0132_0321_2301
Counting sequence:
1, 1, 2, 6, 21, 72, 232, 707, 2066, 5858, 16257, 44428, 120076, 321919, 857942, 2276454, 6020541, 15883440, 41826352, 109988795, 288926426, 758363306, 1989309177, 5215855636, 13670840596, 35821831927, 93844986782, 245813791662, 643797714741, 1685982005688, 4414953608632, 11560489432883, 30269735915426, 79255160764274, 207508631279217, 543296502877084, 1422432416959516, 3724103827216495, 9750085223120102, 25526564159004150, 66830431887613101, 174966380771276736, 458072008961100352, 1199256243181790891, 3139709914723805546, 8219899889268692282, 21520042529640404457, 56340333252768787492, 147501168334898490916, 386163593964391751143, 1010990457983206894382, 2646809468835089195838, 6929441326221781220901, 18141521265229695522504, 47495135980266187457992, 124343913697166631074339, 325536659154429234210866, 852266171852512128450050, 2231262072575889264922977, 5841520478220719893886380, 15293300226777398871871276, 40038381931493733631997791, 104821849026468315844662902, 274427172065440241543072646, 718459681004910464066718621, 1880951898619407261221410512, 4924396070193543540726167632, 12892236422641687803214401947, 33752313419092448753431657466, 88364704277357516226109809098, 231341800298423815462956247257, 605660698388801361238875887668, 1585640298409755130405905325876, 4151260203924013754283307910359, 10868140327529385581052954046142, 28453160806998341886093425510094, 74491342150134037871663065048341, 195020865756740567317767254763480, 510571255346761255259381669499352, 1336692900736890380815863694249235, 3499507447770604251899181294277826, 9161829444388311104303623950643346, 23985980889021106519855578081770577, 62796113229928563372950885342905276, 164402358815271693434372628043419196, 430410963244900736600918098980300367, 1126830530977458955709883869283378182, 2950080629803533009211737909641626902, 7723411358665253829291338661185088141, 20220153446656455993394295677000808928, 52937048982232569180355583575991681632, 138590993501898161606600525463322922123, 362835931527175735757302133638674457226, 949916801087056685901018157102095194714, 2486914471748849602417176900966400617417, 6510826614189202682293361672394685638724, 17045565370878179566348606369412814261316, 44625869498564178260523853942234073070535, 116832043125052039702765748470070036801102, 305870259877067309822858977493537301034590, 800778736507100627715982356061664393706501
Generating function in Maple syntax:
-(5*x^4-11*x^3+13*x^2-6*x+1)/(2*x-1)/(x^2-3*x+1)/(x-1)^2
Generating function in latex syntax:
-\frac{5 x^{4}-11 x^{3}+13 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-5*x**4 + 11*x**3 - 13*x**2 + 6*x - 1)/((x - 1)**2*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^2*F(x)+5*x^4-11*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+5 x^{4}-11 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
1/10*(-5^(1/2)+15)*(3/2-1/2*5^(1/2))^(-n)+1/10*(5^(1/2)+15)*(3/2+1/2*5^(1/2))^(-n)+2*n-3*2^n+1
Explicit closed form in latex syntax:
\frac{\left(-\sqrt{5}+15\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(\sqrt{5}+15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+2 n -3 \,2^{n}+1
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)+2*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) = 21
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+2 n +1, \quad n \geq 5
Specification 1
Strategy pack name: point_and_row_placements
Tree: http://permpal.com/tree/279/
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[19,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[19,x]*F[7,x]
F[7,x] = F[28,x]+F[8,x]
F[8,x] = F[4,x]+F[9,x]
F[9,x] = F[10,x]+F[11,x]+F[26,x]+F[27,x]
F[10,x] = 0
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[19,x]
F[13,x] = F[10,x]+F[14,x]+F[24,x]+F[25,x]
F[14,x] = F[15,x]*F[19,x]
F[15,x] = 2*F[10,x]+F[16,x]+F[17,x]+F[20,x]
F[16,x] = F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[13,x]*F[19,x]
F[19,x] = x
F[20,x] = F[21,x]
F[21,x] = F[22,x]^2*F[19,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[19,x]*F[22,x]
F[24,x] = F[18,x]
F[25,x] = F[19,x]*F[22,x]
F[26,x] = F[15,x]*F[19,x]
F[27,x] = F[19,x]*F[8,x]
F[28,x] = F[29,x]
F[29,x] = F[19,x]*F[30,x]*F[39,x]
F[30,x] = F[22,x]+F[31,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[23,x]
F[33,x] = F[10,x]+F[34,x]+F[38,x]
F[34,x] = F[19,x]*F[35,x]
F[35,x] = F[19,x]+F[36,x]
F[36,x] = F[10,x]+F[34,x]+F[37,x]
F[37,x] = F[19,x]*F[32,x]
F[38,x] = F[19,x]*F[31,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[1,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[19,x]*F[40,x]
F[43,x] = F[32,x]+F[44,x]
F[44,x] = F[10,x]+F[45,x]+F[47,x]
F[45,x] = F[19,x]*F[46,x]
F[46,x] = F[41,x]+F[44,x]
F[47,x] = F[19,x]*F[43,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_{19}\! \left(x \right) F_{4}\! \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_{19}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{28}\! \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_{11}\! \left(x \right)+F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{10}\! \left(x \right) = 0
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)+F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 2 F_{10}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{20}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = x
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22} \left(x \right)^{2} F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{19}\! \left(x \right) F_{22}\! \left(x \right)
F_{24}\! \left(x \right) = F_{18}\! \left(x \right)
F_{25}\! \left(x \right) = F_{19}\! \left(x \right) F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right) F_{19}\! \left(x \right)
F_{27}\! \left(x \right) = F_{19}\! \left(x \right) F_{8}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{19}\! \left(x \right) F_{30}\! \left(x \right) F_{39}\! \left(x \right)
F_{30}\! \left(x \right) = F_{22}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{23}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right)+F_{34}\! \left(x \right)+F_{38}\! \left(x \right)
F_{34}\! \left(x \right) = F_{19}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{19}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right)+F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{19}\! \left(x \right) F_{32}\! \left(x \right)
F_{38}\! \left(x \right) = F_{19}\! \left(x \right) F_{31}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{1}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{19}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{32}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{19}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{19}\! \left(x \right) F_{43}\! \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_19(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_19(x)*F_7(x))
Eq(F_7(x), F_28(x) + F_8(x))
Eq(F_8(x), F_4(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_11(x) + F_26(x) + F_27(x))
Eq(F_10(x), 0)
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_19(x))
Eq(F_13(x), F_10(x) + F_14(x) + F_24(x) + F_25(x))
Eq(F_14(x), F_15(x)*F_19(x))
Eq(F_15(x), 2*F_10(x) + F_16(x) + F_17(x) + F_20(x))
Eq(F_16(x), F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_13(x)*F_19(x))
Eq(F_19(x), x)
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_19(x)*F_22(x)**2)
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_19(x)*F_22(x))
Eq(F_24(x), F_18(x))
Eq(F_25(x), F_19(x)*F_22(x))
Eq(F_26(x), F_15(x)*F_19(x))
Eq(F_27(x), F_19(x)*F_8(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_19(x)*F_30(x)*F_39(x))
Eq(F_30(x), F_22(x) + F_31(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_23(x))
Eq(F_33(x), F_10(x) + F_34(x) + F_38(x))
Eq(F_34(x), F_19(x)*F_35(x))
Eq(F_35(x), F_19(x) + F_36(x))
Eq(F_36(x), F_10(x) + F_34(x) + F_37(x))
Eq(F_37(x), F_19(x)*F_32(x))
Eq(F_38(x), F_19(x)*F_31(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_1(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_19(x)*F_40(x))
Eq(F_43(x), F_32(x) + F_44(x))
Eq(F_44(x), F_10(x) + F_45(x) + F_47(x))
Eq(F_45(x), F_19(x)*F_46(x))
Eq(F_46(x), F_41(x) + F_44(x))
Eq(F_47(x), F_19(x)*F_43(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"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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_and_row_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, 1, 3, 2], "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, 3, 0, 1], "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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [2, 1, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[2, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": 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