0123_0231_1230_2013
Counting sequence:
1, 1, 2, 6, 20, 66, 215, 702, 2321, 7800, 26642, 92335, 324049, 1149436, 4114374, 14842616, 53908122, 196955592, 723354641, 2669022969, 9889187226, 36778761272, 137248752357, 513762060471, 1928602465898, 7258544739102, 27383797830329, 103537195166859, 392272346409896, 1489035313469944, 5662290517721287, 21567437475758602, 82276999773709509, 314333263646082236, 1202531504132990814, 4606407327789712999, 17666740084549318177, 67834364274977513488, 260744845313651121326, 1003297639619915570517, 3864285967259703029537, 14897491215990872314146, 57483334827265079532794, 221992146323358092586921, 857992028119266891027627, 3318663123604512421294712, 12845845196138679961442436, 49758590168407969071227049, 192870788337188750162620391, 748076107200608350717059894, 2903318736882740832318657180, 11274637624502646183581904737, 43808599445137021753380528105, 170316051446295522008821395996, 662494516965397685411460344942, 2578285752521147682609523506315, 10039079017340905777812374765451, 39107782827699461150703268348910, 152416450998250342150396918296096, 594282606864030756751480913710977, 2318143192487476532342192563609721, 9046225767227234901463855725053016, 35315769091023884488496669364592182, 137923553189131742688240490251454496, 538854394347329907404769427451756946, 2106020489339403061302714601128757920, 8233928765616685940801094629124585593, 32203299052441093217536601459572083465, 125990549900829581492093112045512884986, 493078104282430277060606328036868152656, 1930321188573057553288463568427095448725, 7559197591599724094002102619077217541243, 29610816568109967208715742221177144487710, 116024604444634293861127561081366765009326, 454748671656893963672796432396270891699381, 1782835004353506502016897971832677284037715, 6991431491294147460998161289174795140636788, 27424157429111747895691160399217555860252400, 107599393182015226126191336293214965086978091, 422272609225461029926496567641143357532010451, 1657600565652510664663840981818895768619600120, 6508309075295809409387077662527604673803622818, 25559672128514900778313846440650503488551838995, 100401181348885730502856768033032526993388681731, 394472285491488260382889241397830125412307833086, 1550194082649749702915203961551604263276644314620, 6093199701481657162017820315121977967267299986393, 23954793784344740519647922459327806814919898143305, 94194413691861836185220212957087838440315192480596, 370460246152282240801318451708306221832943913978426, 1457269781694819545193359137703371003964167888661455, 5733481108325007549597807586704085078593509825444083, 22561872624663435057642128677256334739825212299989310, 88799095087434393595438539957623239570786744234160736, 349556144355433830957816491244415561748073305876959361, 1376254522618124886755347099943500406648279527173870435, 5419415652827419125433058156190369037427042515102801358, 21344036732737762623416382388169963367046484514787697130, 84075529730525842852134399922914627928683209885708143101, 331230424809071490397501135091962322874053841184312780511, 1305139794253120642750406100547464652618241043044878890900
Generating function in Maple syntax:
1/2*((-x^3+2*x^2-2*x+1)*(1-4*x)^(1/2)+x^3+2*x-1)/x/(-1+x)^3
Generating function in latex syntax:
\frac{\left(-x^{3}+2 x^{2}-2 x +1\right) \sqrt{1-4 x}+x^{3}+2 x -1}{2 x \left(-1+x \right)^{3}}
Generating function in sympy syntax:
(x**3/2 + x + sqrt(1 - 4*x)*(-x**3 + 2*x**2 - 2*x + 1)/2 - 1/2)/(x*(x - 1)**3)
Implicit equation for the generating function in Maple syntax:
x*(-1+x)^6*F(x)^2-(x^3+2*x-1)*(-1+x)^3*F(x)+x^6-4*x^5+9*x^4-11*x^3+10*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(-1+x \right)^{6} F \! \left(x \right)^{2}-\left(x^{3}+2 x -1\right) \left(-1+x \right)^{3} F \! \left(x \right)+x^{6}-4 x^{5}+9 x^{4}-11 x^{3}+10 x^{2}-5 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(6) = 215
a(n+4) = -2*(2*n+1)/(n+5)*a(n)+(10+9*n)/(n+5)*a(n+1)-(10*n+29)/(n+5)*a(n+2)+(23+6*n)/(n+5)*a(n+3)+3/2*(n^2+3*n+8)/(n+5), n >= 7
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 20
a \! \left(5\right) = 66
a \! \left(6\right) = 215
a \! \left(n +4\right) = -\frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +5}+\frac{\left(10+9 n \right) a \! \left(n +1\right)}{n +5}-\frac{\left(10 n +29\right) a \! \left(n +2\right)}{n +5}+\frac{\left(23+6 n \right) a \! \left(n +3\right)}{n +5}+\frac{\frac{3}{2} n^{2}+\frac{9}{2} n +12}{n +5}, \quad n \geq 7
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21155/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[14,x]*F[3,x]
F[3,x] = F[1,x]+F[4,x]+F[64,x]
F[4,x] = F[14,x]*F[5,x]
F[5,x] = F[28,x]+F[6,x]
F[6,x] = F[23,x]+F[7,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[14,x]
F[10,x] = F[11,x]+F[15,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[14,x]
F[14,x] = x
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[17,x]+F[18,x]+F[20,x]
F[17,x] = 0
F[18,x] = F[14,x]*F[19,x]
F[19,x] = F[12,x]+F[16,x]
F[20,x] = F[14,x]*F[21,x]
F[21,x] = F[12,x]+F[22,x]
F[22,x] = F[20,x]
F[23,x] = F[12,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[14,x]*F[26,x]
F[26,x] = F[12,x]+F[27,x]
F[27,x] = F[25,x]
F[28,x] = F[29,x]+F[63,x]
F[29,x] = F[30,x]+F[39,x]
F[30,x] = F[31,x]
F[31,x] = F[14,x]*F[32,x]
F[32,x] = F[11,x]+F[33,x]
F[33,x] = F[30,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[14,x]*F[36,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[12,x]
F[38,x] = F[34,x]
F[39,x] = F[17,x]+F[40,x]+F[49,x]
F[40,x] = F[14,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[24,x]+F[8,x]
F[43,x] = F[39,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[14,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[24,x]
F[48,x] = F[44,x]
F[49,x] = F[14,x]*F[50,x]
F[50,x] = F[21,x]+F[51,x]
F[51,x] = F[52,x]+F[57,x]
F[52,x] = F[17,x]+F[53,x]+F[55,x]
F[53,x] = F[14,x]*F[54,x]
F[54,x] = F[12,x]+F[52,x]
F[55,x] = F[14,x]*F[56,x]
F[56,x] = F[12,x]+F[52,x]
F[57,x] = 2*F[17,x]+F[58,x]+F[60,x]
F[58,x] = F[14,x]*F[59,x]
F[59,x] = F[22,x]+F[57,x]
F[60,x] = F[14,x]*F[61,x]
F[61,x] = F[52,x]+F[62,x]
F[62,x] = F[60,x]
F[63,x] = F[34,x]+F[44,x]
F[64,x] = F[14,x]*F[65,x]
F[65,x] = F[66,x,1]
F[66,x,k[0]] = F[1,x]+F[67,x,k[0]]+F[69,x,k[0]]+F[71,x,k[0]]
F[67,x,k[0]] = F[66,x,k[0]]*F[68,x,k[0]]
F[68,x,k[0]] = k[0]*x
F[69,x,k[0]] = F[14,x]*F[70,x,k[0]]
F[70,x,k[0]] = (F[66,x,k[0]]*k[0]-F[66,x,1])/(-1+k[0])
F[71,x,k[0]] = F[14,x]*F[72,x,k[0]]
F[72,x,k[0]] = F[5,x]+F[73,x,k[0]]
F[73,x,k[0]] = F[106,x,k[0]]+F[74,x,k[0]]
F[74,x,k[0]] = F[75,x,k[0]]+F[99,x,k[0]]
F[75,x,k[0]] = F[76,x,k[0]]+F[79,x,k[0]]
F[76,x,k[0]] = F[77,x,k[0]]
F[77,x,k[0]] = F[68,x,k[0]]*F[78,x,k[0]]
F[78,x,k[0]] = F[1,x]+F[76,x,k[0]]
F[79,x,k[0]] = F[17,x]+F[80,x,k[0]]+F[82,x,k[0]]
F[80,x,k[0]] = F[68,x,k[0]]*F[81,x,k[0]]
F[81,x,k[0]] = F[79,x,k[0]]+F[8,x]
F[82,x,k[0]] = F[14,x]*F[83,x,k[0]]
F[83,x,k[0]] = F[84,x,k[0]]+F[87,x,k[0]]
F[84,x,k[0]] = F[76,x,k[0]]+F[85,x,k[0]]
F[85,x,k[0]] = F[86,x,k[0]]
F[86,x,k[0]] = F[14,x]*F[84,x,k[0]]
F[87,x,k[0]] = F[88,x,k[0]]+F[93,x,k[0]]
F[88,x,k[0]] = F[17,x]+F[89,x,k[0]]+F[91,x,k[0]]
F[89,x,k[0]] = F[68,x,k[0]]*F[90,x,k[0]]
F[90,x,k[0]] = F[12,x]+F[88,x,k[0]]
F[91,x,k[0]] = F[14,x]*F[92,x,k[0]]
F[92,x,k[0]] = F[76,x,k[0]]+F[88,x,k[0]]
F[93,x,k[0]] = 2*F[17,x]+F[94,x,k[0]]+F[96,x,k[0]]
F[94,x,k[0]] = F[14,x]*F[95,x,k[0]]
F[95,x,k[0]] = F[85,x,k[0]]+F[93,x,k[0]]
F[96,x,k[0]] = F[14,x]*F[97,x,k[0]]
F[97,x,k[0]] = F[88,x,k[0]]+F[98,x,k[0]]
F[98,x,k[0]] = F[96,x,k[0]]
F[99,x,k[0]] = F[100,x,k[0]]+F[103,x,k[0]]
F[100,x,k[0]] = F[101,x,k[0]]
F[101,x,k[0]] = F[102,x,k[0]]*F[68,x,k[0]]
F[102,x,k[0]] = F[100,x,k[0]]+F[12,x]
F[103,x,k[0]] = F[104,x,k[0]]
F[104,x,k[0]] = F[105,x,k[0]]*F[68,x,k[0]]
F[105,x,k[0]] = F[103,x,k[0]]+F[24,x]
F[106,x,k[0]] = F[107,x,k[0]]+F[145,x,k[0]]
F[107,x,k[0]] = F[108,x,k[0]]+F[118,x,k[0]]
F[108,x,k[0]] = F[101,x,k[0]]+F[109,x,k[0]]+F[17,x]
F[109,x,k[0]] = F[110,x,k[0]]*F[14,x]
F[110,x,k[0]] = F[111,x,k[0]]+F[112,x,k[0]]
F[111,x,k[0]] = F[100,x,k[0]]+F[76,x,k[0]]
F[112,x,k[0]] = F[108,x,k[0]]+F[113,x,k[0]]
F[113,x,k[0]] = F[114,x,k[0]]
F[114,x,k[0]] = F[115,x,k[0]]*F[14,x]
F[115,x,k[0]] = F[116,x,k[0]]+F[117,x,k[0]]
F[116,x,k[0]] = F[100,x,k[0]]
F[117,x,k[0]] = F[113,x,k[0]]
F[118,x,k[0]] = F[104,x,k[0]]+F[119,x,k[0]]+F[128,x,k[0]]+F[17,x]
F[119,x,k[0]] = F[120,x,k[0]]*F[14,x]
F[120,x,k[0]] = F[121,x,k[0]]+F[122,x,k[0]]
F[121,x,k[0]] = F[103,x,k[0]]+F[79,x,k[0]]
F[122,x,k[0]] = F[118,x,k[0]]+F[123,x,k[0]]
F[123,x,k[0]] = F[124,x,k[0]]
F[124,x,k[0]] = F[125,x,k[0]]*F[14,x]
F[125,x,k[0]] = F[126,x,k[0]]+F[127,x,k[0]]
F[126,x,k[0]] = F[103,x,k[0]]
F[127,x,k[0]] = F[123,x,k[0]]
F[128,x,k[0]] = F[129,x,k[0]]*F[14,x]
F[129,x,k[0]] = F[130,x,k[0]]+F[133,x,k[0]]
F[130,x,k[0]] = F[131,x,k[0]]+F[85,x,k[0]]
F[131,x,k[0]] = F[132,x,k[0]]
F[132,x,k[0]] = F[130,x,k[0]]*F[14,x]
F[133,x,k[0]] = F[134,x,k[0]]+F[139,x,k[0]]
F[134,x,k[0]] = 2*F[17,x]+F[135,x,k[0]]+F[137,x,k[0]]
F[135,x,k[0]] = F[136,x,k[0]]*F[14,x]
F[136,x,k[0]] = F[134,x,k[0]]+F[88,x,k[0]]
F[137,x,k[0]] = F[138,x,k[0]]*F[14,x]
F[138,x,k[0]] = F[134,x,k[0]]+F[85,x,k[0]]
F[139,x,k[0]] = 3*F[17,x]+F[140,x,k[0]]+F[142,x,k[0]]
F[140,x,k[0]] = F[14,x]*F[141,x,k[0]]
F[141,x,k[0]] = F[131,x,k[0]]+F[139,x,k[0]]
F[142,x,k[0]] = F[14,x]*F[143,x,k[0]]
F[143,x,k[0]] = F[134,x,k[0]]+F[144,x,k[0]]
F[144,x,k[0]] = F[142,x,k[0]]
F[145,x,k[0]] = F[113,x,k[0]]+F[123,x,k[0]]
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_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{64}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{28}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{15}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{12}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{20}\! \left(x \right)
F_{17}\! \left(x \right) = 0
F_{18}\! \left(x \right) = F_{14}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right)+F_{16}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{14}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{63}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{39}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{11}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \left(x \right)
F_{39}\! \left(x \right) = F_{17}\! \left(x \right)+F_{40}\! \left(x \right)+F_{49}\! \left(x \right)
F_{40}\! \left(x \right) = F_{14}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{24}\! \left(x \right)+F_{8}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{14}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{24}\! \left(x \right)
F_{48}\! \left(x \right) = F_{44}\! \left(x \right)
F_{49}\! \left(x \right) = F_{14}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{21}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{57}\! \left(x \right)
F_{52}\! \left(x \right) = F_{17}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{14}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{14}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right)+F_{52}\! \left(x \right)
F_{57}\! \left(x \right) = 2 F_{17}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{58}\! \left(x \right) = F_{14}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{22}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{14}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{52}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{34}\! \left(x \right)+F_{44}\! \left(x \right)
F_{64}\! \left(x \right) = F_{14}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x , 1\right)
F_{66}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{67}\! \left(x , y\right)+F_{69}\! \left(x , y\right)+F_{71}\! \left(x , y\right)
F_{67}\! \left(x , y\right) = F_{66}\! \left(x , y\right) F_{68}\! \left(x , y\right)
F_{68}\! \left(x , y\right) = y x
F_{69}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{70}\! \left(x , y\right)
F_{70}\! \left(x , y\right) = \frac{F_{66}\! \left(x , y\right) y -F_{66}\! \left(x , 1\right)}{-1+y}
F_{71}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{72}\! \left(x , y\right)
F_{72}\! \left(x , y\right) = F_{5}\! \left(x \right)+F_{73}\! \left(x , y\right)
F_{73}\! \left(x , y\right) = F_{106}\! \left(x , y\right)+F_{74}\! \left(x , y\right)
F_{74}\! \left(x , y\right) = F_{75}\! \left(x , y\right)+F_{99}\! \left(x , y\right)
F_{75}\! \left(x , y\right) = F_{76}\! \left(x , y\right)+F_{79}\! \left(x , y\right)
F_{76}\! \left(x , y\right) = F_{77}\! \left(x , y\right)
F_{77}\! \left(x , y\right) = F_{68}\! \left(x , y\right) F_{78}\! \left(x , y\right)
F_{78}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{76}\! \left(x , y\right)
F_{79}\! \left(x , y\right) = F_{17}\! \left(x \right)+F_{80}\! \left(x , y\right)+F_{82}\! \left(x , y\right)
F_{80}\! \left(x , y\right) = F_{68}\! \left(x , y\right) F_{81}\! \left(x , y\right)
F_{81}\! \left(x , y\right) = F_{79}\! \left(x , y\right)+F_{8}\! \left(x \right)
F_{82}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{83}\! \left(x , y\right)
F_{83}\! \left(x , y\right) = F_{84}\! \left(x , y\right)+F_{87}\! \left(x , y\right)
F_{84}\! \left(x , y\right) = F_{76}\! \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_{14}\! \left(x \right) F_{84}\! \left(x , y\right)
F_{87}\! \left(x , y\right) = F_{88}\! \left(x , y\right)+F_{93}\! \left(x , y\right)
F_{88}\! \left(x , y\right) = F_{17}\! \left(x \right)+F_{89}\! \left(x , y\right)+F_{91}\! \left(x , y\right)
F_{89}\! \left(x , y\right) = F_{68}\! \left(x , y\right) F_{90}\! \left(x , y\right)
F_{90}\! \left(x , y\right) = F_{12}\! \left(x \right)+F_{88}\! \left(x , y\right)
F_{91}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{92}\! \left(x , y\right)
F_{92}\! \left(x , y\right) = F_{76}\! \left(x , y\right)+F_{88}\! \left(x , y\right)
F_{93}\! \left(x , y\right) = 2 F_{17}\! \left(x \right)+F_{94}\! \left(x , y\right)+F_{96}\! \left(x , y\right)
F_{94}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{95}\! \left(x , y\right)
F_{95}\! \left(x , y\right) = F_{85}\! \left(x , y\right)+F_{93}\! \left(x , y\right)
F_{96}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{97}\! \left(x , y\right)
F_{97}\! \left(x , y\right) = F_{88}\! \left(x , y\right)+F_{98}\! \left(x , y\right)
F_{98}\! \left(x , y\right) = F_{96}\! \left(x , y\right)
F_{99}\! \left(x , y\right) = F_{100}\! \left(x , y\right)+F_{103}\! \left(x , y\right)
F_{100}\! \left(x , y\right) = F_{101}\! \left(x , y\right)
F_{101}\! \left(x , y\right) = F_{102}\! \left(x , y\right) F_{68}\! \left(x , y\right)
F_{102}\! \left(x , y\right) = F_{100}\! \left(x , y\right)+F_{12}\! \left(x \right)
F_{103}\! \left(x , y\right) = F_{104}\! \left(x , y\right)
F_{104}\! \left(x , y\right) = F_{105}\! \left(x , y\right) F_{68}\! \left(x , y\right)
F_{105}\! \left(x , y\right) = F_{103}\! \left(x , y\right)+F_{24}\! \left(x \right)
F_{106}\! \left(x , y\right) = F_{107}\! \left(x , y\right)+F_{145}\! \left(x , y\right)
F_{107}\! \left(x , y\right) = F_{108}\! \left(x , y\right)+F_{118}\! \left(x , y\right)
F_{108}\! \left(x , y\right) = F_{101}\! \left(x , y\right)+F_{109}\! \left(x , y\right)+F_{17}\! \left(x \right)
F_{109}\! \left(x , y\right) = F_{110}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{110}\! \left(x , y\right) = F_{111}\! \left(x , y\right)+F_{112}\! \left(x , y\right)
F_{111}\! \left(x , y\right) = F_{100}\! \left(x , y\right)+F_{76}\! \left(x , y\right)
F_{112}\! \left(x , y\right) = F_{108}\! \left(x , y\right)+F_{113}\! \left(x , y\right)
F_{113}\! \left(x , y\right) = F_{114}\! \left(x , y\right)
F_{114}\! \left(x , y\right) = F_{115}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{115}\! \left(x , y\right) = F_{116}\! \left(x , y\right)+F_{117}\! \left(x , y\right)
F_{116}\! \left(x , y\right) = F_{100}\! \left(x , y\right)
F_{117}\! \left(x , y\right) = F_{113}\! \left(x , y\right)
F_{118}\! \left(x , y\right) = F_{104}\! \left(x , y\right)+F_{119}\! \left(x , y\right)+F_{128}\! \left(x , y\right)+F_{17}\! \left(x \right)
F_{119}\! \left(x , y\right) = F_{120}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{120}\! \left(x , y\right) = F_{121}\! \left(x , y\right)+F_{122}\! \left(x , y\right)
F_{121}\! \left(x , y\right) = F_{103}\! \left(x , y\right)+F_{79}\! \left(x , y\right)
F_{122}\! \left(x , y\right) = F_{118}\! \left(x , y\right)+F_{123}\! \left(x , y\right)
F_{123}\! \left(x , y\right) = F_{124}\! \left(x , y\right)
F_{124}\! \left(x , y\right) = F_{125}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{125}\! \left(x , y\right) = F_{126}\! \left(x , y\right)+F_{127}\! \left(x , y\right)
F_{126}\! \left(x , y\right) = F_{103}\! \left(x , y\right)
F_{127}\! \left(x , y\right) = F_{123}\! \left(x , y\right)
F_{128}\! \left(x , y\right) = F_{129}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{129}\! \left(x , y\right) = F_{130}\! \left(x , y\right)+F_{133}\! \left(x , y\right)
F_{130}\! \left(x , y\right) = F_{131}\! \left(x , y\right)+F_{85}\! \left(x , y\right)
F_{131}\! \left(x , y\right) = F_{132}\! \left(x , y\right)
F_{132}\! \left(x , y\right) = F_{130}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{133}\! \left(x , y\right) = F_{134}\! \left(x , y\right)+F_{139}\! \left(x , y\right)
F_{134}\! \left(x , y\right) = 2 F_{17}\! \left(x \right)+F_{135}\! \left(x , y\right)+F_{137}\! \left(x , y\right)
F_{135}\! \left(x , y\right) = F_{136}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{136}\! \left(x , y\right) = F_{134}\! \left(x , y\right)+F_{88}\! \left(x , y\right)
F_{137}\! \left(x , y\right) = F_{138}\! \left(x , y\right) F_{14}\! \left(x \right)
F_{138}\! \left(x , y\right) = F_{134}\! \left(x , y\right)+F_{85}\! \left(x , y\right)
F_{139}\! \left(x , y\right) = 3 F_{17}\! \left(x \right)+F_{140}\! \left(x , y\right)+F_{142}\! \left(x , y\right)
F_{140}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{141}\! \left(x , y\right)
F_{141}\! \left(x , y\right) = F_{131}\! \left(x , y\right)+F_{139}\! \left(x , y\right)
F_{142}\! \left(x , y\right) = F_{14}\! \left(x \right) F_{143}\! \left(x , y\right)
F_{143}\! \left(x , y\right) = F_{134}\! \left(x , y\right)+F_{144}\! \left(x , y\right)
F_{144}\! \left(x , y\right) = F_{142}\! \left(x , y\right)
F_{145}\! \left(x , y\right) = F_{113}\! \left(x , y\right)+F_{123}\! \left(x , y\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_14(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_64(x))
Eq(F_4(x), F_14(x)*F_5(x))
Eq(F_5(x), F_28(x) + F_6(x))
Eq(F_6(x), F_23(x) + F_7(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_14(x))
Eq(F_10(x), F_11(x) + F_15(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_17(x) + F_18(x) + F_20(x))
Eq(F_17(x), 0)
Eq(F_18(x), F_14(x)*F_19(x))
Eq(F_19(x), F_12(x) + F_16(x))
Eq(F_20(x), F_14(x)*F_21(x))
Eq(F_21(x), F_12(x) + F_22(x))
Eq(F_22(x), F_20(x))
Eq(F_23(x), F_12(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_14(x)*F_26(x))
Eq(F_26(x), F_12(x) + F_27(x))
Eq(F_27(x), F_25(x))
Eq(F_28(x), F_29(x) + F_63(x))
Eq(F_29(x), F_30(x) + F_39(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_14(x)*F_32(x))
Eq(F_32(x), F_11(x) + F_33(x))
Eq(F_33(x), F_30(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_14(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_12(x))
Eq(F_38(x), F_34(x))
Eq(F_39(x), F_17(x) + F_40(x) + F_49(x))
Eq(F_40(x), F_14(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_24(x) + F_8(x))
Eq(F_43(x), F_39(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_14(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_24(x))
Eq(F_48(x), F_44(x))
Eq(F_49(x), F_14(x)*F_50(x))
Eq(F_50(x), F_21(x) + F_51(x))
Eq(F_51(x), F_52(x) + F_57(x))
Eq(F_52(x), F_17(x) + F_53(x) + F_55(x))
Eq(F_53(x), F_14(x)*F_54(x))
Eq(F_54(x), F_12(x) + F_52(x))
Eq(F_55(x), F_14(x)*F_56(x))
Eq(F_56(x), F_12(x) + F_52(x))
Eq(F_57(x), 2*F_17(x) + F_58(x) + F_60(x))
Eq(F_58(x), F_14(x)*F_59(x))
Eq(F_59(x), F_22(x) + F_57(x))
Eq(F_60(x), F_14(x)*F_61(x))
Eq(F_61(x), F_52(x) + F_62(x))
Eq(F_62(x), F_60(x))
Eq(F_63(x), F_34(x) + F_44(x))
Eq(F_64(x), F_14(x)*F_65(x))
Eq(F_65(x), F_66(x, 1))
Eq(F_66(x, k_0), F_1(x) + F_67(x, k_0) + F_69(x, k_0) + F_71(x, k_0))
Eq(F_67(x, k_0), F_66(x, k_0)*F_68(x, k_0))
Eq(F_68(x, k_0), k_0*x)
Eq(F_69(x, k_0), F_14(x)*F_70(x, k_0))
Eq(F_70(x, k_0), (-k_0*F_66(x, k_0) + F_66(x, 1))/(1 - k_0))
Eq(F_71(x, k_0), F_14(x)*F_72(x, k_0))
Eq(F_72(x, k_0), F_5(x) + F_73(x, k_0))
Eq(F_73(x, k_0), F_106(x, k_0) + F_74(x, k_0))
Eq(F_74(x, k_0), F_75(x, k_0) + F_99(x, k_0))
Eq(F_75(x, k_0), F_76(x, k_0) + F_79(x, k_0))
Eq(F_76(x, k_0), F_77(x, k_0))
Eq(F_77(x, k_0), F_68(x, k_0)*F_78(x, k_0))
Eq(F_78(x, k_0), F_1(x) + F_76(x, k_0))
Eq(F_79(x, k_0), F_17(x) + F_80(x, k_0) + F_82(x, k_0))
Eq(F_80(x, k_0), F_68(x, k_0)*F_81(x, k_0))
Eq(F_81(x, k_0), F_79(x, k_0) + F_8(x))
Eq(F_82(x, k_0), F_14(x)*F_83(x, k_0))
Eq(F_83(x, k_0), F_84(x, k_0) + F_87(x, k_0))
Eq(F_84(x, k_0), F_76(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_14(x)*F_84(x, k_0))
Eq(F_87(x, k_0), F_88(x, k_0) + F_93(x, k_0))
Eq(F_88(x, k_0), F_17(x) + F_89(x, k_0) + F_91(x, k_0))
Eq(F_89(x, k_0), F_68(x, k_0)*F_90(x, k_0))
Eq(F_90(x, k_0), F_12(x) + F_88(x, k_0))
Eq(F_91(x, k_0), F_14(x)*F_92(x, k_0))
Eq(F_92(x, k_0), F_76(x, k_0) + F_88(x, k_0))
Eq(F_93(x, k_0), 2*F_17(x) + F_94(x, k_0) + F_96(x, k_0))
Eq(F_94(x, k_0), F_14(x)*F_95(x, k_0))
Eq(F_95(x, k_0), F_85(x, k_0) + F_93(x, k_0))
Eq(F_96(x, k_0), F_14(x)*F_97(x, k_0))
Eq(F_97(x, k_0), F_88(x, k_0) + F_98(x, k_0))
Eq(F_98(x, k_0), F_96(x, k_0))
Eq(F_99(x, k_0), F_100(x, k_0) + F_103(x, k_0))
Eq(F_100(x, k_0), F_101(x, k_0))
Eq(F_101(x, k_0), F_102(x, k_0)*F_68(x, k_0))
Eq(F_102(x, k_0), F_100(x, k_0) + F_12(x))
Eq(F_103(x, k_0), F_104(x, k_0))
Eq(F_104(x, k_0), F_105(x, k_0)*F_68(x, k_0))
Eq(F_105(x, k_0), F_103(x, k_0) + F_24(x))
Eq(F_106(x, k_0), F_107(x, k_0) + F_145(x, k_0))
Eq(F_107(x, k_0), F_108(x, k_0) + F_118(x, k_0))
Eq(F_108(x, k_0), F_101(x, k_0) + F_109(x, k_0) + F_17(x))
Eq(F_109(x, k_0), F_110(x, k_0)*F_14(x))
Eq(F_110(x, k_0), F_111(x, k_0) + F_112(x, k_0))
Eq(F_111(x, k_0), F_100(x, k_0) + F_76(x, k_0))
Eq(F_112(x, k_0), F_108(x, k_0) + F_113(x, k_0))
Eq(F_113(x, k_0), F_114(x, k_0))
Eq(F_114(x, k_0), F_115(x, k_0)*F_14(x))
Eq(F_115(x, k_0), F_116(x, k_0) + F_117(x, k_0))
Eq(F_116(x, k_0), F_100(x, k_0))
Eq(F_117(x, k_0), F_113(x, k_0))
Eq(F_118(x, k_0), F_104(x, k_0) + F_119(x, k_0) + F_128(x, k_0) + F_17(x))
Eq(F_119(x, k_0), F_120(x, k_0)*F_14(x))
Eq(F_120(x, k_0), F_121(x, k_0) + F_122(x, k_0))
Eq(F_121(x, k_0), F_103(x, k_0) + F_79(x, k_0))
Eq(F_122(x, k_0), F_118(x, k_0) + F_123(x, k_0))
Eq(F_123(x, k_0), F_124(x, k_0))
Eq(F_124(x, k_0), F_125(x, k_0)*F_14(x))
Eq(F_125(x, k_0), F_126(x, k_0) + F_127(x, k_0))
Eq(F_126(x, k_0), F_103(x, k_0))
Eq(F_127(x, k_0), F_123(x, k_0))
Eq(F_128(x, k_0), F_129(x, k_0)*F_14(x))
Eq(F_129(x, k_0), F_130(x, k_0) + F_133(x, k_0))
Eq(F_130(x, k_0), F_131(x, k_0) + F_85(x, k_0))
Eq(F_131(x, k_0), F_132(x, k_0))
Eq(F_132(x, k_0), F_130(x, k_0)*F_14(x))
Eq(F_133(x, k_0), F_134(x, k_0) + F_139(x, k_0))
Eq(F_134(x, k_0), F_135(x, k_0) + F_137(x, k_0) + 2*F_17(x))
Eq(F_135(x, k_0), F_136(x, k_0)*F_14(x))
Eq(F_136(x, k_0), F_134(x, k_0) + F_88(x, k_0))
Eq(F_137(x, k_0), F_138(x, k_0)*F_14(x))
Eq(F_138(x, k_0), F_134(x, k_0) + F_85(x, k_0))
Eq(F_139(x, k_0), F_140(x, k_0) + F_142(x, k_0) + 3*F_17(x))
Eq(F_140(x, k_0), F_14(x)*F_141(x, k_0))
Eq(F_141(x, k_0), F_131(x, k_0) + F_139(x, k_0))
Eq(F_142(x, k_0), F_14(x)*F_143(x, k_0))
Eq(F_143(x, k_0), F_134(x, k_0) + F_144(x, k_0))
Eq(F_144(x, k_0), F_142(x, k_0))
Eq(F_145(x, k_0), F_113(x, k_0) + F_123(x, k_0))
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:
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