0123_0231_0321_1032_1203
Counting sequence:
1, 1, 2, 6, 19, 52, 141, 385, 1059, 2910, 7990, 21933, 60216, 165327, 453915, 1246237, 3421578, 9394046, 25791651, 70811788, 194415973, 533775113, 1465496227, 4023565630, 11046825006, 30329402781, 83270321792, 228621267031, 627686823059, 1723333760477, 4731466618186, 12990389251590, 35665519071699, 97920795598820, 268844600058365, 738121239094673, 2026534895937571, 5563914797371102, 15275901705153190, 41940464835255757, 115148855010237448, 316144774795694751, 867985344897564171, 2383080850992458653, 6542822843440012618, 17963524293694188558, 49319416522743856067, 135407997137719627260, 371766881718750146261, 1020697575212733351961, 2802357044900293100707, 7693958717855817119806, 21124003759547969442974, 57996611522988790972477, 159231506793693816072656, 437175070921859522403495, 1200277799814817025227139, 3295400155572461987309405, 9047623964237672802974090, 24840533936318747356678038, 68200461102319301297934451, 187246494237726908448599252, 514091093192429518482166765, 1411453139220042543461986977, 3875188639902018521084241443, 10639451340994555142540666206, 29210945674183481645087058966, 80199562912831158386142138093, 220190402705575963989668105688, 604539621947076361968315694703, 1659782397476214456565458822907, 4556984367871693519596313070365, 12511342788429931677420200429066, 34350282057848178670589899089566, 94309771333649094980109241918947, 258930420251761757915395848232556, 710901549051182586759146277925189, 1951802387498470675018972281953577, 5358734363329480365896918495513251, 14712572420577957506457752265445950, 40393826705054416397922985438135054, 110902511758978338538849828338969565, 304486306886872098966279310501406240, 835976657436712870147567522970560503, 2295200000697273845657276710193788531, 6301543226521934849644966121751443677, 17301083576011190323098759929886822090, 47500664860048030532161457615817891622, 130414557691351297070062308002207097363, 358057237892177117535645811335670154436, 983057320260169756806881473550960944861, 2699014550316453863835285462304892710833, 7410228674043149664844838922138702696483, 20345014070106823889166044122113811647198, 55857871021273477038002646242184274770822, 153359528004142629302733988044251768908045, 421053369912650952137063263732377354621096, 1156015166596044399446264257674944324503871, 3173875714799089029537405359774780819908587, 8713974819771103622998148493297784892404381, 23924489798243889437262761935227202383351754
Generating function in Maple syntax:
(x-1)*(x^2+x-1)/(x+1)/(2*x^4-5*x^3+5*x^2-4*x+1)
Generating function in latex syntax:
\frac{\left(x -1\right) \left(x^{2}+x -1\right)}{\left(x +1\right) \left(2 x^{4}-5 x^{3}+5 x^{2}-4 x +1\right)}
Generating function in sympy syntax:
(x - 1)*(x**2 + x - 1)/((x + 1)*(2*x**4 - 5*x**3 + 5*x**2 - 4*x + 1))
Implicit equation for the generating function in Maple syntax:
(x+1)*(2*x^4-5*x^3+5*x^2-4*x+1)*F(x)-(x-1)*(x^2+x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x +1\right) \left(2 x^{4}-5 x^{3}+5 x^{2}-4 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{2}+x -1\right) = 0
Explicit closed form in Maple syntax:
-5102/72539*Sum(_alpha^(-n+3),_alpha = RootOf(2*_Z^5-3*_Z^4+_Z^2-3*_Z+1))-3531/72539*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^5-3*_Z^4+_Z^2-3*_Z+1))+4942/72539*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^5-3*_Z^4+_Z^2-3*_Z+1))+14794/72539*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^5-3*_Z^4+_Z^2-3*_Z+1))+2889/72539*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^5-3*_Z^4+_Z^2-3*_Z+1))
Explicit closed form in latex syntax:
-\frac{5102 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{72539}-\frac{3531 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{72539}+\frac{4942 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{72539}+\frac{14794 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{72539}+\frac{2889 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+\textit{\_Z}^{2}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{72539}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+1) = 2/3*a(n)+1/3*a(n+3)-a(n+4)+1/3*a(n+5), 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 +1\right) = \frac{2 a \! \left(n \right)}{3}+\frac{a \! \left(n +3\right)}{3}-a \! \left(n +4\right)+\frac{a \! \left(n +5\right)}{3}, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/17071/
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[20,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[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[4,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[16,x]
F[16,x] = F[13,x]*F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[4,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[57,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[35,x]
F[25,x] = F[15,x]+F[26,x]
F[26,x] = F[22,x]+F[27,x]+F[33,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[30,x]+F[4,x]
F[30,x] = F[31,x]
F[31,x] = x^2
F[32,x] = F[18,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[15,x]
F[35,x] = F[36,x]+F[41,x]
F[36,x] = F[22,x]+F[23,x]+F[37,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[2,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[2,x]*F[4,x]
F[41,x] = F[22,x]+F[42,x]+F[49,x]+F[55,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[18,x]
F[45,x] = F[46,x]
F[46,x] = 2*F[22,x]+F[42,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[39,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[54,x]
F[51,x] = F[39,x]+F[52,x]
F[52,x] = F[53,x]
F[53,x] = F[39,x]*F[4,x]
F[54,x] = F[47,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[36,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[62,x]
F[59,x] = F[2,x]+F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[38,x]*F[4,x]
F[62,x] = F[36,x]+F[63,x]
F[63,x] = F[47,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_{20}\! \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_{14}\! \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_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{57}\! \left(x \right)
F_{22}\! \left(x \right) = 0
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{35}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{33}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = x^{2}
F_{32}\! \left(x \right) = F_{18}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)
F_{36}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{22}\! \left(x \right)+F_{42}\! \left(x \right)+F_{49}\! \left(x \right)+F_{55}\! \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_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{18}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{39}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{54}\! \left(x \right)
F_{51}\! \left(x \right) = F_{39}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{54}\! \left(x \right) = F_{47}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{36}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{62}\! \left(x \right)
F_{59}\! \left(x \right) = F_{2}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{62}\! \left(x \right) = F_{36}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{47}\! \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_20(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_14(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_4(x))
Eq(F_13(x), F_1(x) + F_4(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_13(x)*F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_4(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_57(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_35(x))
Eq(F_25(x), F_15(x) + F_26(x))
Eq(F_26(x), F_22(x) + F_27(x) + F_33(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_30(x) + F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), x**2)
Eq(F_32(x), F_18(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_15(x))
Eq(F_35(x), F_36(x) + F_41(x))
Eq(F_36(x), F_22(x) + F_23(x) + F_37(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_2(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_2(x)*F_4(x))
Eq(F_41(x), F_22(x) + F_42(x) + F_49(x) + F_55(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_18(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), 2*F_22(x) + F_42(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_39(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_54(x))
Eq(F_51(x), F_39(x) + F_52(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_39(x)*F_4(x))
Eq(F_54(x), F_47(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_36(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_62(x))
Eq(F_59(x), F_2(x) + F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_38(x)*F_4(x))
Eq(F_62(x), F_36(x) + F_63(x))
Eq(F_63(x), F_47(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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 0, 3, 2], "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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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