0132_1032_2301_2310
Counting sequence:
1, 1, 2, 6, 20, 60, 168, 448, 1152, 2880, 7040, 16896, 39936, 93184, 215040, 491520, 1114112, 2506752, 5603328, 12451840, 27525120, 60555264, 132644864, 289406976, 629145600, 1363148800, 2944401408, 6341787648, 13623099392, 29192355840, 62411243520, 133143986176, 283467841536, 602369163264, 1277752770560, 2705829396480, 5720896438272, 12077448036352, 25460566130688, 53601191854080, 112699941847040, 236669877878784, 496429499940864, 1040137999876096, 2177033022996480, 4551978138992640, 9508576557006848, 19843985858101248, 41376821576466432, 86201711617638400, 179440297653043200, 373235819118329856, 775745035814567936, 1611162766691794944, 3343922723322593280, 6935543426150563840, 14375490010566623232, 29777800736173719552, 61645271699447349248, 127541941447132446720, 263730794178816245760, 545043641302886907904, 1125827849248586072064, 2324289753287403503616, 4796153459164483420160, 9892066509526747054080, 20392875573485909311488, 42021682999910358581248, 86552123193845216182272, 178195547752034268610560, 366721272185345886126080, 754398045638425822887936, 1551297389622678452699136, 3188777967557727930548224, 6552283494981632734003200, 13458744476178488859033600, 27635288657753163790548992, 56725066192230178306916352, 116396889069771015227179008, 238762849373889262004469760, 489614956943924815756001280, 1003710661735045872299802624, 2056987282074291540762558464, 4214315407176597303025729536, 8631730352048452307402096640, 17674495482765878534204293120, 36180731929426621940606435328, 74044288599756807692403867648, 151492912307548411140780326912, 309871866083621750060687032320, 633670557609204028213989212160, 1295504251112150457681933500416, 2647953744031428408009226715136, 5411035911716397181584071983104, 11054804550818445854849179320320, 22580026316565336214160025845760, 46110790583301844479442579095552, 94142864107574599145528598986752, 192167908178348150833140851539968, 392179404445608471088042554163200, 800204441394069809694793898393600
Generating function in Maple syntax:
(4*x^5+2*x^3-8*x^2+5*x-1)/(2*x-1)^3
Generating function in latex syntax:
\frac{4 x^{5}+2 x^{3}-8 x^{2}+5 x -1}{\left(2 x -1\right)^{3}}
Generating function in sympy syntax:
(4*x**5 + 2*x**3 - 8*x**2 + 5*x - 1)/(2*x - 1)**3
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^3*F(x)+4*x^5+2*x^3-8*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{3} F \! \left(x \right)+4 x^{5}+2 x^{3}-8 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/16*2^n*n*(n+1))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{2^{n} n \left(n +1\right)}{16} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 60
a(n+3) = 8*a(n)-12*a(n+1)+6*a(n+2), 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) = 20
a \! \left(5\right) = 60
a \! \left(n +3\right) = 8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15540/
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[37,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[26,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]+F[23,x]+F[25,x]
F[22,x] = 0
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[10,x]+F[21,x]
F[25,x] = F[12,x]*F[17,x]
F[26,x] = F[22,x]+F[25,x]+F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[10,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[29,x]
F[32,x] = F[21,x]+F[33,x]
F[33,x] = 2*F[22,x]+F[34,x]+F[36,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[30,x]+F[33,x]
F[36,x] = F[12,x]*F[32,x]
F[37,x] = F[38,x]+F[54,x]
F[38,x] = F[39,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[1,x]+F[14,x]
F[42,x] = F[18,x]+F[43,x]
F[43,x] = F[22,x]+F[23,x]+F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[17,x]+F[46,x]
F[46,x] = F[47,x]+F[50,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[18,x]+F[47,x]
F[50,x] = 2*F[22,x]+F[51,x]+F[53,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[21,x]+F[50,x]
F[53,x] = F[12,x]*F[46,x]
F[54,x] = F[22,x]+F[55,x]+F[67,x]
F[55,x] = F[12,x]*F[56,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[10,x]+F[26,x]
F[58,x] = F[21,x]+F[59,x]
F[59,x] = F[22,x]+F[34,x]+F[53,x]+F[60,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[32,x]+F[62,x]
F[62,x] = F[50,x]+F[63,x]
F[63,x] = 3*F[22,x]+F[64,x]+F[66,x]
F[64,x] = F[12,x]*F[65,x]
F[65,x] = F[33,x]+F[63,x]
F[66,x] = F[12,x]*F[62,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[17,x]+F[69,x]
F[69,x] = F[43,x]+F[59,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_{37}\! \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_{26}\! \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_{9}\! \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_{12}\! \left(x \right) F_{20}\! \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_{23}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{21}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{54}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{17}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{50}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{18}\! \left(x \right)+F_{47}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{21}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{46}\! \left(x \right)
F_{54}\! \left(x \right) = F_{22}\! \left(x \right)+F_{55}\! \left(x \right)+F_{67}\! \left(x \right)
F_{55}\! \left(x \right) = F_{12}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{10}\! \left(x \right)+F_{26}\! \left(x \right)
F_{58}\! \left(x \right) = F_{21}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{22}\! \left(x \right)+F_{34}\! \left(x \right)+F_{53}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{32}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{50}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = 3 F_{22}\! \left(x \right)+F_{64}\! \left(x \right)+F_{66}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{33}\! \left(x \right)+F_{63}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{17}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{43}\! \left(x \right)+F_{59}\! \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_37(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_26(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_9(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_25(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_10(x) + F_21(x))
Eq(F_25(x), F_12(x)*F_17(x))
Eq(F_26(x), F_22(x) + F_25(x) + F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_10(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_12(x)*F_29(x))
Eq(F_32(x), F_21(x) + F_33(x))
Eq(F_33(x), 2*F_22(x) + F_34(x) + F_36(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_30(x) + F_33(x))
Eq(F_36(x), F_12(x)*F_32(x))
Eq(F_37(x), F_38(x) + F_54(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_1(x) + F_14(x))
Eq(F_42(x), F_18(x) + F_43(x))
Eq(F_43(x), F_22(x) + F_23(x) + F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_17(x) + F_46(x))
Eq(F_46(x), F_47(x) + F_50(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_18(x) + F_47(x))
Eq(F_50(x), 2*F_22(x) + F_51(x) + F_53(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_21(x) + F_50(x))
Eq(F_53(x), F_12(x)*F_46(x))
Eq(F_54(x), F_22(x) + F_55(x) + F_67(x))
Eq(F_55(x), F_12(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_10(x) + F_26(x))
Eq(F_58(x), F_21(x) + F_59(x))
Eq(F_59(x), F_22(x) + F_34(x) + F_53(x) + F_60(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_32(x) + F_62(x))
Eq(F_62(x), F_50(x) + F_63(x))
Eq(F_63(x), 3*F_22(x) + F_64(x) + F_66(x))
Eq(F_64(x), F_12(x)*F_65(x))
Eq(F_65(x), F_33(x) + F_63(x))
Eq(F_66(x), F_12(x)*F_62(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_17(x) + F_69(x))
Eq(F_69(x), F_43(x) + F_59(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": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 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": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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