0132_0213_0231_1302_2103
Counting sequence:
1, 1, 2, 6, 19, 55, 151, 411, 1128, 3116, 8619, 23819, 65773, 181589, 501386, 1384510, 3823239, 10557539, 29153387, 80503319, 222299736, 613853436, 1695081727, 4680761991, 12925352809, 35691781625, 98558492146, 272157233686, 751528953323, 2075255395071, 5730564240559, 15824253033859, 43696741466248, 120663213053004, 333196721256771, 920082038660915, 2540694142038069, 7015816472927789, 19373320057421242, 53497056471859214, 147725585634079871, 407926156879636171, 1126438245293003427, 3110521595782011551, 8589325369815653624, 23718372638404206620, 65495388332959607991, 180857513214862024911, 499415927118082539793, 1379076068368262673201, 3808150079075045701474, 10515741196146061623974, 29037934589800151221123, 80184708763138742243719, 221420276967213505066631, 611425043608506975556395, 1688375559240315497088424, 4662242835550719873081964, 12874213997402543414177947, 35550569092426343255212059, 98168553284135850939374781, 271080466499572581011232581, 748555589944718985783082602, 2067044809509982785088878686, 5707891707598762386424406839, 15761645609123584144396796915, 43523858726485174771603750811, 120185818500227810695217524711, 331878454512578467886862939288, 916441806064239101302175933756, 2530642084421460483326862888239, 6988058943915188109039241166039, 19296671032322984892618216897273, 53285399553466080059886289245609, 147141120912331608695937900048466, 406312228955203162163832262720950, 1121981580505341910307802924521179, 3098215060448144345255688273919439, 8555342375999012086835005424004127, 23624532752731897424456624287224979, 65236260953230448554982026049078216, 180141964613706499790240692730123916, 497440027075600358170714143248008627, 1373619861799522625376037837824952771, 3793083431228145158366525432029341829, 10474136488831075439059009804714761885, 28923048273457266824078277553260158106, 79867464236385793359860200755409143086, 220544244964809286233651979906224171183, 609005988259968320870679035865606363611, 1681695633434827292577975324506201328979, 4643797036535090504252974239077079325679, 12823278176970908123838388457145960283448, 35409916047207457465271262967088856184924, 97780157083550478551482981863440141954151, 270007957842583349879182140685977207351455, 745593988318381744744237093752209504658209, 2058866708442019601252240861999801405877089, 5685308880630053559355989523845507071577282, 15699285891426226560018884301077836482907974, 43351659984641098593048790126835225701912051
Generating function in Maple syntax:
(x-1)^3/(x^4+4*x^3-5*x^2+4*x-1)
Generating function in latex syntax:
\frac{\left(x -1\right)^{3}}{x^{4}+4 x^{3}-5 x^{2}+4 x -1}
Generating function in sympy syntax:
(x - 1)**3/(x**4 + 4*x**3 - 5*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^4+4*x^3-5*x^2+4*x-1)*F(x)-(x-1)^3 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}+4 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x -1\right)^{3} = 0
Explicit closed form in Maple syntax:
1/170*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^4+4*_Z^3-5*_Z^2+4*_Z-1))+1/170*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^4+4*_Z^3-5*_Z^2+4*_Z-1))+13/85*Sum(_alpha^(-n),_alpha = RootOf(_Z^4+4*_Z^3-5*_Z^2+4*_Z-1))+11/170*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^4+4*_Z^3-5*_Z^2+4*_Z-1))
Explicit closed form in latex syntax:
\frac{\left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{170}+\frac{\left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{170}+\frac{13 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{85}+\frac{11 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+4 \textit{\_Z}^{3}-5 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{170}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = a(n)+4*a(n+1)-5*a(n+2)+4*a(n+3), n >= 4
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(n +4\right) = a \! \left(n \right)+4 a \! \left(n +1\right)-5 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18173/
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[14,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[7,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]+F[17,x]+F[46,x]
F[16,x] = 0
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[29,x]
F[19,x] = F[20,x]+F[7,x]
F[20,x] = F[16,x]+F[21,x]+F[26,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]
F[23,x] = F[11,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[23,x]*F[4,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[13,x]+F[28,x]
F[28,x] = F[26,x]
F[29,x] = F[30,x]+F[37,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[36,x]
F[33,x] = F[2,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[33,x]*F[4,x]
F[36,x] = F[30,x]
F[37,x] = 2*F[16,x]+F[38,x]+F[43,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]
F[40,x] = F[34,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[36,x]+F[45,x]
F[45,x] = F[43,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[59,x]
F[48,x] = F[2,x]+F[49,x]
F[49,x] = F[16,x]+F[50,x]+F[58,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[11,x]+F[53,x]
F[53,x] = F[16,x]+F[21,x]+F[54,x]
F[54,x] = F[13,x]*F[4,x]
F[55,x] = F[34,x]+F[56,x]
F[56,x] = 2*F[16,x]+F[38,x]+F[57,x]
F[57,x] = F[36,x]*F[4,x]
F[58,x] = F[4,x]*F[48,x]
F[59,x] = F[30,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_{14}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{46}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{29}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{26}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{30}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{43}\! \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_{40}\! \left(x \right) = F_{34}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{36}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{43}\! \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_{59}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{16}\! \left(x \right)+F_{50}\! \left(x \right)+F_{58}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{11}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{55}\! \left(x \right) = F_{34}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{59}\! \left(x \right) = F_{30}\! \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_14(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_7(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_46(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_29(x))
Eq(F_19(x), F_20(x) + F_7(x))
Eq(F_20(x), F_16(x) + F_21(x) + F_26(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_11(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_23(x)*F_4(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_13(x) + F_28(x))
Eq(F_28(x), F_26(x))
Eq(F_29(x), F_30(x) + F_37(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_36(x))
Eq(F_33(x), F_2(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_33(x)*F_4(x))
Eq(F_36(x), F_30(x))
Eq(F_37(x), 2*F_16(x) + F_38(x) + F_43(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_34(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_4(x)*F_40(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_36(x) + F_45(x))
Eq(F_45(x), F_43(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_59(x))
Eq(F_48(x), F_2(x) + F_49(x))
Eq(F_49(x), F_16(x) + F_50(x) + F_58(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_11(x) + F_53(x))
Eq(F_53(x), F_16(x) + F_21(x) + F_54(x))
Eq(F_54(x), F_13(x)*F_4(x))
Eq(F_55(x), F_34(x) + F_56(x))
Eq(F_56(x), 2*F_16(x) + F_38(x) + F_57(x))
Eq(F_57(x), F_36(x)*F_4(x))
Eq(F_58(x), F_4(x)*F_48(x))
Eq(F_59(x), F_30(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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