0132_0213_0321_2103_2130
Counting sequence:
1, 1, 2, 6, 19, 54, 140, 344, 822, 1937, 4533, 10572, 24615, 57265, 133171, 309635, 719868, 1673548, 3890589, 9044588, 21026186, 48880026, 113632352, 264163251, 614105139, 1427621334, 3318817045, 7715313679, 17935928357, 41695974837, 96931381558, 225338123442, 523847582135, 1217797881170, 2831036602776, 6581361628220, 15299809560378, 35567742027573, 82684968590289, 192219231276208, 446855498675731, 1038813002065181, 2414947240120407, 5614071214906711, 13051132166544624, 30340201309940984, 70532410811640545, 163967961981584424, 381179265631413302, 886134283742711742, 2060012281946893188, 4788947543986669527, 11132952351808934095, 25880974249400356570, 60165965588569871201, 139868900618717834715, 325155744928414118473, 755895399136376557353, 1757243608171019269994, 4085095771168718813918, 9496695496300494459291, 22077138554735065020206, 51323120442772924956132, 119311779715149139287456, 277366236814636632970494, 644798271456384545292889, 1498974120455029509225325, 3484692055266956070060884, 8100926196347193737025087, 18832368598962698580179017, 43779945459460664336547963, 101776025376803789586311147, 236600553810452738666016740, 550029556137211301161976100, 1278663586167532215740206181, 2972532200038626783562683300, 6910298983918027220369613858, 16064496137844360309723681378, 37345422645735653271994499944, 86817574645436266416905751163, 201826374782681853016451935211, 469189397702908679487538803486, 1090733018188798598846618291437, 2535646633943260290581229202791, 5894663263155092353537989829229, 13703429539767555078298129373789, 31856608726935740818399637665950, 74057630364427204652140644079754, 172163103179177687397920786281407, 400230657535614393707880708350922, 930423396612915010979941196569968, 2162971977946693932921982959289732, 5028299798149866170513947193080450, 11689378835169125656677816857232125, 27174508887154338561927539144825481, 63173068789274630542940930913092920, 146859567428684340161645531306860203, 341407073594658097960982271239220533, 793675154715880244102596682017034399, 1845070884387008876547471034879522623, 4289269417323924239398202011843719896
Generating function in Maple syntax:
-(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{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:
(-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)+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)+3 x^{4}-3 x^{3}+6 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
1/13800*(-460*2^(2/3)*((-39/46*I+13/46*3^(1/2))*23^(1/2)+I*3^(1/2)-1)*(11+3*23^(1/2)*3^(1/2))^(1/3)-4600-437*2^(1/3)*((-3/437*I-1/437*3^(1/2))*23^(1/2)+I*3^(1/2)+1)*(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*(460*((-39/46*I-13/46*3^(1/2))*23^(1/2)+I*3^(1/2)+1)*2^(2/3)*(11+3*23^(1/2)*3^(1/2))^(1/3)-4600+437*2^(1/3)*((-3/437*I+1/437*3^(1/2))*23^(1/2)+I*3^(1/2)-1)*(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*((260*2^(2/3)*23^(1/2)*3^(1/2)-920*2^(2/3))*(11+3*23^(1/2)*3^(1/2))^(1/3)-4600+(-2*23^(1/2)*3^(1/2)*2^(1/3)+874*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)-3*n+2
Explicit closed form in latex syntax:
\frac{\left(-460 \,2^{\frac{2}{3}} \left(\left(-\frac{39 \,\mathrm{I}}{46}+\frac{13 \sqrt{3}}{46}\right) \sqrt{23}+\mathrm{I} \sqrt{3}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-4600-437 \,2^{\frac{1}{3}} \left(\left(-\frac{3 \,\mathrm{I}}{437}-\frac{\sqrt{3}}{437}\right) \sqrt{23}+\mathrm{I} \sqrt{3}+1\right) \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(460 \left(\left(-\frac{39 \,\mathrm{I}}{46}-\frac{13 \sqrt{3}}{46}\right) \sqrt{23}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-4600+437 \,2^{\frac{1}{3}} \left(\left(-\frac{3 \,\mathrm{I}}{437}+\frac{\sqrt{3}}{437}\right) \sqrt{23}+\mathrm{I} \sqrt{3}-1\right) \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(260 \,2^{\frac{2}{3}} \sqrt{23}\, \sqrt{3}-920 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-4600+\left(-2 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}+874 \,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}-3 n +2
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+3) = a(n)-2*a(n+1)+3*a(n+2)+3*n+1, 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) = 19
a \! \left(n +3\right) = a \! \left(n \right)-2 a \! \left(n +1\right)+3 a \! \left(n +2\right)+3 n +1, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18281/
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[16,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[11,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[4,x]
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[18,x]+F[19,x]+F[33,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[18,x]+F[23,x]+F[28,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]
F[25,x] = F[11,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[25,x]*F[4,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[13,x]+F[30,x]
F[30,x] = F[26,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[30,x]*F[4,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[42,x]
F[35,x] = F[2,x]+F[36,x]
F[36,x] = F[18,x]+F[37,x]+F[41,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]
F[39,x] = F[11,x]+F[40,x]
F[40,x] = F[15,x]+F[18,x]+F[23,x]
F[41,x] = F[35,x]*F[4,x]
F[42,x] = F[36,x]+F[43,x]
F[43,x] = 2*F[18,x]+F[44,x]+F[48,x]
F[44,x] = F[4,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[14,x]+F[47,x]
F[47,x] = F[32,x]
F[48,x] = F[4,x]*F[42,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_{16}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{33}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{28}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \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_{42}\! \left(x \right)
F_{35}\! \left(x \right) = F_{2}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{18}\! \left(x \right)+F_{37}\! \left(x \right)+F_{41}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{23}\! \left(x \right)
F_{41}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{36}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{48}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{32}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \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_16(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_11(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_33(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_18(x) + F_23(x) + F_28(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_11(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_25(x)*F_4(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_13(x) + F_30(x))
Eq(F_30(x), F_26(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_30(x)*F_4(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_42(x))
Eq(F_35(x), F_2(x) + F_36(x))
Eq(F_36(x), F_18(x) + F_37(x) + F_41(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_11(x) + F_40(x))
Eq(F_40(x), F_15(x) + F_18(x) + F_23(x))
Eq(F_41(x), F_35(x)*F_4(x))
Eq(F_42(x), F_36(x) + F_43(x))
Eq(F_43(x), 2*F_18(x) + F_44(x) + F_48(x))
Eq(F_44(x), F_4(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_14(x) + F_47(x))
Eq(F_47(x), F_32(x))
Eq(F_48(x), F_4(x)*F_42(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, 3, 2, 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, 3, 2, 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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