0132_0231_1032_2301_3201
Counting sequence:
1, 1, 2, 6, 19, 51, 127, 303, 703, 1599, 3583, 7935, 17407, 37887, 81919, 176127, 376831, 802815, 1703935, 3604479, 7602175, 15990783, 33554431, 70254591, 146800639, 306184191, 637534207, 1325400063, 2751463423, 5704253439, 11811160063, 24427626495, 50465865727, 104152956927, 214748364799, 442381631487, 910533066751, 1872605741055, 3848290697215, 7902739824639, 16217796509695, 33260226740223, 68169720922111, 139637976727551, 285873023221759, 584940185976831, 1196268651020287, 2445313860173823, 4996180836614143, 10203467905761279, 20829148276588543, 42502721483309055, 86694292826882047, 176766285374291967, 360287970189639679, 734086739261390847, 1495195076287004671, 3044433348102455295, 6196953087261802495, 12610078956637388799, 25652503477502345215, 52169698083459825663, 106068778423829921791, 215596321361480384511, 438110171750601850879, 890055401556485865471, 1807780919223536058367, 3670902070668200771583, 7452484605778658852863, 15126330140441832325119, 30695382138652693889023, 62276207992843446255615, 126323303416763009466367, 256188381695678252843007, 519460313115660973506559, 1053087725679930882654207, 2134509650257079636590591, 4325687698308595015745535, 8764712192206061516619775, 17756097975589866003496959, 35965543133535217947508735, 72837780631781407776047103, 147488949992984759314153471, 298604677444813406152425471, 604462909807314587353087999, 1223432929450004724802650111, 2475880078570760549798248447, 5009788596483023299982393343, 10135634071649051000736579583, 20503381900664110803016744959, 41470991316060239209120661503, 83870437661584513624415666175, 169597785382097097661180018687, 342909390882050336147057410047, 693246421999812953943509565439, 1401348124471050471185808621567, 2832406809884950068969196224511, 5724234741655598391133550411775, 11567311727082593288657416749055, 23372307941707979590095465349119, 47219984858501545205752194400255
Generating function in Maple syntax:
(4*x^5-x^4-5*x^2+4*x-1)/(x-1)/(2*x-1)^2
Generating function in latex syntax:
\frac{4 x^{5}-x^{4}-5 x^{2}+4 x -1}{\left(x -1\right) \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(4*x**5 - x**4 - 5*x**2 + 4*x - 1)/((x - 1)*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(x-1)*(2*x-1)^2*F(x)+4*x^5-x^4-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+4 x^{5}-x^{4}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,-1+1/8*(3*n-2)*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ -1+\frac{\left(3 n -2\right) 2^{n}}{8} & \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) = 51
a(n+2) = -4*a(n)+4*a(n+1)-1, 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) = 51
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18922/
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[34,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[14,x]+F[24,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[10,x]+F[17,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[11,x]+F[21,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[11,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[27,x]*F[4,x]
F[30,x] = F[21,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[28,x]+F[31,x]
F[34,x] = F[35,x]+F[56,x]
F[35,x] = F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[1,x]+F[14,x]
F[39,x] = F[18,x]+F[40,x]
F[40,x] = F[41,x]+F[42,x]+F[47,x]
F[41,x] = 0
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[11,x]+F[44,x]
F[44,x] = F[41,x]+F[42,x]+F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[18,x]+F[44,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[46,x]+F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[18,x]+F[50,x]
F[53,x] = F[54,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[44,x]+F[53,x]
F[56,x] = F[41,x]+F[57,x]+F[76,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[62,x]
F[59,x] = F[11,x]+F[60,x]
F[60,x] = F[25,x]+F[41,x]+F[61,x]
F[61,x] = F[17,x]*F[4,x]
F[62,x] = F[44,x]+F[63,x]
F[63,x] = F[41,x]+F[64,x]+F[69,x]+F[75,x]
F[64,x] = F[4,x]*F[65,x]
F[65,x] = F[28,x]+F[66,x]
F[66,x] = 2*F[41,x]+F[64,x]+F[67,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[44,x]+F[66,x]
F[69,x] = F[4,x]*F[70,x]
F[70,x] = F[68,x]+F[71,x]
F[71,x] = F[53,x]+F[72,x]
F[72,x] = F[73,x]
F[73,x] = F[4,x]*F[74,x]
F[74,x] = F[66,x]+F[72,x]
F[75,x] = F[4,x]*F[49,x]
F[76,x] = F[4,x]*F[77,x]
F[77,x] = F[46,x]+F[78,x]
F[78,x] = F[40,x]+F[79,x]
F[79,x] = 2*F[41,x]+F[64,x]+F[69,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_{34}\! \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_{14}\! \left(x \right)+F_{24}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{56}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{41}\! \left(x \right) = 0
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{11}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{18}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{46}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{18}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{44}\! \left(x \right)+F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{41}\! \left(x \right)+F_{57}\! \left(x \right)+F_{76}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{62}\! \left(x \right)
F_{59}\! \left(x \right) = F_{11}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{25}\! \left(x \right)+F_{41}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{62}\! \left(x \right) = F_{44}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{41}\! \left(x \right)+F_{64}\! \left(x \right)+F_{69}\! \left(x \right)+F_{75}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{28}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{64}\! \left(x \right)+F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{44}\! \left(x \right)+F_{66}\! \left(x \right)
F_{69}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{68}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{53}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{4}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{66}\! \left(x \right)+F_{72}\! \left(x \right)
F_{75}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{76}\! \left(x \right) = F_{4}\! \left(x \right) F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{46}\! \left(x \right)+F_{78}\! \left(x \right)
F_{78}\! \left(x \right) = F_{40}\! \left(x \right)+F_{79}\! \left(x \right)
F_{79}\! \left(x \right) = 2 F_{41}\! \left(x \right)+F_{64}\! \left(x \right)+F_{69}\! \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_34(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_14(x) + F_24(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_10(x) + F_17(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_11(x) + F_21(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_11(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_27(x)*F_4(x))
Eq(F_30(x), F_21(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_28(x) + F_31(x))
Eq(F_34(x), F_35(x) + F_56(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_1(x) + F_14(x))
Eq(F_39(x), F_18(x) + F_40(x))
Eq(F_40(x), F_41(x) + F_42(x) + F_47(x))
Eq(F_41(x), 0)
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_11(x) + F_44(x))
Eq(F_44(x), F_41(x) + F_42(x) + F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_18(x) + F_44(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_46(x) + F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_18(x) + F_50(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_44(x) + F_53(x))
Eq(F_56(x), F_41(x) + F_57(x) + F_76(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_62(x))
Eq(F_59(x), F_11(x) + F_60(x))
Eq(F_60(x), F_25(x) + F_41(x) + F_61(x))
Eq(F_61(x), F_17(x)*F_4(x))
Eq(F_62(x), F_44(x) + F_63(x))
Eq(F_63(x), F_41(x) + F_64(x) + F_69(x) + F_75(x))
Eq(F_64(x), F_4(x)*F_65(x))
Eq(F_65(x), F_28(x) + F_66(x))
Eq(F_66(x), 2*F_41(x) + F_64(x) + F_67(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_44(x) + F_66(x))
Eq(F_69(x), F_4(x)*F_70(x))
Eq(F_70(x), F_68(x) + F_71(x))
Eq(F_71(x), F_53(x) + F_72(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_4(x)*F_74(x))
Eq(F_74(x), F_66(x) + F_72(x))
Eq(F_75(x), F_4(x)*F_49(x))
Eq(F_76(x), F_4(x)*F_77(x))
Eq(F_77(x), F_46(x) + F_78(x))
Eq(F_78(x), F_40(x) + F_79(x))
Eq(F_79(x), 2*F_41(x) + F_64(x) + F_69(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:
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