0231_1032_1203
Counting sequence:
1, 1, 2, 6, 21, 74, 255, 863, 2891, 9638, 32068, 106627, 354480, 1178459, 3917863, 13025489, 43305654, 143978586, 478687745, 1591501602, 5291294519, 17592063707, 58488653747, 194458283338, 646518954760, 2149493188071, 7146458609432, 23759959271131, 78995163239531, 262636637720769, 873192745587690, 2903119601274398, 9652053870227181, 32090356826458538, 106691385595173023, 354718765571584487, 1179339849667813435, 3920972375885662414, 13036127267969118108, 43341446420747116139, 144098085208022034544, 479085491495697756491, 1592824136631173971935, 5295691009790522752017, 17606678996270107116254, 58537241826343110228882, 194619819067621142358617, 647056007289843391447778, 2151278726779629554344055, 7152395013962848016649091, 23779696140230066742244387, 79060782775246479150734706, 262854804206688076856263616, 873918087694027657675210639, 2905531159317322454687507000, 9660071620716447031691570283, 32117013585666328792313319171, 106780011801338312588085190913, 355013422710715388034905626578, 1180319501549235191286689893622, 3924229442086910067452773805765, 13046956094454881016807393390794, 43377449214617921697870677736703, 144217784343312209474512155761839, 479483457360260285288147233202859, 1594147260887452859394344429993846, 5300090024765039484518032105064500, 17621304480350077450787434532874707, 58585867435897200390786665051495664, 194781485504887626777684435182803707, 647593502596907098522965337260492887, 2153065747078959099318769891912586193, 7158336352441989899603791687889721350, 23799449414961738994777038677123330186, 79126456842461258768251912739341149617, 263073151957282352964190592802918315106, 874644032381352444297811787424620556151, 2907944720655237594558537592891009918059, 9668096031437525150824630045295661596435, 32143692488087004474888549261945204373018, 106868711627295913352170128223656018103736, 355308324614881106945019137777608491694263, 1181299967205640626120463261039051181635608, 3927489213861222973792759948246205807902907, 13057793916209458704046597782490931291285595, 43413481915222783895402470091510682132230849, 144337582910061218444451564579387289244563898, 479881753806382898745608400947142751051843214, 1595471484234183476852500485950171099546472349, 5304492693905317504024471457338167360420506026, 17635942113500567069988582205207121324874742815, 58634533437684194213629810602174084681448197047, 194943286234936150040075759557133345038010553499, 648131444389594733380446716155334988006256353182, 2154854251817060109337255616018761725389559409932, 7164282626261371639119150135303818899534610449083, 23819219098307731765668216048703879333966303657520, 79192185463690723329059434116012600186644022874731, 263291681084588892025840279107459660742977070159247, 875370580095089517895065411709546517742542294936081, 2910360286885894358389565438887311039589014157987886
Generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)/(7*x^3-11*x^2+6*x-1)
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(x^{2}-3 x +1\right)}{7 x^{3}-11 x^{2}+6 x -1}
Generating function in sympy syntax:
(2*x - 1)*(x**2 - 3*x + 1)/(7*x**3 - 11*x**2 + 6*x - 1)
Implicit equation for the generating function in Maple syntax:
(7*x^3-11*x^2+6*x-1)*F(x)-(2*x-1)*(x^2-3*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(7 x^{3}-11 x^{2}+6 x -1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x^{2}-3 x +1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/96600*(-2231*((I-287/2231*23^(1/2))*3^(1/2)-861/2231*I*23^(1/2)+1)*2^(1/3)*(173+21*23^(1/2)*3^(1/2))^(2/3)+23000+2990*2^(2/3)*((I+28/299*23^(1/2))*3^(1/2)-84/299*I*23^(1/2)-1)*(173+21*23^(1/2)*3^(1/2))^(1/3))*(173/4200*((I-21/173*23^(1/2))*3^(1/2)-63/173*I*23^(1/2)+1)*2^(1/3)*(173+21*23^(1/2)*3^(1/2))^(2/3)-1/84*I*3^(1/2)*(692+84*23^(1/2)*3^(1/2))^(1/3)+1/84*(692+84*23^(1/2)*3^(1/2))^(1/3)+11/21)^(-n)+1/96600*(-2990*2^(2/3)*((I-28/299*23^(1/2))*3^(1/2)-84/299*I*23^(1/2)+1)*(173+21*23^(1/2)*3^(1/2))^(1/3)+23000+2231*((I+287/2231*23^(1/2))*3^(1/2)-861/2231*I*23^(1/2)-1)*2^(1/3)*(173+21*23^(1/2)*3^(1/2))^(2/3))*(-173/4200*((I+21/173*23^(1/2))*3^(1/2)-63/173*I*23^(1/2)-1)*2^(1/3)*(173+21*23^(1/2)*3^(1/2))^(2/3)+1/84*I*3^(1/2)*(692+84*23^(1/2)*3^(1/2))^(1/3)+1/84*(692+84*23^(1/2)*3^(1/2))^(1/3)+11/21)^(-n)-41/6900*(40/41*2^(2/3)*(23^(1/2)*3^(1/2)-299/28)*(173+21*23^(1/2)*3^(1/2))^(1/3)-11500/287+2^(1/3)*(23^(1/2)*3^(1/2)-2231/287)*(173+21*23^(1/2)*3^(1/2))^(2/3))*(1/2100*2^(1/3)*(21*23^(1/2)*3^(1/2)-173)*(173+21*23^(1/2)*3^(1/2))^(2/3)-1/42*(692+84*23^(1/2)*3^(1/2))^(1/3)+11/21)^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-2231 \left(\left(\mathrm{I}-\frac{287 \sqrt{23}}{2231}\right) \sqrt{3}-\frac{861 \,\mathrm{I} \sqrt{23}}{2231}+1\right) 2^{\frac{1}{3}} \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+23000+2990 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{28 \sqrt{23}}{299}\right) \sqrt{3}-\frac{84 \,\mathrm{I} \sqrt{23}}{299}-1\right) \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{173 \left(\left(\mathrm{I}-\frac{21 \sqrt{23}}{173}\right) \sqrt{3}-\frac{63 \,\mathrm{I} \sqrt{23}}{173}+1\right) 2^{\frac{1}{3}} \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{4200}-\frac{\mathrm{I} \sqrt{3}\, \left(692+84 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{\left(692+84 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{11}{21}\right)^{-n}}{96600}\\+\\\frac{\left(-2990 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{28 \sqrt{23}}{299}\right) \sqrt{3}-\frac{84 \,\mathrm{I} \sqrt{23}}{299}+1\right) \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+23000+2231 \left(\left(\mathrm{I}+\frac{287 \sqrt{23}}{2231}\right) \sqrt{3}-\frac{861 \,\mathrm{I} \sqrt{23}}{2231}-1\right) 2^{\frac{1}{3}} \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{173 \left(\left(\mathrm{I}+\frac{21 \sqrt{23}}{173}\right) \sqrt{3}-\frac{63 \,\mathrm{I} \sqrt{23}}{173}-1\right) 2^{\frac{1}{3}} \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{4200}+\frac{\mathrm{I} \sqrt{3}\, \left(692+84 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{\left(692+84 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{11}{21}\right)^{-n}}{96600}\\-\\\frac{41 \left(\frac{40 \,2^{\frac{2}{3}} \left(\sqrt{23}\, \sqrt{3}-\frac{299}{28}\right) \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{41}-\frac{11500}{287}+2^{\frac{1}{3}} \left(\sqrt{23}\, \sqrt{3}-\frac{2231}{287}\right) \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(21 \sqrt{23}\, \sqrt{3}-173\right) \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{2100}-\frac{\left(692+84 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{42}+\frac{11}{21}\right)^{-n}}{6900} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+3) = 7*a(n)-11*a(n+1)+6*a(n+2), n >= 4
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(n +3\right) = 7 a \! \left(n \right)-11 a \! \left(n +1\right)+6 a \! \left(n +2\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/352/
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[7,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[7,x]*F[8,x]
F[7,x] = x
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[10,x]+F[5,x]
F[10,x] = F[0,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[7,x]*F[8,x]
F[13,x] = F[14,x]
F[14,x] = F[0,x]*F[15,x]
F[15,x] = F[16,x]+F[27,x]
F[16,x] = F[17,x]*F[26,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[7,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[21,x]*F[24,x]*F[7,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[24,x]*F[7,x]
F[26,x] = F[25,x]
F[27,x] = F[28,x]
F[28,x] = F[24,x]^2*F[19,x]*F[7,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_{7}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right) F_{8}\! \left(x \right)
F_{7}\! \left(x \right) = x
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{5}\! \left(x \right)
F_{10}\! \left(x \right) = F_{0}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{7}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{0}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{27}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{26}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{7}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{21}\! \left(x \right) F_{24}\! \left(x \right) F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{24}\! \left(x \right) F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{25}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{24} \left(x \right)^{2} F_{19}\! \left(x \right) F_{7}\! \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_7(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_7(x)*F_8(x))
Eq(F_7(x), x)
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_5(x))
Eq(F_10(x), F_0(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_7(x)*F_8(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_0(x)*F_15(x))
Eq(F_15(x), F_16(x) + F_27(x))
Eq(F_16(x), F_17(x)*F_26(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_7(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_21(x)*F_24(x)*F_7(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_24(x)*F_7(x))
Eq(F_26(x), F_25(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_19(x)*F_24(x)**2*F_7(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], 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