0132_0213_0231_2103_2130
Counting sequence:
1, 1, 2, 6, 19, 55, 145, 360, 865, 2044, 4790, 11179, 26037, 60583, 140898, 327613, 761678, 1770762, 4116603, 9570027, 22247705, 51719736, 120233901, 279510016, 649782066, 1510560155, 3511626441, 8163541175, 18977930898, 44118336889, 102562690154, 238429327694, 554280939779, 1288546854223, 2995508010929, 6963711264248, 16188664625241, 37634079358292, 87488620088782, 203386366175147, 472815937706317, 1099163700857591, 2555245593335442, 5940225315997621, 13809348462179734, 32102840347879570, 74630049435277035, 173493816072251875, 403324189694081305, 937614986373017384, 2179690395803141605, 5067165404357471544, 11779730407839149002, 27384550810605645723, 63661357020496110913, 147994699848116190503, 344045936313961995618, 799809766266149716977, 1859332126018641350418, 4322422781837586613142, 10048413859741626855795, 23359728141568348691751, 54304779487059378977041, 126243296037783066403656, 293480057280798789948881, 682258359253889616016620, 1586058259237854334556006, 3687138116486582561583915, 8571556190237928631656613, 19926450596978475106358279, 46323377526946150617345794, 107688787577119430270977709, 250346058274444464684600094, 581983977196040684129190938, 1352948602616352553289350619, 3145223911731420756294270363, 7311758507157597846433301081, 16997776300626304580000713432, 39515035799295138803429808797, 91861313303790405096721300912, 213551644613407242263304998882, 496447343031936055399902203931, 1154100053172784086769817915257, 2682957118067887391772954337111, 6237118590890030057179129385074, 14499541589707099474761297396585, 33707344705409125701698587757050, 78360069527703208212752297863390, 182165060761998472709621015472995, 423482387935998127405057038449599, 984477111811700645008681382266545, 2288631620325104152925551085373784, 5320423025287909296164347530038217, 12368482947025220227650621801633988, 28753234410824946243548721430199678, 66843160363712307571509268217369643, 155391495216512250455080983593343933, 361241399332937082465773135775492567, 839784367929499054058666708357159858, 1952261800339135247699534837113838757, 4538458064491344717447044230402689510
Generating function in Maple syntax:
-(x^5+3*x^4-3*x^3+6*x^2-4*x+1)/(x^3-2*x^2+3*x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{x^{5}+3 x^{4}-3 x^{3}+6 x^{2}-4 x +1}{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-x**5 - 3*x**4 + 3*x**3 - 6*x**2 + 4*x - 1)/((x - 1)**2*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3-2*x^2+3*x-1)*(x-1)^2*F(x)+x^5+3*x^4-3*x^3+6*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{5}+3 x^{4}-3 x^{3}+6 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/13800*(-667*((I-41/667*23^(1/2))*3^(1/2)-123/667*I*23^(1/2)+1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3)-9200+115*((I-31/23*23^(1/2))*3^(1/2)+93/23*I*23^(1/2)-1)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3))*(11/600*2^(1/3)*((I-3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)+1)*(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)+1/13800*(-115*((I+31/23*23^(1/2))*3^(1/2)+93/23*I*23^(1/2)+1)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)-9200+667*((I+41/667*23^(1/2))*3^(1/2)-123/667*I*23^(1/2)-1)*2^(1/3)*(11+3*23^(1/2)*3^(1/2))^(2/3))*(-11/600*2^(1/3)*((I+3/11*23^(1/2))*3^(1/2)-9/11*I*23^(1/2)-1)*(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)+1/13800*((310*2^(2/3)*23^(1/2)*3^(1/2)+230*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)-9200+(-82*23^(1/2)*3^(1/2)*2^(1/3)+1334*2^(1/3))*(11+3*23^(1/2)*3^(1/2))^(2/3))*(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)-4*n+4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-667 \left(\left(\mathrm{I}-\frac{41 \sqrt{23}}{667}\right) \sqrt{3}-\frac{123 \,\mathrm{I} \sqrt{23}}{667}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}-9200+115 \left(\left(\mathrm{I}-\frac{31 \sqrt{23}}{23}\right) \sqrt{3}+\frac{93 \,\mathrm{I} \sqrt{23}}{23}-1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \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}}{13800}\\+\\\frac{\left(-115 \left(\left(\mathrm{I}+\frac{31 \sqrt{23}}{23}\right) \sqrt{3}+\frac{93 \,\mathrm{I} \sqrt{23}}{23}+1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-9200+667 \left(\left(\mathrm{I}+\frac{41 \sqrt{23}}{667}\right) \sqrt{3}-\frac{123 \,\mathrm{I} \sqrt{23}}{667}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) \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}}{13800}\\+\\\frac{\left(\left(310 \,2^{\frac{2}{3}} \sqrt{23}\, \sqrt{3}+230 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-9200+\left(-82 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}+1334 \,2^{\frac{1}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \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}}{13800}\\-4 n +4 & \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) = 55
a(n+3) = a(n)-2*a(n+1)+3*a(n+2)+4*n, n >= 6
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) = 55
a \! \left(n +3\right) = a \! \left(n \right)-2 a \! \left(n +1\right)+3 a \! \left(n +2\right)+4 n, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18199/
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[14,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[7,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]+F[17,x]+F[29,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]+F[7,x]
F[20,x] = F[16,x]+F[21,x]+F[26,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]
F[23,x] = F[11,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[23,x]*F[4,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[13,x]+F[28,x]
F[28,x] = F[20,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]+F[39,x]
F[31,x] = F[2,x]+F[32,x]
F[32,x] = F[16,x]+F[33,x]+F[38,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]
F[35,x] = F[11,x]+F[36,x]
F[36,x] = F[16,x]+F[21,x]+F[37,x]
F[37,x] = F[13,x]*F[4,x]
F[38,x] = F[31,x]*F[4,x]
F[39,x] = F[15,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_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{29}\! \left(x \right)
F_{16}\! \left(x \right) = 0
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_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{26}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{20}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{39}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right)+F_{33}\! \left(x \right)+F_{38}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{11}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{15}\! \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_14(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_7(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_29(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x) + F_7(x))
Eq(F_20(x), F_16(x) + F_21(x) + F_26(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_11(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_23(x)*F_4(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_13(x) + F_28(x))
Eq(F_28(x), F_20(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x) + F_39(x))
Eq(F_31(x), F_2(x) + F_32(x))
Eq(F_32(x), F_16(x) + F_33(x) + F_38(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_11(x) + F_36(x))
Eq(F_36(x), F_16(x) + F_21(x) + F_37(x))
Eq(F_37(x), F_13(x)*F_4(x))
Eq(F_38(x), F_31(x)*F_4(x))
Eq(F_39(x), F_15(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_placements", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 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, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 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, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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