0213_0231_1032_1230_2031
Counting sequence:
1, 1, 2, 6, 19, 57, 164, 460, 1269, 3460, 9353, 25120, 67135, 178741, 474465, 1256489, 3321185, 8765274, 23104634, 60840194, 160072476, 420860915, 1105874286, 2904412088, 7624805916, 20009876913, 52496152275, 137688166548, 361050129322, 946571332849, 2481220449032, 6502989758327, 17041376901142, 44652648044597, 116989463689381, 306484509399027, 802855004078353, 2102986537770113, 5508205306301108, 14426501752108476, 37782604523876768, 98947547136459396, 259120937340107757, 678556662748706726, 1776877404030358924, 4652838163684516302, 12183408522154150819, 31901502633587617105, 83530660417071456824, 218712693910433599080, 572659042754348001433, 1499384393941443364340, 3925772919880306314965, 10278582269655412638136, 26911479717446950573895, 70459356777716992693809, 184474725422132680604365, 482983727694465106478469, 1264520408057189430954689, 3310679657052544846394014, 8667756034226302051882506, 22693140452220178593361810, 59412948488481819315298016, 155548687825191070413400183, 407240048809257035065712746, 1066187577034732215368104408, 2791360151606109152964100080, 7307979980369122142059078193, 19132782272455965047123760827, 50090837540779640029045042344, 131140824579454923406213958462, 343334179924800520855230456797, 898867628547442765158867730272, 2353282452402583387064802620867, 6161011685442549466853329954086, 16129826893663880316702381599597, 42228641696705517242965836859233, 110556499674865376914299773520183, 289441790645832327307160922443013, 757771041951343028218628752536921, 1983876379097970838939233797555848, 5193869820914631668891976786870916, 13597760342195716104919923513810512, 35599474573916426450970346128890696, 93200810692455539514321564947408329, 244003299963613630308156506551714922, 638809885320106697511461451273408724, 1672428206752027748724945213013901250, 4378479037413825450894602933012676379, 11463018907524402677199511553434514825, 30010600937044525469348301437983357388, 78568837957633857526300766569426035060, 205696038596118395539416691975870394045, 538519569955186407174360340033421229156, 1409863350376028562156806061479135943825, 3691072059903374710060880986685868473504, 9663356499435573526017148509114966747295, 25299005970350940736529424060509704239789, 66233681245982601043812760632075454219369, 173402083876889304176233078243336542494209, 453972677575759719936286154836923495480161
Generating function in Maple syntax:
(x^6-5*x^5+10*x^4-20*x^3+18*x^2-7*x+1)/(2*x-1)/(x^2-3*x+1)/(x^3-2*x^2+3*x-1)
Generating function in latex syntax:
\frac{x^{6}-5 x^{5}+10 x^{4}-20 x^{3}+18 x^{2}-7 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}
Generating function in sympy syntax:
(x**6 - 5*x**5 + 10*x**4 - 20*x**3 + 18*x**2 - 7*x + 1)/((2*x - 1)*(x**2 - 3*x + 1)*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x^3-2*x^2+3*x-1)*F(x)-x^6+5*x^5-10*x^4+20*x^3-18*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)-x^{6}+5 x^{5}-10 x^{4}+20 x^{3}-18 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-1/40*(((-2/5*I+8/115*23^(1/2))*3^(1/2)+24/115*I*23^(1/2)-2/5)*(11+3*23^(1/2)*3^(1/2))^(1/3)+2^(1/3)*((I-1/23*23^(1/2))*3^(1/2)+3/23*I*23^(1/2)-1))*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)*(11/600*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/40*(((-2/5*I-8/115*23^(1/2))*3^(1/2)+24/115*I*23^(1/2)+2/5)*(11+3*23^(1/2)*3^(1/2))^(1/3)+((I+1/23*23^(1/2))*3^(1/2)+3/23*I*23^(1/2)+1)*2^(1/3))*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)*(-11/600*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(44+12*23^(1/2)*3^(1/2))^(1/3)+1/12*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)-1/460*((-8/5*23^(1/2)*3^(1/2)+46/5)*(11+3*23^(1/2)*3^(1/2))^(1/3)+2^(1/3)*(23^(1/2)*3^(1/2)+23))*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)*(1/300*2^(1/3)*(3*23^(1/2)*3^(1/2)-11)*(11+3*23^(1/2)*3^(1/2))^(2/3)-1/6*(44+12*23^(1/2)*3^(1/2))^(1/3)+2/3)^(-n)+1/4600*(460*5^(1/2)+2300)*(3/2-1/2*5^(1/2))^(-n)+1/4600*(-460*5^(1/2)+2300)*(3/2+1/2*5^(1/2))^(-n)-2^(n-1))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -\frac{\left(\left(\left(-\frac{2 \,\mathrm{I}}{5}+\frac{8 \sqrt{23}}{115}\right) \sqrt{3}+\frac{24 \,\mathrm{I} \sqrt{23}}{115}-\frac{2}{5}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{\sqrt{23}}{23}\right) \sqrt{3}+\frac{3 \,\mathrm{I} \sqrt{23}}{23}-1\right)\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\frac{11 \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{40}\\+\\\frac{\left(\left(\left(-\frac{2 \,\mathrm{I}}{5}-\frac{8 \sqrt{23}}{115}\right) \sqrt{3}+\frac{24 \,\mathrm{I} \sqrt{23}}{115}+\frac{2}{5}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(\mathrm{I}+\frac{\sqrt{23}}{23}\right) \sqrt{3}+\frac{3 \,\mathrm{I} \sqrt{23}}{23}+1\right) 2^{\frac{1}{3}}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}} \left(-\frac{11 \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{40}\\-\\\frac{\left(\left(-\frac{8 \sqrt{23}\, \sqrt{3}}{5}+\frac{46}{5}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+2^{\frac{1}{3}} \left(\sqrt{23}\, \sqrt{3}+23\right)\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{460}\\+\frac{\left(460 \sqrt{5}+2300\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{4600}+\frac{\left(-460 \sqrt{5}+2300\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{4600}-\\2^{n -1} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 57
a(6) = 164
a(n+6) = -2*a(n)+11*a(n+1)-25*a(n+2)+34*a(n+3)-24*a(n+4)+8*a(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) = 19
a \! \left(5\right) = 57
a \! \left(6\right) = 164
a \! \left(n +6\right) = -2 a \! \left(n \right)+11 a \! \left(n +1\right)-25 a \! \left(n +2\right)+34 a \! \left(n +3\right)-24 a \! \left(n +4\right)+8 a \! \left(n +5\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21843/
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[0,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[4,x]*F[8,x]
F[8,x] = F[1,x]+F[9,x]
F[9,x] = F[4,x]*F[8,x]
F[10,x] = F[11,x]+F[23,x]
F[11,x] = F[12,x]+F[15,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[4,x]*F[8,x]
F[15,x] = F[16,x]*F[19,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[21,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[19,x]*F[4,x]
F[21,x] = F[22,x]
F[22,x] = F[18,x]*F[19,x]*F[4,x]*F[8,x]
F[23,x] = F[24,x]*F[8,x]
F[24,x] = F[15,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[32,x]*F[35,x]*F[4,x]
F[29,x] = F[1,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[29,x]*F[4,x]*F[8,x]
F[32,x] = F[10,x]+F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[20,x]
F[35,x] = F[9,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_{0}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{1}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{23}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{15}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right) F_{19}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{8}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{32}\! \left(x \right) F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{32}\! \left(x \right) = F_{10}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{20}\! \left(x \right)
F_{35}\! \left(x \right) = F_{9}\! \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_0(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_10(x)*F_4(x)*F_8(x))
Eq(F_8(x), F_1(x) + F_9(x))
Eq(F_9(x), F_4(x)*F_8(x))
Eq(F_10(x), F_11(x) + F_23(x))
Eq(F_11(x), F_12(x) + F_15(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_4(x)*F_8(x))
Eq(F_15(x), F_16(x)*F_19(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_21(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_19(x)*F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_18(x)*F_19(x)*F_4(x)*F_8(x))
Eq(F_23(x), F_24(x)*F_8(x))
Eq(F_24(x), F_15(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_32(x)*F_35(x)*F_4(x))
Eq(F_29(x), F_1(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_29(x)*F_4(x)*F_8(x))
Eq(F_32(x), F_10(x) + F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_20(x))
Eq(F_35(x), F_9(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": 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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