0132_1032_1230_2130_3120
Counting sequence:
1, 1, 2, 6, 19, 53, 139, 351, 863, 2079, 4927, 11519, 26623, 60927, 138239, 311295, 696319, 1548287, 3424255, 7536639, 16515071, 36044799, 78381055, 169869311, 367001599, 790626303, 1698693119, 3640655871, 7784628223, 16609443839, 35366371327, 75161927679, 159450660863, 337691803647, 714038312959, 1507533520895, 3178275799039, 6691559047167, 14070312861695, 29549374996479, 61984968015871, 129879811031039, 271854249967615, 568447511560191, 1187472557998079, 2478299209007103, 5167704650547199, 10766417859182591, 22412445020585983, 46619293017702399, 96897760732643327, 201254608348119039, 417708865438613503, 866379978315399167, 1795810351413985279, 3719973292208029695, 7701155362803548159, 15933735481636814847, 32948334873842548735, 68094426365841899519, 140656423562035331071, 290392103972849582079, 599230952019408715775, 1235931852938539958271, 2547956525181131816959, 5250404531979581128703, 10814403713212224634879, 22265220096967428800511, 45821712279094526214143, 94262862216655808757759, 193838386726539968380927, 398449671992126314905599, 818740288967524738924543, 1681752763711952401727487, 3453230490598428062515199, 7088272090787337465757695, 14544888767238507258183679, 29835911438770418460131327, 61182980152059123388645375, 125426053785017776875765759, 257047852395560528271900671, 526638310169622834231377919, 1078664062551152881131585535, 2208707472435927502188183551, 4521382565358713113401098239, 9253118223330371703201071103, 18931778335165092875898716159, 38724311853895801724188229631, 79189476888036669459953352703, 161899345762791139076651089919, 330916846751473214733972144127, 676224746459638975163646607359, 1381541083842484386787422633983, 2821884319551024336632553668607, 5762610882873445179655423262719, 11765382133368254132641276624895, 24016036762136377333043009945599, 49012522035386775843806126276607, 100005748133630160127450851311615, 204012518474230669303375672115199, 416106309524916301041292827164671
Generating function in Maple syntax:
(6*x^6-4*x^5+x^4-10*x^3+13*x^2-6*x+1)/(x-1)/(2*x-1)^3
Generating function in latex syntax:
\frac{6 x^{6}-4 x^{5}+x^{4}-10 x^{3}+13 x^{2}-6 x +1}{\left(x -1\right) \left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(6*x**6 - 4*x**5 + x**4 - 10*x**3 + 13*x**2 - 6*x + 1)/((x - 1)*(2*x - 1)**3)
Implicit equation for the generating function in Maple syntax:
-(x-1)*(2*x-1)^3*F(x)+6*x^6-4*x^5+x^4-10*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(2 x -1\right)^{3} F \! \left(x \right)+6 x^{6}-4 x^{5}+x^{4}-10 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,-1+1/32*(n^2+5*n+4)*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(n^{2}+5 n +4\right) 2^{n}}{32} & \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) = 53
a(6) = 139
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2)+1, 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) = 53
a \! \left(6\right) = 139
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)+1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19759/
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[12,x]*F[4,x]
F[4,x] = F[28,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[19,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[12,x]
F[18,x] = F[14,x]
F[19,x] = F[20,x]+F[21,x]+F[27,x]
F[20,x] = 0
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[10,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[10,x]*F[12,x]
F[26,x] = F[19,x]
F[27,x] = F[12,x]*F[13,x]
F[28,x] = F[29,x]+F[36,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[31,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[1,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[32,x]
F[35,x] = F[29,x]
F[36,x] = F[20,x]+F[37,x]+F[48,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[44,x]
F[39,x] = F[10,x]+F[40,x]
F[40,x] = F[20,x]+F[41,x]+F[42,x]
F[41,x] = F[12,x]*F[39,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[33,x]+F[40,x]
F[44,x] = F[45,x]
F[45,x] = F[20,x]+F[37,x]+F[46,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[29,x]+F[45,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[47,x]+F[50,x]
F[50,x] = F[51,x]+F[61,x]
F[51,x] = F[20,x]+F[52,x]+F[57,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[12,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[33,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[29,x]
F[60,x] = F[51,x]
F[61,x] = F[20,x]+F[62,x]+F[67,x]+F[71,x]
F[62,x] = F[12,x]*F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[24,x]+F[65,x]
F[65,x] = F[66,x]
F[66,x] = F[12,x]*F[40,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[69,x]+F[70,x]
F[69,x] = F[45,x]
F[70,x] = F[61,x]
F[71,x] = F[12,x]*F[50,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{28}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{27}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{26}\! \left(x \right) = F_{19}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{36}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{35}\! \left(x \right) = F_{29}\! \left(x \right)
F_{36}\! \left(x \right) = F_{20}\! \left(x \right)+F_{37}\! \left(x \right)+F_{48}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{20}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{20}\! \left(x \right)+F_{37}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{29}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{61}\! \left(x \right)
F_{51}\! \left(x \right) = F_{20}\! \left(x \right)+F_{52}\! \left(x \right)+F_{57}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{29}\! \left(x \right)
F_{60}\! \left(x \right) = F_{51}\! \left(x \right)
F_{61}\! \left(x \right) = F_{20}\! \left(x \right)+F_{62}\! \left(x \right)+F_{67}\! \left(x \right)+F_{71}\! \left(x \right)
F_{62}\! \left(x \right) = F_{12}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{24}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{70}\! \left(x \right)
F_{69}\! \left(x \right) = F_{45}\! \left(x \right)
F_{70}\! \left(x \right) = F_{61}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_28(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_19(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_12(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_27(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_10(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_10(x)*F_12(x))
Eq(F_26(x), F_19(x))
Eq(F_27(x), F_12(x)*F_13(x))
Eq(F_28(x), F_29(x) + F_36(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_1(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_32(x))
Eq(F_35(x), F_29(x))
Eq(F_36(x), F_20(x) + F_37(x) + F_48(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_44(x))
Eq(F_39(x), F_10(x) + F_40(x))
Eq(F_40(x), F_20(x) + F_41(x) + F_42(x))
Eq(F_41(x), F_12(x)*F_39(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_33(x) + F_40(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_20(x) + F_37(x) + F_46(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_29(x) + F_45(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_47(x) + F_50(x))
Eq(F_50(x), F_51(x) + F_61(x))
Eq(F_51(x), F_20(x) + F_52(x) + F_57(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_12(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_33(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_29(x))
Eq(F_60(x), F_51(x))
Eq(F_61(x), F_20(x) + F_62(x) + F_67(x) + F_71(x))
Eq(F_62(x), F_12(x)*F_63(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_24(x) + F_65(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_12(x)*F_40(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_70(x))
Eq(F_69(x), F_45(x))
Eq(F_70(x), F_61(x))
Eq(F_71(x), F_12(x)*F_50(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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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