0132_0321_1320_3120
Counting sequence:
1, 1, 2, 6, 20, 65, 200, 586, 1656, 4565, 12378, 33200, 88410, 234297, 618862, 1630844, 4290508, 11274007, 29597728, 77651064, 203618576, 533730091, 1398621602, 3664233406, 9598274690, 25138981295, 65835448710, 172400921866, 451434428676, 1182036585165, 3094943765928, 8103331587590, 21216124743160, 55547190130497, 145429740621082, 380750621673324, 996839304274618, 2609801650896037, 6832634367897998, 17888238891759864, 46832357185297676, 122609382419956931, 320996889586211552, 840383485361944756, 2200157964546146160, 5760099204369529175, 15080157240748499970, 39480407702248074746, 103361136234739918146, 270603141739460052315, 708448570458616967878, 1854743132586344292230, 4855781953200322772260, 12712604978814437731897, 33282037486842617817352, 87133516488912670485954, 228118529994293903148728, 597222109522766057951917, 1563547870631598308664218, 4093421646487216943927464, 10716717357060428674862426, 28056731001154821384117521, 73453476799325540084373102, 192303701702664808082733556, 503457632920354902591255180, 1318069206281771936545849455, 3450750004371704980755888480, 9034180843726831153140964976, 23651792600595764773505260816, 61921197105634415757051280515, 162111799011455387677001458978, 424414200519027557632658802870, 1111130803726218905938386310210, 2908978213020812401617322794247, 7615803840058584781783227348422, 19938433316599674909471649743234, 52199496128629905878110302803588, 136660055107068974587816420447397, 357780669268134881611253282030888, 936681952848451397697772072559614, 2452265189579450766385720229403576, 6420113616494363811266703203086361, 16808075661112566487029018554647002, 44004113369261187289049610810355556, 115204264451506698658578330575336250, 301608679994930315243602414313397885, 789621775552627060186062979160254990, 2067256646701636491542254656758067056, 5412148164629653666896037258295247436, 14169187847342067014056529652490175515, 37095415377706032385094896767900173728, 97117058286395000160870850788660025260, 254255759482716908136803035872979147760, 665650220164231604328109017380075792047, 1742694901014929665004665537366844855042, 4562434482890460911000170636919651900914, 11944608547676260108624412457790496973698, 31271391160177933496130198905248611138227, 81869564932936768542280448595548880538758, 214337303638790828455739675556585118530814, 561142345983752629474995635424580651011876
Generating function in Maple syntax:
-(3*x^6-19*x^5+39*x^4-43*x^3+26*x^2-8*x+1)/(2*x-1)/(x^2-3*x+1)/(x-1)^4
Generating function in latex syntax:
-\frac{3 x^{6}-19 x^{5}+39 x^{4}-43 x^{3}+26 x^{2}-8 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-3*x**6 + 19*x**5 - 39*x**4 + 43*x**3 - 26*x**2 + 8*x - 1)/((x - 1)**4*(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)^4*F(x)+3*x^6-19*x^5+39*x^4-43*x^3+26*x^2-8*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)^{4} F \! \left(x \right)+3 x^{6}-19 x^{5}+39 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
-1/6*n^3+1/2*n^2-4/3*n+2-2/5*(3/2+1/2*5^(1/2))^(-n)*5^(1/2)+2/5*(3/2-1/2*5^(1/2))^(-n)*5^(1/2)-2^n
Explicit closed form in latex syntax:
-\frac{n^{3}}{6}+\frac{n^{2}}{2}-\frac{4 n}{3}+2-\frac{2 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}+\frac{2 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}}{5}-2^{n}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(6) = 200
a(n+3) = -1/6*n^3+1/2*n^2+2*a(n)-7*a(n+1)+5*a(n+2)+2/3*n+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) = 20
a \! \left(5\right) = 65
a \! \left(6\right) = 200
a \! \left(n +3\right) = -\frac{n^{3}}{6}+\frac{n^{2}}{2}+2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+\frac{2 n}{3}+1, \quad n \geq 7
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/21267/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[20,x]*F[3,x]
F[3,x] = F[0,x]+F[4,x]
F[4,x] = F[5,x]
F[5,x] = F[20,x]*F[6,x]
F[6,x] = F[50,x]+F[7,x]
F[7,x] = F[47,x]+F[8,x]
F[8,x] = F[46,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[20,x]
F[12,x] = F[13,x]+F[26,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[20,x]
F[16,x] = F[17,x]+F[21,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[20,x]
F[20,x] = x
F[21,x] = F[20,x]+F[22,x]
F[22,x] = F[23,x]+F[24,x]+F[25,x]
F[23,x] = 0
F[24,x] = F[18,x]*F[20,x]
F[25,x] = F[20,x]*F[21,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[28,x]
F[28,x] = F[20,x]*F[29,x]
F[29,x] = F[1,x]+F[27,x]
F[30,x] = F[23,x]+F[31,x]+F[33,x]
F[31,x] = F[20,x]*F[32,x]
F[32,x] = F[14,x]+F[30,x]
F[33,x] = F[20,x]*F[34,x]
F[34,x] = F[35,x]+F[40,x]
F[35,x] = F[27,x]+F[36,x]
F[36,x] = F[23,x]+F[37,x]+F[39,x]
F[37,x] = F[20,x]*F[38,x]
F[38,x] = F[18,x]+F[36,x]
F[39,x] = F[20,x]*F[35,x]
F[40,x] = F[41,x]+F[43,x]
F[41,x] = F[42,x]
F[42,x] = F[20,x]*F[27,x]
F[43,x] = 2*F[23,x]+F[44,x]+F[45,x]
F[44,x] = F[20,x]*F[36,x]
F[45,x] = F[20,x]*F[40,x]
F[46,x] = F[20,x]*F[7,x]
F[47,x] = F[48,x]
F[48,x] = F[20,x]*F[49,x]*F[8,x]
F[49,x] = F[17,x]+F[35,x]
F[50,x] = F[51,x]
F[51,x] = F[20,x]*F[52,x]*F[56,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[1,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[20,x]*F[52,x]
F[56,x] = F[57,x]+F[8,x]
F[57,x] = F[58,x]
F[58,x] = F[17,x]^2*F[20,x]*F[29,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_{20}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{0}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{20}\! \left(x \right) F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{50}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{47}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{46}\! \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_{20}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{26}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{21}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = x
F_{21}\! \left(x \right) = F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{18}\! \left(x \right) F_{20}\! \left(x \right)
F_{25}\! \left(x \right) = F_{20}\! \left(x \right) F_{21}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{20}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{23}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)
F_{31}\! \left(x \right) = F_{20}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{20}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)
F_{35}\! \left(x \right) = F_{27}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{23}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{20}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{18}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{20}\! \left(x \right) F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{20}\! \left(x \right) F_{27}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{20}\! \left(x \right) F_{36}\! \left(x \right)
F_{45}\! \left(x \right) = F_{20}\! \left(x \right) F_{40}\! \left(x \right)
F_{46}\! \left(x \right) = F_{20}\! \left(x \right) F_{7}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{20}\! \left(x \right) F_{49}\! \left(x \right) F_{8}\! \left(x \right)
F_{49}\! \left(x \right) = F_{17}\! \left(x \right)+F_{35}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{20}\! \left(x \right) F_{52}\! \left(x \right) F_{56}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{1}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{20}\! \left(x \right) F_{52}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{8}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{17} \left(x \right)^{2} F_{20}\! \left(x \right) F_{29}\! \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_20(x)*F_3(x))
Eq(F_3(x), F_0(x) + F_4(x))
Eq(F_4(x), F_5(x))
Eq(F_5(x), F_20(x)*F_6(x))
Eq(F_6(x), F_50(x) + F_7(x))
Eq(F_7(x), F_47(x) + F_8(x))
Eq(F_8(x), F_46(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_20(x))
Eq(F_12(x), F_13(x) + F_26(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_20(x))
Eq(F_16(x), F_17(x) + F_21(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_20(x))
Eq(F_20(x), x)
Eq(F_21(x), F_20(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_25(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_18(x)*F_20(x))
Eq(F_25(x), F_20(x)*F_21(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_20(x)*F_29(x))
Eq(F_29(x), F_1(x) + F_27(x))
Eq(F_30(x), F_23(x) + F_31(x) + F_33(x))
Eq(F_31(x), F_20(x)*F_32(x))
Eq(F_32(x), F_14(x) + F_30(x))
Eq(F_33(x), F_20(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_40(x))
Eq(F_35(x), F_27(x) + F_36(x))
Eq(F_36(x), F_23(x) + F_37(x) + F_39(x))
Eq(F_37(x), F_20(x)*F_38(x))
Eq(F_38(x), F_18(x) + F_36(x))
Eq(F_39(x), F_20(x)*F_35(x))
Eq(F_40(x), F_41(x) + F_43(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_20(x)*F_27(x))
Eq(F_43(x), 2*F_23(x) + F_44(x) + F_45(x))
Eq(F_44(x), F_20(x)*F_36(x))
Eq(F_45(x), F_20(x)*F_40(x))
Eq(F_46(x), F_20(x)*F_7(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_20(x)*F_49(x)*F_8(x))
Eq(F_49(x), F_17(x) + F_35(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_20(x)*F_52(x)*F_56(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_1(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_20(x)*F_52(x))
Eq(F_56(x), F_57(x) + F_8(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_17(x)**2*F_20(x)*F_29(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "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": "row_and_col_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:
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{"patt": [2, 0, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"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": [0, 3, 2, 1], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 2], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 2], [0, 0], [0, 2], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}], "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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 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