0123_0132_0231_2013_2103
Counting sequence:
1, 1, 2, 6, 19, 55, 158, 457, 1323, 3828, 11075, 32043, 92710, 268237, 776087, 2245444, 6496719, 18796887, 54384830, 157351041, 455262067, 1317204820, 3811054475, 11026482739, 31902803382, 92304036357, 267062271167, 772688383908, 2235610953271, 6468269018239, 18714572869262, 54146671496089, 156661979655483, 453268413208884, 1311436602963347, 3794365354992123, 10978196288430790, 31763096716226941, 91899824570234279, 265892769571337476, 769304677576232031, 2225820912297827175, 6439943598459924446, 18632619238234410225, 53909555940838960579, 155975935781202320404, 451283489879236821083, 1305693645738186064003, 3777749318897726011734, 10930121290713465613365, 31624001844719042956367, 91497382881239115279204, 264728389380425234107975, 765935788949474672957455, 2216073743230477015322798, 6411742219502806841226889, 18551024493176892854272779, 53673478746488825752237044, 155292896185295198084892131, 449307258795750554210984907, 1299975837694934116052536678, 3761206046659642190116409389, 10882256819875493562718661303, 31485516088356706730144647876, 91096703538513911007948101103, 263569108166936013419136978039, 762581653139007542709752128766, 2206369258327114436726015610465, 6383664338176207794306568658707, 18469787062479524031613194623188, 53438435365916593173985046431787, 154612847711622671078955081989587, 447339681893910468126106588783350, 1294283068701941919647334777224037, 3744735219635160632973807906867935, 10834601953991360915526302120946468, 31347636779742805952899768010419543, 90697778824507378209120617724048031, 262414903601762050505514689403617486, 759242205540264385213222986507643513, 2196707270668045861552395505567409115, 6355709413667397300454992909001818804, 18388905381414674810239029194453334323, 53204421271912718853366885685493002395, 153935777261658276350771613680029310470, 445380721275891459819426013743501597661, 1288615229109841540814240198921461487303, 3728336520576949715324837810380941485188, 10787155775172790524799125259708366163583, 31210361263161104940701018366587079980167, 90300601055464423727953605460814756747294, 261265753453577747292758048535273922588881, 755917381831591028031722576878181170979683, 2187087594152124184258316893020899103844756, 6327876907531937019328744199255063985732347, 18308377892106868929791360858500655195311651, 52971431957092918138561972162288699807892118, 153261671794235759524552044026875572199829397, 443430339209828098294676881635339327182219375, 1282972209749434371636602545203719808962612964, 3712009633626946535664234842029571244105278311
Generating function in Maple syntax:
-(x^4-x^3-2*x+1)/(2*x^3-x^2+3*x-1)
Generating function in latex syntax:
-\frac{x^{4}-x^{3}-2 x +1}{2 x^{3}-x^{2}+3 x -1}
Generating function in sympy syntax:
(-x**4 + x**3 + 2*x - 1)/(2*x**3 - x**2 + 3*x - 1)
Implicit equation for the generating function in Maple syntax:
(2*x^3-x^2+3*x-1)*F(x)+x^4-x^3-2*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{3}-x^{2}+3 x -1\right) F \! \left(x \right)+x^{4}-x^{3}-2 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,n = 1,1,1/487832*(((-14348*I+51*211^(1/2))*3^(1/2)+153*I*211^(1/2)-14348)*(28+3*211^(1/2)*3^(1/2))^(1/3)+121958+((1055*I+144*211^(1/2))*3^(1/2)-432*I*211^(1/2)-1055)*(28+3*211^(1/2)*3^(1/2))^(2/3))*(1/3468*((28*I+3*211^(1/2))*3^(1/2)-9*I*211^(1/2)-28)*(28+3*211^(1/2)*3^(1/2))^(2/3)-1/12*I*3^(1/2)*(28+3*211^(1/2)*3^(1/2))^(1/3)-1/12*(28+3*211^(1/2)*3^(1/2))^(1/3)+1/6)^(-n)+1/487832*(((14348*I+51*211^(1/2))*3^(1/2)-153*I*211^(1/2)-14348)*(28+3*211^(1/2)*3^(1/2))^(1/3)+121958+((-1055*I+144*211^(1/2))*3^(1/2)+432*I*211^(1/2)-1055)*(28+3*211^(1/2)*3^(1/2))^(2/3))*(1/3468*((-28*I+3*211^(1/2))*3^(1/2)+9*I*211^(1/2)-28)*(28+3*211^(1/2)*3^(1/2))^(2/3)+1/12*I*3^(1/2)*(28+3*211^(1/2)*3^(1/2))^(1/3)-1/12*(28+3*211^(1/2)*3^(1/2))^(1/3)+1/6)^(-n)-3/14348*(1/6*(28+3*211^(1/2)*3^(1/2))^(1/3)+1/6+14/867*(28+3*211^(1/2)*3^(1/2))^(2/3)-1/578*(28+3*211^(1/2)*3^(1/2))^(2/3)*211^(1/2)*3^(1/2))^(-n)*((48/17*211^(1/2)*3^(1/2)-1055/51)*(28+3*211^(1/2)*3^(1/2))^(2/3)+(28+3*211^(1/2)*3^(1/2))^(1/3)*211^(1/2)*3^(1/2)-844/3*(28+3*211^(1/2)*3^(1/2))^(1/3)-3587/3))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(\left(-14348 \,\mathrm{I}+51 \sqrt{211}\right) \sqrt{3}+153 \,\mathrm{I} \sqrt{211}-14348\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}+121958+\left(\left(1055 \,\mathrm{I}+144 \sqrt{211}\right) \sqrt{3}-432 \,\mathrm{I} \sqrt{211}-1055\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(28 \,\mathrm{I}+3 \sqrt{211}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{211}-28\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{3468}-\frac{\mathrm{I} \sqrt{3}\, \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{6}\right)^{-n}}{487832}\\+\\\frac{\left(\left(\left(14348 \,\mathrm{I}+51 \sqrt{211}\right) \sqrt{3}-153 \,\mathrm{I} \sqrt{211}-14348\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}+121958+\left(\left(-1055 \,\mathrm{I}+144 \sqrt{211}\right) \sqrt{3}+432 \,\mathrm{I} \sqrt{211}-1055\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(-28 \,\mathrm{I}+3 \sqrt{211}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{211}-28\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{3468}+\frac{\mathrm{I} \sqrt{3}\, \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{6}\right)^{-n}}{487832}\\-\\\frac{3 \left(\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{6}+\frac{14 \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{867}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{211}\, \sqrt{3}}{578}\right)^{-n} \left(\left(\frac{48 \sqrt{211}\, \sqrt{3}}{17}-\frac{1055}{51}\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{211}\, \sqrt{3}-\frac{844 \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{3587}{3}\right)}{14348} & \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(n+3) = 2*a(n)-a(n+1)+3*a(n+2), 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) = 19
a \! \left(n +3\right) = 2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16149/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[15,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[4,x]
F[15,x] = F[16,x]+F[2,x]
F[16,x] = F[17,x]+F[18,x]+F[42,x]
F[17,x] = 0
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]+F[26,x]
F[20,x] = F[21,x]+F[7,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[25,x]
F[24,x] = F[7,x]
F[25,x] = F[21,x]
F[26,x] = F[27,x]+F[37,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[2,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[2,x]*F[4,x]
F[33,x] = F[27,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[31,x]
F[37,x] = F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[41,x]
F[40,x] = F[16,x]
F[41,x] = F[37,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[54,x]
F[44,x] = F[2,x]+F[45,x]
F[45,x] = F[17,x]+F[32,x]+F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[4,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[7,x]
F[51,x] = F[31,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[16,x]*F[4,x]
F[54,x] = F[16,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[31,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{15}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \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_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{2}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{42}\! \left(x \right)
F_{17}\! \left(x \right) = 0
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{26}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{7}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{37}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{2}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{27}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{16}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{54}\! \left(x \right)
F_{44}\! \left(x \right) = F_{2}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{17}\! \left(x \right)+F_{32}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{51}\! \left(x \right) = F_{31}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{54}\! \left(x \right) = F_{16}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{31}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_15(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_4(x))
Eq(F_15(x), F_16(x) + F_2(x))
Eq(F_16(x), F_17(x) + F_18(x) + F_42(x))
Eq(F_17(x), 0)
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x) + F_26(x))
Eq(F_20(x), F_21(x) + F_7(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_25(x))
Eq(F_24(x), F_7(x))
Eq(F_25(x), F_21(x))
Eq(F_26(x), F_27(x) + F_37(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_2(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_2(x)*F_4(x))
Eq(F_33(x), F_27(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_31(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_41(x))
Eq(F_40(x), F_16(x))
Eq(F_41(x), F_37(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_54(x))
Eq(F_44(x), F_2(x) + F_45(x))
Eq(F_45(x), F_17(x) + F_32(x) + F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_4(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_7(x))
Eq(F_51(x), F_31(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_16(x)*F_4(x))
Eq(F_54(x), F_16(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_31(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:
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