0132_0213_1203_2301_3012
Counting sequence:
1, 1, 2, 6, 19, 54, 138, 326, 729, 1570, 3296, 6802, 13879, 28110, 56662, 113870, 228405, 457610, 916172, 1833466, 3668243, 7338006, 14677762, 29357526, 58717329, 117437234, 234877368, 469757986, 939519599, 1879043230, 3758090926, 7516186782, 15032378989, 30064763930, 60129534372, 120259075850, 240518159435, 481036327270, 962072663642, 1924145337126, 3848290684873, 7696581381186, 15393162774672, 30786325562546, 61572651139239, 123145302293614, 246290604603398, 492581209224046, 985162418466469, 1970324836952490, 3940649673925756, 7881299347873562, 15762598695770499, 31525197391565750, 63050394783157682, 126100789566343030, 252201579132715265, 504403158265461330, 1008806316530955112, 2017612633061944386, 4035225266123924703, 8070450532247887166, 16140901064495813982, 32281802128991669566, 64563604257983382749, 129127208515966811194, 258254417031933670228, 516508834063867390506, 1033017668127734833339, 2066035336255469721350, 4132070672510939499786, 8264141345021879059142, 16528282690043758180409, 33056565380087516425570, 66113130760175032918592, 132226261520350065907410, 264452523040700131887895, 528905046081400263851790, 1057810092162800527782582, 2115620184325601055647246, 4231240368651202111379733, 8462480737302404222847946, 16924961474604808445787692, 33849922949209616891670586, 67699845898419233783439859, 135399691796838467566981974, 270799383593676935134069858, 541598767187353870268249366, 1083197534374707740536612209, 2166395068749415481073341810, 4332790137498830962146805016, 8665580274997661924293735522, 17331160549995323848587600719, 34662321099990647697175335390, 69324642199981295394350809102, 138649284399962590788701760990, 277298568799925181577403669325, 554597137599850363154807490650, 1109194275199700726309615138052, 2218388550399401452619230437706, 4436777100798802905238461041963
Generating function in Maple syntax:
(x^5-4*x^4+8*x^3-10*x^2+5*x-1)/(2*x-1)/(-1+x)^4
Generating function in latex syntax:
\frac{x^{5}-4 x^{4}+8 x^{3}-10 x^{2}+5 x -1}{\left(2 x -1\right) \left(-1+x \right)^{4}}
Generating function in sympy syntax:
(x**5 - 4*x**4 + 8*x**3 - 10*x**2 + 5*x - 1)/((x - 1)**4*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(-1+x)^4*F(x)-x^5+4*x^4-8*x^3+10*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(-1+x \right)^{4} F \! \left(x \right)-x^{5}+4 x^{4}-8 x^{3}+10 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-3-1/6*n^3-n^2-11/6*n+7*2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -3-\frac{n^{3}}{6}-n^{2}-\frac{11 n}{6}+7 \,2^{-1+n} & \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) = 54
a(n+1) = 2*a(n)+1/6*n*(n+4)*(-1+n), 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) = 19
a \! \left(5\right) = 54
a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{n \left(n +4\right) \left(-1+n \right)}{6}, \quad n \geq 6
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/21626/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[12,x]*F[3,x]
F[3,x] = F[1,x]+F[4,x]+F[60,x]
F[4,x] = F[12,x]*F[5,x]
F[5,x] = F[28,x]+F[6,x]
F[6,x] = F[19,x]+F[7,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[12,x]
F[10,x] = F[11,x]+F[13,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = x
F[13,x] = F[14,x]+F[8,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[12,x]*F[18,x]
F[18,x] = F[1,x]+F[16,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[21,x]
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[1,x]+F[20,x]
F[23,x] = F[24,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[20,x]
F[27,x] = F[23,x]
F[28,x] = F[29,x]+F[44,x]
F[29,x] = F[16,x]+F[30,x]
F[30,x] = F[31,x]+F[32,x]+F[34,x]
F[31,x] = 0
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[30,x]+F[8,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[40,x]
F[36,x] = F[16,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[39,x]
F[39,x] = F[12,x]+F[37,x]
F[40,x] = F[30,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[14,x]+F[41,x]
F[44,x] = F[45,x]+F[53,x]
F[45,x] = F[31,x]+F[46,x]+F[48,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[20,x]+F[45,x]
F[48,x] = F[12,x]*F[49,x]
F[49,x] = F[16,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[52,x]
F[52,x] = F[20,x]+F[50,x]
F[53,x] = 2*F[31,x]+F[54,x]+F[56,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[23,x]+F[53,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[59,x]
F[58,x] = F[45,x]
F[59,x] = F[53,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[3,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[63,x]+F[65,x]+F[66,x]
F[65,x] = F[18,x]^2
F[66,x] = F[67,x]
F[67,x] = F[12,x]*F[18,x]*F[68,x]
F[68,x] = F[69,x]+F[76,x]
F[69,x] = F[18,x]*F[70,x]
F[70,x] = F[18,x]+F[71,x]
F[71,x] = F[16,x]+F[72,x]
F[72,x] = F[31,x]+F[73,x]+F[75,x]
F[73,x] = F[12,x]*F[74,x]
F[74,x] = F[16,x]+F[72,x]
F[75,x] = F[12,x]*F[71,x]
F[76,x] = F[16,x]+F[77,x]
F[77,x] = F[31,x]+F[78,x]+F[79,x]
F[78,x] = F[12,x]*F[76,x]
F[79,x] = F[12,x]*F[80,x]
F[80,x] = F[16,x]+F[77,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_{12}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{60}\! \left(x \right)
F_{4}\! \left(x \right) = F_{12}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{28}\! \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_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{13}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{8}\! \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_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{44}\! \left(x \right)
F_{29}\! \left(x \right) = F_{16}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{30}\! \left(x \right)+F_{8}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{40}\! \left(x \right)
F_{36}\! \left(x \right) = F_{16}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{30}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{14}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{53}\! \left(x \right)
F_{45}\! \left(x \right) = F_{31}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{20}\! \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_{16}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{20}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{54}\! \left(x \right)+F_{56}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{23}\! \left(x \right)+F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{45}\! \left(x \right)
F_{59}\! \left(x \right) = F_{53}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{3}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{63}\! \left(x \right)+F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{18} \left(x \right)^{2}
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{76}\! \left(x \right)
F_{69}\! \left(x \right) = F_{18}\! \left(x \right) F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{18}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{16}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{31}\! \left(x \right)+F_{73}\! \left(x \right)+F_{75}\! \left(x \right)
F_{73}\! \left(x \right) = F_{12}\! \left(x \right) F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{16}\! \left(x \right)+F_{72}\! \left(x \right)
F_{75}\! \left(x \right) = F_{12}\! \left(x \right) F_{71}\! \left(x \right)
F_{76}\! \left(x \right) = F_{16}\! \left(x \right)+F_{77}\! \left(x \right)
F_{77}\! \left(x \right) = F_{31}\! \left(x \right)+F_{78}\! \left(x \right)+F_{79}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{76}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{80}\! \left(x \right)
F_{80}\! \left(x \right) = F_{16}\! \left(x \right)+F_{77}\! \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_12(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_4(x) + F_60(x))
Eq(F_4(x), F_12(x)*F_5(x))
Eq(F_5(x), F_28(x) + F_6(x))
Eq(F_6(x), F_19(x) + F_7(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_12(x))
Eq(F_10(x), F_11(x) + F_13(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_8(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))
Eq(F_17(x), F_12(x)*F_18(x))
Eq(F_18(x), F_1(x) + F_16(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_1(x) + F_20(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_20(x))
Eq(F_27(x), F_23(x))
Eq(F_28(x), F_29(x) + F_44(x))
Eq(F_29(x), F_16(x) + F_30(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_34(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_30(x) + F_8(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_40(x))
Eq(F_36(x), F_16(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_39(x))
Eq(F_39(x), F_12(x) + F_37(x))
Eq(F_40(x), F_30(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_14(x) + F_41(x))
Eq(F_44(x), F_45(x) + F_53(x))
Eq(F_45(x), F_31(x) + F_46(x) + F_48(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_20(x) + F_45(x))
Eq(F_48(x), F_12(x)*F_49(x))
Eq(F_49(x), F_16(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_52(x))
Eq(F_52(x), F_20(x) + F_50(x))
Eq(F_53(x), 2*F_31(x) + F_54(x) + F_56(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_23(x) + F_53(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_59(x))
Eq(F_58(x), F_45(x))
Eq(F_59(x), F_53(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_3(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_63(x) + F_65(x) + F_66(x))
Eq(F_65(x), F_18(x)**2)
Eq(F_66(x), F_67(x))
Eq(F_67(x), F_12(x)*F_18(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_76(x))
Eq(F_69(x), F_18(x)*F_70(x))
Eq(F_70(x), F_18(x) + F_71(x))
Eq(F_71(x), F_16(x) + F_72(x))
Eq(F_72(x), F_31(x) + F_73(x) + F_75(x))
Eq(F_73(x), F_12(x)*F_74(x))
Eq(F_74(x), F_16(x) + F_72(x))
Eq(F_75(x), F_12(x)*F_71(x))
Eq(F_76(x), F_16(x) + F_77(x))
Eq(F_77(x), F_31(x) + F_78(x) + F_79(x))
Eq(F_78(x), F_12(x)*F_76(x))
Eq(F_79(x), F_12(x)*F_80(x))
Eq(F_80(x), F_16(x) + F_77(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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