28,x] = F[29,x]
F[29,x] = F[20,x]*F[30,x]
F[30,x] = F[1,x]+F[28,x]
F[31,x,k[0]] = F[32,x,k[0]]
F[32,x,k[0]] = F[14,x,k[0]]*F[27,x,k[0]]
F[33,x,k[0]] = F[34,x]+F[43,x,k[0]]
F[34,x] = F[35,x]
F[35,x] = F[20,x]*F[36,x]
F[36,x] = F[37,x]+F[40,x]
F[37,x] = F[28,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[20,x]*F[28,x]
F[40,x] = F[34,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[20,x]*F[34,x]
F[43,x,k[0]] = F[44,x,k[0]]
F[44,x,k[0]] = F[14,x,k[0]]*F[33,x,k[0]]
F[45,x,k[0]] = F[46,x,k[0]]+F[74,x,k[0]]
F[46,x,k[0]] = F[47,x,k[0]]+F[56,x,k[0]]
F[47,x,k[0]] = F[48,x]+F[51,x,k[0]]
F[48,x] = F[49,x]
F[49,x] = F[20,x]*F[50,x]
F[50,x] = F[1,x]+F[48,x]
F[51,x,k[0]] = F[52,x]+F[53,x,k[0]]+F[55,x,k[0]]
F[52,x] = 0
F[53,x,k[0]] = F[20,x]*F[54,x,k[0]]
F[54,x,k[0]] = F[12,x,k[0]]+F[51,x,k[0]]
F[55,x,k[0]] = F[14,x,k[0]]*F[47,x,k[0]]
F[56,x,k[0]] = F[57,x]+F[70,x,k[0]]
F[57,x] = F[52,x]+F[58,x]+F[60,x]
F[58,x] = F[20,x]*F[59,x]
F[59,x] = F[16,x]+F[57,x]
F[60,x] = F[20,x]*F[61,x]
F[61,x] = F[62,x]+F[66,x]
F[62,x] = F[48,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[20,x]*F[65,x]
F[65,x] = F[20,x]+F[63,x]
F[66,x] = F[57,x]+F[67,x]
F[67,x] = F[68,x]
F[68,x] = F[20,x]*F[69,x]
F[69,x] = F[22,x]+F[67,x]
F[70,x,k[0]] = 2*F[52,x]+F[71,x,k[0]]+F[73,x,k[0]]
F[71,x,k[0]] = F[20,x]*F[72,x,k[0]]
F[72,x,k[0]] = F[24,x,k[0]]+F[70,x,k[0]]
F[73,x,k[0]] = F[14,x,k[0]]*F[56,x,k[0]]
F[74,x,k[0]] = F[75,x,k[0]]+F[88,x,k[0]]
F[75,x,k[0]] = F[76,x]+F[84,x,k[0]]
F[76,x] = F[52,x]+F[77,x]+F[79,x]
F[77,x] = F[20,x]*F[78,x]
F[78,x] = F[28,x]+F[76,x]
F[79,x] = F[20,x]*F[80,x]
F[80,x] = F[48,x]+F[81,x]
F[81,x] = F[82,x]
F[82,x] = F[20,x]*F[83,x]
F[83,x] = F[28,x]+F[81,x]
F[84,x,k[0]] = 2*F[52,x]+F[85,x,k[0]]+F[87,x,k[0]]
F[85,x,k[0]] = F[20,x]*F[86,x,k[0]]
F[86,x,k[0]] = F[31,x,k[0]]+F[84,x,k[0]]
F[87,x,k[0]] = F[14,x,k[0]]*F[75,x,k[0]]
F[88,x,k[0]] = F[102,x,k[0]]+F[89,x]
F[89,x] = 2*F[52,x]+F[90,x]+F[92,x]
F[90,x] = F[20,x]*F[91,x]
F[91,x] = F[34,x]+F[89,x]
F[92,x] = F[20,x]*F[93,x]
F[93,x] = F[94,x]+F[98,x]
F[94,x] = F[76,x]+F[95,x]
F[95,x] = F[96,x]
F[96,x] = F[20,x]*F[97,x]
F[97,x] = F[38,x]+F[95,x]
F[98,x] = F[89,x]+F[99,x]
F[99,x] = F[100,x]
F[100,x] = F[101,x]*F[20,x]
F[101,x] = F[41,x]+F[99,x]
F[102,x,k[0]] = 3*F[52,x]+F[103,x,k[0]]+F[105,x,k[0]]
F[103,x,k[0]] = F[104,x,k[0]]*F[20,x]
F[104,x,k[0]] = F[102,x,k[0]]+F[43,x,k[0]]
F[105,x,k[0]] = F[14,x,k[0]]*F[88,x,k[0]]
F[106,x,k[0]] = F[107,x,k[0]]*F[20,x]
F[107,x,k[0]] = 1/(-1+k[0])*(F[6,x,k[0]]*k[0]-F[6,x,1])
F[108,x,k[0]] = F[14,x,k[0]]*F[6,x,k[0]]
F[109,x] = F[110,x]*F[20,x]
F[110,x] = F[111,x]+F[114,x]
F[111,x] = F[112,x]+F[113,x]
F[112,x] = F[1,x]+F[16,x]
F[113,x] = F[28,x]+F[34,x]
F[114,x] = F[115,x]+F[116,x]
F[115,x] = F[48,x]+F[57,x]
F[116,x] = F[76,x]+F[89,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_{20}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{109}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{20}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x , 1\right)
F_{6}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{106}\! \left(x , y\right)+F_{108}\! \left(x , y\right)+F_{7}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{20}\! \left(x \right) F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = F_{45}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right)+F_{26}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right)+F_{15}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = F_{11}\! \left(x , y\right) F_{14}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = y x
F_{15}\! \left(x , y\right) = F_{16}\! \left(x \right)+F_{24}\! \left(x , y\right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{20}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = x
F_{21}\! \left(x \right) = F_{16}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{24}\! \left(x , y\right) = F_{25}\! \left(x , y\right)
F_{25}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{15}\! \left(x , y\right)
F_{26}\! \left(x , y\right) = F_{27}\! \left(x , y\right)+F_{33}\! \left(x , y\right)
F_{27}\! \left(x , y\right) = F_{28}\! \left(x \right)+F_{31}\! \left(x , y\right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{20}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x , y\right) = F_{32}\! \left(x , y\right)
F_{32}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{27}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = F_{34}\! \left(x \right)+F_{43}\! \left(x , y\right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{20}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{37}\! \left(x \right) = F_{28}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{20}\! \left(x \right) F_{28}\! \left(x \right)
F_{40}\! \left(x \right) = F_{34}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{20}\! \left(x \right) F_{34}\! \left(x \right)
F_{43}\! \left(x , y\right) = F_{44}\! \left(x , y\right)
F_{44}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{33}\! \left(x , y\right)
F_{45}\! \left(x , y\right) = F_{46}\! \left(x , y\right)+F_{74}\! \left(x , y\right)
F_{46}\! \left(x , y\right) = F_{47}\! \left(x , y\right)+F_{56}\! \left(x , y\right)
F_{47}\! \left(x , y\right) = F_{48}\! \left(x \right)+F_{51}\! \left(x , y\right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{20}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{1}\! \left(x \right)+F_{48}\! \left(x \right)
F_{51}\! \left(x , y\right) = F_{52}\! \left(x \right)+F_{53}\! \left(x , y\right)+F_{55}\! \left(x , y\right)
F_{52}\! \left(x \right) = 0
F_{53}\! \left(x , y\right) = F_{20}\! \left(x \right) F_{54}\! \left(x , y\right)
F_{54}\! \left(x , y\right) = F_{12}\! \left(x , y\right)+F_{51}\! \left(x , y\right)
F_{55}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{47}\! \left(x , y\right)
F_{56}\! \left(x , y\right) = F_{57}\! \left(x \right)+F_{70}\! \left(x , y\right)
F_{57}\! \left(x \right) = F_{52}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{58}\! \left(x \right) = F_{20}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{16}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{20}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{66}\! \left(x \right)
F_{62}\! \left(x \right) = F_{48}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{20}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{20}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{57}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{20}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{22}\! \left(x \right)+F_{67}\! \left(x \right)
F_{70}\! \left(x , y\right) = 2 F_{52}\! \left(x \right)+F_{71}\! \left(x , y\right)+F_{73}\! \left(x , y\right)
F_{71}\! \left(x , y\right) = F_{20}\! \left(x \right) F_{72}\! \left(x , y\right)
F_{72}\! \left(x , y\right) = F_{24}\! \left(x , y\right)+F_{70}\! \left(x , y\right)
F_{73}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{56}\! \left(x , y\right)
F_{74}\! \left(x , y\right) = F_{75}\! \left(x , y\right)+F_{88}\! \left(x , y\right)
F_{75}\! \left(x , y\right) = F_{76}\! \left(x \right)+F_{84}\! \left(x , y\right)
F_{76}\! \left(x \right) = F_{52}\! \left(x \right)+F_{77}\! \left(x \right)+F_{79}\! \left(x \right)
F_{77}\! \left(x \right) = F_{20}\! \left(x \right) F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{28}\! \left(x \right)+F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{20}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{48}\! \left(x \right)+F_{81}\! \left(x \right)
F_{81}\! \left(x \right) = F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{20}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{28}\! \left(x \right)+F_{81}\! \left(x \right)
F_{84}\! \left(x , y\right) = 2 F_{52}\! \left(x \right)+F_{85}\! \left(x , y\right)+F_{87}\! \left(x , y\right)
F_{85}\! \left(x , y\right) = F_{20}\! \left(x \right) F_{86}\! \left(x , y\right)
F_{86}\! \left(x , y\right) = F_{31}\! \left(x , y\right)+F_{84}\! \left(x , y\right)
F_{87}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{75}\! \left(x , y\right)
F_{88}\! \left(x , y\right) = F_{102}\! \left(x , y\right)+F_{89}\! \left(x \right)
F_{89}\! \left(x \right) = 2 F_{52}\! \left(x \right)+F_{90}\! \left(x \right)+F_{92}\! \left(x \right)
F_{90}\! \left(x \right) = F_{20}\! \left(x \right) F_{91}\! \left(x \right)
F_{91}\! \left(x \right) = F_{34}\! \left(x \right)+F_{89}\! \left(x \right)
F_{92}\! \left(x \right) = F_{20}\! \left(x \right) F_{93}\! \left(x \right)
F_{93}\! \left(x \right) = F_{94}\! \left(x \right)+F_{98}\! \left(x \right)
F_{94}\! \left(x \right) = F_{76}\! \left(x \right)+F_{95}\! \left(x \right)
F_{95}\! \left(x \right) = F_{96}\! \left(x \right)
F_{96}\! \left(x \right) = F_{20}\! \left(x \right) F_{97}\! \left(x \right)
F_{97}\! \left(x \right) = F_{38}\! \left(x \right)+F_{95}\! \left(x \right)
F_{98}\! \left(x \right) = F_{89}\! \left(x \right)+F_{99}\! \left(x \right)
F_{99}\! \left(x \right) = F_{100}\! \left(x \right)
F_{100}\! \left(x \right) = F_{101}\! \left(x \right) F_{20}\! \left(x \right)
F_{101}\! \left(x \right) = F_{41}\! \left(x \right)+F_{99}\! \left(x \right)
F_{102}\! \left(x , y\right) = 3 F_{52}\! \left(x \right)+F_{103}\! \left(x , y\right)+F_{105}\! \left(x , y\right)
F_{103}\! \left(x , y\right) = F_{104}\! \left(x , y\right) F_{20}\! \left(x \right)
F_{104}\! \left(x , y\right) = F_{102}\! \left(x , y\right)+F_{43}\! \left(x , y\right)
F_{105}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{88}\! \left(x , y\right)
F_{106}\! \left(x , y\right) = F_{107}\! \left(x , y\right) F_{20}\! \left(x \right)
F_{107}\! \left(x , y\right) = \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}
F_{108}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{6}\! \left(x , y\right)
F_{109}\! \left(x \right) = F_{110}\! \left(x \right) F_{20}\! \left(x \right)
F_{110}\! \left(x \right) = F_{111}\! \left(x \right)+F_{114}\! \left(x \right)
F_{111}\! \left(x \right) = F_{112}\! \left(x \right)+F_{113}\! \left(x \right)
F_{112}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{113}\! \left(x \right) = F_{28}\! \left(x \right)+F_{34}\! \left(x \right)
F_{114}\! \left(x \right) = F_{115}\! \left(x \right)+F_{116}\! \left(x \right)
F_{115}\! \left(x \right) = F_{48}\! \left(x \right)+F_{57}\! \left(x \right)
F_{116}\! \left(x \right) = F_{76}\! \left(x \right)+F_{89}\! \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_20(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_109(x) + F_4(x))
Eq(F_4(x), F_20(x)*F_5(x))
Eq(F_5(x), F_6(x, 1))
Eq(F_6(x, k_0), F_1(x) + F_106(x, k_0) + F_108(x, k_0) + F_7(x, k_0))
Eq(F_7(x, k_0), F_20(x)*F_8(x, k_0))
Eq(F_8(x, k_0), F_45(x, k_0) + F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0) + F_26(x, k_0))
Eq(F_10(x, k_0), F_11(x, k_0) + F_15(x, k_0))
Eq(F_11(x, k_0), F_1(x) + F_12(x, k_0))
Eq(F_12(x, k_0), F_13(x, k_0))
Eq(F_13(x, k_0), F_11(x, k_0)*F_14(x, k_0))
Eq(F_14(x, k_0), k_0*x)
Eq(F_15(x, k_0), F_16(x) + F_24(x, k_0))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_20(x))
Eq(F_18(x), F_19(x) + F_21(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), x)
Eq(F_21(x), F_16(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_16(x)*F_20(x))
Eq(F_24(x, k_0), F_25(x, k_0))
Eq(F_25(x, k_0), F_14(x, k_0)*F_15(x, k_0))
Eq(F_26(x, k_0), F_27(x, k_0) + F_33(x, k_0))
Eq(F_27(x, k_0), F_28(x) + F_31(x, k_0))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_20(x)*F_30(x))
Eq(F_30(x), F_1(x) + F_28(x))
Eq(F_31(x, k_0), F_32(x, k_0))
Eq(F_32(x, k_0), F_14(x, k_0)*F_27(x, k_0))
Eq(F_33(x, k_0), F_34(x) + F_43(x, k_0))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_20(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_40(x))
Eq(F_37(x), F_28(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_20(x)*F_28(x))
Eq(F_40(x), F_34(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_20(x)*F_34(x))
Eq(F_43(x, k_0), F_44(x, k_0))
Eq(F_44(x, k_0), F_14(x, k_0)*F_33(x, k_0))
Eq(F_45(x, k_0), F_46(x, k_0) + F_74(x, k_0))
Eq(F_46(x, k_0), F_47(x, k_0) + F_56(x, k_0))
Eq(F_47(x, k_0), F_48(x) + F_51(x, k_0))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_20(x)*F_50(x))
Eq(F_50(x), F_1(x) + F_48(x))
Eq(F_51(x, k_0), F_52(x) + F_53(x, k_0) + F_55(x, k_0))
Eq(F_52(x), 0)
Eq(F_53(x, k_0), F_20(x)*F_54(x, k_0))
Eq(F_54(x, k_0), F_12(x, k_0) + F_51(x, k_0))
Eq(F_55(x, k_0), F_14(x, k_0)*F_47(x, k_0))
Eq(F_56(x, k_0), F_57(x) + F_70(x, k_0))
Eq(F_57(x), F_52(x) + F_58(x) + F_60(x))
Eq(F_58(x), F_20(x)*F_59(x))
Eq(F_59(x), F_16(x) + F_57(x))
Eq(F_60(x), F_20(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_66(x))
Eq(F_62(x), F_48(x) + F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_20(x)*F_65(x))
Eq(F_65(x), F_20(x) + F_63(x))
Eq(F_66(x), F_57(x) + F_67(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_20(x)*F_69(x))
Eq(F_69(x), F_22(x) + F_67(x))
Eq(F_70(x, k_0), 2*F_52(x) + F_71(x, k_0) + F_73(x, k_0))
Eq(F_71(x, k_0), F_20(x)*F_72(x, k_0))
Eq(F_72(x, k_0), F_24(x, k_0) + F_70(x, k_0))
Eq(F_73(x, k_0), F_14(x, k_0)*F_56(x, k_0))
Eq(F_74(x, k_0), F_75(x, k_0) + F_88(x, k_0))
Eq(F_75(x, k_0), F_76(x) + F_84(x, k_0))
Eq(F_76(x), F_52(x) + F_77(x) + F_79(x))
Eq(F_77(x), F_20(x)*F_78(x))
Eq(F_78(x), F_28(x) + F_76(x))
Eq(F_79(x), F_20(x)*F_80(x))
Eq(F_80(x), F_48(x) + F_81(x))
Eq(F_81(x), F_82(x))
Eq(F_82(x), F_20(x)*F_83(x))
Eq(F_83(x), F_28(x) + F_81(x))
Eq(F_84(x, k_0), 2*F_52(x) + F_85(x, k_0) + F_87(x, k_0))
Eq(F_85(x, k_0), F_20(x)*F_86(x, k_0))
Eq(F_86(x, k_0), F_31(x, k_0) + F_84(x, k_0))
Eq(F_87(x, k_0), F_14(x, k_0)*F_75(x, k_0))
Eq(F_88(x, k_0), F_102(x, k_0) + F_89(x))
Eq(F_89(x), 2*F_52(x) + F_90(x) + F_92(x))
Eq(F_90(x), F_20(x)*F_91(x))
Eq(F_91(x), F_34(x) + F_89(x))
Eq(F_92(x), F_20(x)*F_93(x))
Eq(F_93(x), F_94(x) + F_98(x))
Eq(F_94(x), F_76(x) + F_95(x))
Eq(F_95(x), F_96(x))
Eq(F_96(x), F_20(x)*F_97(x))
Eq(F_97(x), F_38(x) + F_95(x))
Eq(F_98(x), F_89(x) + F_99(x))
Eq(F_99(x), F_100(x))
Eq(F_100(x), F_101(x)*F_20(x))
Eq(F_101(x), F_41(x) + F_99(x))
Eq(F_102(x, k_0), F_103(x, k_0) + F_105(x, k_0) + 3*F_52(x))
Eq(F_103(x, k_0), F_104(x, k_0)*F_20(x))
Eq(F_104(x, k_0), F_102(x, k_0) + F_43(x, k_0))
Eq(F_105(x, k_0), F_14(x, k_0)*F_88(x, k_0))
Eq(F_106(x, k_0), F_107(x, k_0)*F_20(x))
Eq(F_107(x, k_0), (k_0*F_6(x, k_0) - F_6(x, 1))/(k_0 - 1))
Eq(F_108(x, k_0), F_14(x, k_0)*F_6(x, k_0))
Eq(F_109(x), F_110(x)*F_20(x))
Eq(F_110(x), F_111(x) + F_114(x))
Eq(F_111(x), F_112(x) + F_113(x))
Eq(F_112(x), F_1(x) + F_16(x))
Eq(F_113(x), F_28(x) + F_34(x))
Eq(F_114(x), F_115(x) + F_116(x))
Eq(F_115(x), F_48(x) + F_57(x))
Eq(F_116(x), F_76(x) + F_89(x))
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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 1, 2], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 3, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"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]]}]]}], "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": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 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