0132_1032_3120
Counting sequence:
1, 1, 2, 6, 21, 73, 240, 744, 2192, 6192, 16896, 44800, 115968, 294144, 733184, 1800192, 4362240, 10448896, 24772608, 58195968, 135593984, 313589760, 720371712, 1644691456, 3733979136, 8433696768, 18958254080, 42429579264, 94573166592, 210000412672, 464661774336, 1024752353280, 2252978782208, 4938943954944, 10797547782144, 23545010716672, 51217485004800, 111158048587776, 240724327006208, 520240798629888, 1122120335622144, 2415833204654080, 5191893906358272, 11139152300998656, 23860501834366976, 51031633179967488, 108983592545157120, 232419165925801984, 494991338731732992, 1052839558200164352, 2236600164942872576, 4745668107341660160, 10057945342801870848, 21293300513184415744, 45031492674077589504, 95136290328388042752, 200792989386313564160, 423387904568727699456, 891928899001471991808, 1877316497470136516608, 3947963519744033685504, 8295630513616453632000, 17417185170095637266432, 36540117706257104633856, 76601257687583519932416, 160467073775694782660608, 335915209582250934927360, 702710268743891659259904, 1469043357798007562043392, 3069113938751574069215232, 6407956169348929202159616, 13370937974387631393341440, 27883508046008999517093888, 58114327381909392058220544, 121053142421880485407883264, 252018532091304356119314432, 524395186090259752754872320, 1090588037920398735296167936, 2266962585368607445544337408, 4709937214286732307493552128, 9780889901455883252283736064, 20302020193834549131104747520, 42121393407012909705112584192, 87352144022074645647569453056, 181074118187698772416665747456, 375193754894859008257585840128, 777097516848283633501129932800, 1608867494962819256799171969024, 3329613820976054755942352289792, 6888111149528108024273475469312, 14244357062037228133126772883456, 29445951259451876563819293573120, 60848776236004117997185377763328, 125697027253929478317501101113344, 259566315677201394519748770791424, 535827490724205427426787130867712, 1105752801890487369145395739361280, 2281132303350541666472217821577216, 4704409833771987837629452683051008, 9698953198671211679526463312232448, 19989978455811520753889480565325824
Generating function in Maple syntax:
(x^3+7*x^2-5*x+1)*(x-1)^2/(2*x-1)^4
Generating function in latex syntax:
\frac{\left(x^{3}+7 x^{2}-5 x +1\right) \left(x -1\right)^{2}}{\left(2 x -1\right)^{4}}
Generating function in sympy syntax:
(x - 1)**2*(x**3 + 7*x**2 - 5*x + 1)/(2*x - 1)**4
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^4*F(x)+(x^3+7*x^2-5*x+1)*(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{4} F \! \left(x \right)+\left(x^{3}+7 x^{2}-5 x +1\right) \left(x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/64*2^n*(n^3+n^2-8*n+36))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{2^{n} \left(n^{3}+n^{2}-8 n +36\right)}{64} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 73
a(n+4) = -16*a(n)+32*a(n+1)-24*a(n+2)+8*a(n+3), 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) = 21
a \! \left(5\right) = 73
a \! \left(n +4\right) = -16 a \! \left(n \right)+32 a \! \left(n +1\right)-24 a \! \left(n +2\right)+8 a \! \left(n +3\right), \quad n \geq 6
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/286/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[13,x]*F[3,x]
F[3,x] = F[1,x]+F[4,x]+F[57,x]
F[4,x] = F[13,x]*F[5,x]
F[5,x] = F[1,x]+F[49,x]+F[51,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[13,x]*F[8,x]
F[8,x] = F[33,x]+F[9,x]
F[9,x] = F[10,x]*F[27,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[14,x]
F[13,x] = x
F[14,x] = F[15,x]+F[18,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[13,x]*F[15,x]
F[18,x] = F[19,x]+F[22,x]
F[19,x] = F[20,x]
F[20,x] = F[13,x]*F[21,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[23,x]+F[24,x]+F[26,x]
F[23,x] = 0
F[24,x] = F[13,x]*F[25,x]
F[25,x] = F[16,x]+F[22,x]
F[26,x] = F[13,x]*F[18,x]
F[27,x] = F[15,x]+F[28,x]
F[28,x] = F[19,x]+F[29,x]
F[29,x] = F[23,x]+F[30,x]+F[32,x]
F[30,x] = F[13,x]*F[31,x]
F[31,x] = F[16,x]+F[29,x]
F[32,x] = F[13,x]*F[28,x]
F[33,x] = F[15,x]^2*F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[13,x]*F[36,x]
F[36,x] = F[10,x]+F[37,x]
F[37,x] = F[19,x]+F[38,x]
F[38,x] = F[23,x]+F[24,x]+F[39,x]
F[39,x] = F[13,x]*F[40,x]
F[40,x] = F[18,x]+F[41,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[43,x]
F[43,x] = F[13,x]*F[44,x]
F[44,x] = F[19,x]+F[42,x]
F[45,x] = 2*F[23,x]+F[46,x]+F[48,x]
F[46,x] = F[13,x]*F[47,x]
F[47,x] = F[22,x]+F[45,x]
F[48,x] = F[13,x]*F[41,x]
F[49,x] = F[13,x]*F[50,x]
F[50,x] = 2*F[49,x]+F[1,x]+F[6,x]
F[51,x] = F[13,x]*F[52,x]
F[52,x] = 2*F[51,x]+F[1,x]+F[49,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[13,x]*F[55,x]
F[55,x] = F[56,x]+F[8,x]
F[56,x] = F[10,x]*F[13,x]*F[27,x]
F[57,x] = F[58,x]
F[58,x] = F[13,x]*F[59,x]
F[59,x] = F[3,x]+F[60,x]
F[60,x] = F[13,x]*F[27,x]*F[61,x]
F[61,x] = F[1,x]+F[62,x]+F[71,x]
F[62,x] = F[63,x]
F[63,x] = F[13,x]*F[64,x]
F[64,x] = F[65,x]+F[66,x]+F[69,x]
F[65,x] = F[1,x]+F[34,x]
F[66,x] = F[67,x]
F[67,x] = F[15,x]^2*F[13,x]*F[68,x]
F[68,x] = F[10,x]+F[34,x]
F[69,x] = F[70,x]
F[70,x] = F[13,x]*F[15,x]*F[68,x]
F[71,x] = F[72,x]
F[72,x] = F[13,x]*F[15,x]*F[61,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_{13}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{57}\! \left(x \right)
F_{4}\! \left(x \right) = F_{13}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{33}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{27}\! \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_{13}\! \left(x \right) F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{13}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right)+F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{27}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{19}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{13}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{28}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15} \left(x \right)^{2} F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{19}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{13}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{18}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{19}\! \left(x \right)+F_{42}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{13}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{22}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{13}\! \left(x \right) F_{41}\! \left(x \right)
F_{49}\! \left(x \right) = F_{13}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{49}\! \left(x \right)+F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{51}\! \left(x \right)+F_{1}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{13}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{8}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{27}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{13}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{3}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{13}\! \left(x \right) F_{27}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{1}\! \left(x \right)+F_{62}\! \left(x \right)+F_{71}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{13}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)+F_{69}\! \left(x \right)
F_{65}\! \left(x \right) = F_{1}\! \left(x \right)+F_{34}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{15} \left(x \right)^{2} F_{13}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{10}\! \left(x \right)+F_{34}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{68}\! \left(x \right)
F_{71}\! \left(x \right) = F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{61}\! \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_13(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_57(x))
Eq(F_4(x), F_13(x)*F_5(x))
Eq(F_5(x), F_1(x) + F_49(x) + F_51(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_13(x)*F_8(x))
Eq(F_8(x), F_33(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_27(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_14(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_18(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_13(x)*F_15(x))
Eq(F_18(x), F_19(x) + F_22(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_13(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_26(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_13(x)*F_25(x))
Eq(F_25(x), F_16(x) + F_22(x))
Eq(F_26(x), F_13(x)*F_18(x))
Eq(F_27(x), F_15(x) + F_28(x))
Eq(F_28(x), F_19(x) + F_29(x))
Eq(F_29(x), F_23(x) + F_30(x) + F_32(x))
Eq(F_30(x), F_13(x)*F_31(x))
Eq(F_31(x), F_16(x) + F_29(x))
Eq(F_32(x), F_13(x)*F_28(x))
Eq(F_33(x), F_15(x)**2*F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_13(x)*F_36(x))
Eq(F_36(x), F_10(x) + F_37(x))
Eq(F_37(x), F_19(x) + F_38(x))
Eq(F_38(x), F_23(x) + F_24(x) + F_39(x))
Eq(F_39(x), F_13(x)*F_40(x))
Eq(F_40(x), F_18(x) + F_41(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_13(x)*F_44(x))
Eq(F_44(x), F_19(x) + F_42(x))
Eq(F_45(x), 2*F_23(x) + F_46(x) + F_48(x))
Eq(F_46(x), F_13(x)*F_47(x))
Eq(F_47(x), F_22(x) + F_45(x))
Eq(F_48(x), F_13(x)*F_41(x))
Eq(F_49(x), F_13(x)*F_50(x))
Eq(F_50(x), F_1(x) + 2*F_49(x) + F_6(x))
Eq(F_51(x), F_13(x)*F_52(x))
Eq(F_52(x), F_1(x) + F_49(x) + 2*F_51(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_13(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_8(x))
Eq(F_56(x), F_10(x)*F_13(x)*F_27(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_13(x)*F_59(x))
Eq(F_59(x), F_3(x) + F_60(x))
Eq(F_60(x), F_13(x)*F_27(x)*F_61(x))
Eq(F_61(x), F_1(x) + F_62(x) + F_71(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_13(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_66(x) + F_69(x))
Eq(F_65(x), F_1(x) + F_34(x))
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_13(x)*F_15(x)**2*F_68(x))
Eq(F_68(x), F_10(x) + F_34(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_13(x)*F_15(x)*F_68(x))
Eq(F_71(x), F_72(x))
Eq(F_72(x), F_13(x)*F_15(x)*F_61(x))
Pack JSON:
{"expansion_strats": [[{"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": "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:
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