0231_0321_1203
Counting sequence:
1, 1, 2, 6, 21, 75, 264, 914, 3127, 10621, 35932, 121324, 409301, 1380417, 4655382, 15700590, 52954137, 178609067, 602449564, 2032105066, 6854506171, 23121097405, 77990499392, 263072412420, 887378656761, 2993247393297, 10096624106970, 34057260581510, 114879675815149, 387504427399195, 1307103942204640, 4409035282060786, 14872261779525647, 50166114764968077, 169216969606951764, 570791318860945740, 1925354948480451309, 6494478027713432177, 21906737193892147150, 73894334925215036494, 249255408786025068049, 840771608191619373211, 2836034333598837850964, 9566320583419065833514, 32268470244057082002867, 108845837102287793373165, 367151469061950259551000, 1238450682386370123090116, 4177458683799070940663089, 14091123129114640195989601, 47531230364978207661938290, 160329154696014762603264454, 540811539027143267239040261, 1824229169669368728430258731, 6153367343934502529367666552, 20756125545488465762193785106, 70013168982141870067828154151, 236163720448656802433045923805, 796611604173743248325222540044, 2687076773260049744467395218284, 9063866942890852799629974742597, 30573627361885179111726346802401, 103128906895149512417755747032902, 347867503960204757787390956827374, 1173403306160657581480457278953801, 3958045242036385428084044008063947, 13351012440271689031719318284909772, 45034738685448000174149077534917290, 151908156594101329695146565011742187, 512406393673036877063907808219281757, 1728414840676189177557228307016112304, 5830172883002603495740829485222009156, 19665947691354575701405808426135187881, 66335854246557796409167447402384346865, 223759649302589019908752751999397428682, 754771024277805277437124785612167884230, 2545942938617109401137600475099677652061, 8587803768561980831373825084492839642747, 28967803028368994988189037450813820132560, 97712248079335888552693742194494948236530, 329596394154135941464553721771750753238399, 1111772425409812327423859839066422922964845, 3750156093405508784730988936194568915881860, 12649774723205995604821669769137480487598156, 42669370704127252993101489590941008004381917, 143929456146456643148642707161850341212602257, 485493177067432888228825834740488937840999294, 1637632985559172220528995223506912729054725710, 5523953625035088439981006912213679525686538689, 18633029452029034455229217953849658865650548027, 62851683798843632125556527152378116188439652100, 212007079499337851094452297440323192573539073514, 715128044965209203130260474132044374727509666659, 2412221902699996987577592194906205393500007419661, 8136744949149180095061301117802551146127465574664, 27446321705892690653786566572394825405776226901124, 92580089444996704690903586181571585243166725902177, 312284941256933226623959360449210849715904495948225, 1053378594906041109717422208562660763249627341342690, 3553185945310420612366565746539287530048310664054406, 11985368245571421772507772988430609959407728908322549
Generating function in Maple syntax:
-(x^2-3*x+1)*(2*x-1)^2/(x-1)/(x^4-13*x^3+16*x^2-7*x+1)
Generating function in latex syntax:
-\frac{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2}}{\left(x -1\right) \left(x^{4}-13 x^{3}+16 x^{2}-7 x +1\right)}
Generating function in sympy syntax:
(2*x - 1)**2*(-x**2 + 3*x - 1)/((x - 1)*(x**4 - 13*x**3 + 16*x**2 - 7*x + 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^4-13*x^3+16*x^2-7*x+1)*F(x)+(x^2-3*x+1)*(2*x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{4}-13 x^{3}+16 x^{2}-7 x +1\right) F \! \left(x \right)+\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
-619/1192*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^5-14*_Z^4+29*_Z^3-23*_Z^2+8*_Z-1))+8187/1192*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^5-14*_Z^4+29*_Z^3-23*_Z^2+8*_Z-1))-5797/596*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^5-14*_Z^4+29*_Z^3-23*_Z^2+8*_Z-1))+5323/1192*Sum(_alpha^(-n),_alpha = RootOf(_Z^5-14*_Z^4+29*_Z^3-23*_Z^2+8*_Z-1))-701/1192*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^5-14*_Z^4+29*_Z^3-23*_Z^2+8*_Z-1))
Explicit closed form in latex syntax:
-\frac{619 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-14 \textit{\_Z}^{4}+29 \textit{\_Z}^{3}-23 \textit{\_Z}^{2}+8 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{1192}+\frac{8187 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-14 \textit{\_Z}^{4}+29 \textit{\_Z}^{3}-23 \textit{\_Z}^{2}+8 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{1192}-\frac{5797 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-14 \textit{\_Z}^{4}+29 \textit{\_Z}^{3}-23 \textit{\_Z}^{2}+8 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{596}+\frac{5323 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-14 \textit{\_Z}^{4}+29 \textit{\_Z}^{3}-23 \textit{\_Z}^{2}+8 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{1192}-\frac{701 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-14 \textit{\_Z}^{4}+29 \textit{\_Z}^{3}-23 \textit{\_Z}^{2}+8 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{1192}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(n+4) = -a(n)+13*a(n+1)-16*a(n+2)+7*a(n+3)-1, 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) = 21
a \! \left(n +4\right) = -a \! \left(n \right)+13 a \! \left(n +1\right)-16 a \! \left(n +2\right)+7 a \! \left(n +3\right)-1, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14753/
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[10,x]*F[4,x]
F[4,x] = F[16,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = x
F[11,x] = F[12,x]+F[13,x]+F[14,x]
F[12,x] = 0
F[13,x] = F[10,x]*F[6,x]
F[14,x] = F[10,x]*F[15,x]
F[15,x] = F[9,x]
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[12,x]+F[18,x]+F[54,x]
F[18,x] = F[10,x]*F[19,x]
F[19,x] = F[20,x]+F[26,x]
F[20,x] = F[21,x]+F[6,x]
F[21,x] = F[22,x]
F[22,x] = F[10,x]*F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[22,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = 2*F[12,x]+F[28,x]+F[32,x]
F[28,x] = F[10,x]*F[29,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[21,x]
F[31,x] = F[27,x]
F[32,x] = F[10,x]*F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]+F[49,x]
F[35,x] = F[12,x]+F[36,x]+F[48,x]
F[36,x] = F[10,x]*F[37,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[10,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = x^2
F[41,x] = F[35,x]+F[42,x]
F[42,x] = 2*F[12,x]+F[43,x]+F[47,x]
F[43,x] = F[10,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[39,x]
F[46,x] = F[42,x]
F[47,x] = F[10,x]*F[35,x]
F[48,x] = F[10,x]*F[2,x]
F[49,x] = 2*F[12,x]+F[32,x]+F[50,x]
F[50,x] = F[10,x]*F[51,x]
F[51,x] = F[52,x]+F[53,x]
F[52,x] = F[25,x]
F[53,x] = F[49,x]
F[54,x] = F[10,x]*F[55,x]
F[55,x] = F[16,x]+F[56,x]
F[56,x] = F[35,x]+F[57,x]
F[57,x] = F[12,x]+F[50,x]+F[58,x]+F[59,x]
F[58,x] = F[10,x]*F[17,x]
F[59,x] = F[10,x]*F[60,x]
F[60,x] = F[56,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{16}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \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) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{12}\! \left(x \right) = 0
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{9}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)+F_{54}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right) F_{19}\! \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_{6}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = 2 F_{12}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)
F_{32}\! \left(x \right) = F_{10}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{49}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right)+F_{36}\! \left(x \right)+F_{48}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = x^{2}
F_{41}\! \left(x \right) = F_{35}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = 2 F_{12}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{39}\! \left(x \right)
F_{46}\! \left(x \right) = F_{42}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \left(x \right) F_{35}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right) F_{2}\! \left(x \right)
F_{49}\! \left(x \right) = 2 F_{12}\! \left(x \right)+F_{32}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{10}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{53}\! \left(x \right)
F_{52}\! \left(x \right) = F_{25}\! \left(x \right)
F_{53}\! \left(x \right) = F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{10}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{16}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{35}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right)+F_{50}\! \left(x \right)+F_{58}\! \left(x \right)+F_{59}\! \left(x \right)
F_{58}\! \left(x \right) = F_{10}\! \left(x \right) F_{17}\! \left(x \right)
F_{59}\! \left(x \right) = F_{10}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{56}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_16(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_13(x) + F_14(x))
Eq(F_12(x), 0)
Eq(F_13(x), F_10(x)*F_6(x))
Eq(F_14(x), F_10(x)*F_15(x))
Eq(F_15(x), F_9(x))
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_12(x) + F_18(x) + F_54(x))
Eq(F_18(x), F_10(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_26(x))
Eq(F_20(x), F_21(x) + F_6(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_10(x)*F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_22(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), 2*F_12(x) + F_28(x) + F_32(x))
Eq(F_28(x), F_10(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_21(x))
Eq(F_31(x), F_27(x))
Eq(F_32(x), F_10(x)*F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x) + F_49(x))
Eq(F_35(x), F_12(x) + F_36(x) + F_48(x))
Eq(F_36(x), F_10(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_10(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), x**2)
Eq(F_41(x), F_35(x) + F_42(x))
Eq(F_42(x), 2*F_12(x) + F_43(x) + F_47(x))
Eq(F_43(x), F_10(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_39(x))
Eq(F_46(x), F_42(x))
Eq(F_47(x), F_10(x)*F_35(x))
Eq(F_48(x), F_10(x)*F_2(x))
Eq(F_49(x), 2*F_12(x) + F_32(x) + F_50(x))
Eq(F_50(x), F_10(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_53(x))
Eq(F_52(x), F_25(x))
Eq(F_53(x), F_49(x))
Eq(F_54(x), F_10(x)*F_55(x))
Eq(F_55(x), F_16(x) + F_56(x))
Eq(F_56(x), F_35(x) + F_57(x))
Eq(F_57(x), F_12(x) + F_50(x) + F_58(x) + F_59(x))
Eq(F_58(x), F_10(x)*F_17(x))
Eq(F_59(x), F_10(x)*F_60(x))
Eq(F_60(x), F_56(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:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": 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