0132_0321_1032_1302_3102
Counting sequence:
1, 1, 2, 6, 19, 59, 181, 554, 1697, 5202, 15950, 48905, 149945, 459731, 1409530, 4321607, 13250022, 40624502, 124554523, 381883539, 1170852981, 3589829250, 11006398129, 33745560434, 103463715926, 317219224561, 972592522177, 2981963705011, 9142685487714, 28031427004599, 85944212012006, 263504515027214, 807903497096787, 2477027995341067, 7594555181097525, 23284867392382170, 71391284486324545, 218885313569565906, 671101253341713310, 2057593014771546777, 6308569673138339657, 19342042393772185427, 59302603180471610698, 181821478434605854119, 557463724139385003686, 1709180930695835089510, 5240340003045393175499, 16066855681766768478979, 49260897451062772610101, 151033659961105797268146, 463068429963303398461777, 1419765440921576485476466, 4352993589727070275344934, 13346256111083651687609089, 40919553064126124836697681, 125459140678207749639077299, 384657084476161170764378802, 1179356656181859427190941207, 3615901483719209767639821350, 11086335479117886116744074110, 33990647950156446091409630595, 104215152991509891042192427131, 319523126742684669761613966965, 979656274474203570876909683722, 3003621133469796841404164411617, 9209087052771128546252457502802, 28235013863931323656644112073198, 86568408282827760661276231857449, 265418297605146520194187136398041, 813771144705087524184729696637395, 2495018172936197136685796790858394, 7649712973710655609987903872264199, 23453980902789020585357878638119526, 71909785645403062955317280355218006, 220475035474805345090267106195266427, 675975332582904054686326630856896371, 2072536916828320912226001509932637429, 6354387600511203413360629685464445154, 19482519925060167461618888551468868081, 59733306573843648277418762581443448434, 183142012838658683763215109273852138742, 561512475876942998001336406207066292497, 1721594382842230722919654717811680755425, 5278399583918533120375250579155951511475, 16183546162316103437001939778188071283970, 49618669868373981286442604469960377031031, 152130588365086349332984948742765613952870, 466431606846815554191646985243149649480942, 1430076924067372638425449688006969832431027, 4384608544381195241352081990349090509915627, 13443187400564531280032371446906210208590773, 41216743902551071829940055157300816491684154, 126370326270772730322054978767684094694241921, 387450774848644405448276888727659231978040530, 1187922096593768624914460256704116697360874814, 3642163080269994849580590914758496660856374969, 11166853399998790626081100783717796159852676201, 34237515484293085897265444619994505929982423891, 104972047590177531174913632720131403602363585002, 321843761715994513379851130297208550369214517351, 986771329448606337528642500497517472217335521574
Generating function in Maple syntax:
-(3*x^3-5*x^2+4*x-1)/(2*x^4-7*x^3+8*x^2-5*x+1)
Generating function in latex syntax:
-\frac{3 x^{3}-5 x^{2}+4 x -1}{2 x^{4}-7 x^{3}+8 x^{2}-5 x +1}
Generating function in sympy syntax:
(-3*x**3 + 5*x**2 - 4*x + 1)/(2*x**4 - 7*x**3 + 8*x**2 - 5*x + 1)
Implicit equation for the generating function in Maple syntax:
(2*x^4-7*x^3+8*x^2-5*x+1)*F(x)+3*x^3-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{4}-7 x^{3}+8 x^{2}-5 x +1\right) F \! \left(x \right)+3 x^{3}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
154/751*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))-207/751*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))+199/751*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))+5/751*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))
Explicit closed form in latex syntax:
\frac{154 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{751}-\frac{207 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{751}+\frac{199 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{751}+\frac{5 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-7 \textit{\_Z}^{3}+8 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{751}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = -2*a(n)+7*a(n+1)-8*a(n+2)+5*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) = -2 a \! \left(n \right)+7 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19375/
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[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[11,x]+F[14,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[16,x]
F[19,x] = F[13,x]*F[4,x]
F[20,x] = F[21,x]+F[41,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[31,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[10,x]+F[28,x]
F[28,x] = F[11,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[28,x]*F[4,x]
F[31,x] = F[21,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[21,x]+F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[35,x]*F[4,x]
F[38,x] = F[36,x]+F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[38,x]*F[4,x]
F[41,x] = F[15,x]+F[42,x]+F[67,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[51,x]
F[44,x] = F[11,x]+F[45,x]
F[45,x] = F[15,x]+F[19,x]+F[46,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[17,x]+F[48,x]
F[48,x] = F[14,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[48,x]
F[51,x] = F[52,x]+F[55,x]
F[52,x] = F[15,x]+F[42,x]+F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[21,x]+F[52,x]
F[55,x] = 2*F[15,x]+F[56,x]+F[63,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = F[52,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[58,x]
F[61,x] = F[62,x]+F[65,x]
F[62,x] = 2*F[15,x]+F[60,x]+F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[36,x]+F[62,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[61,x]
F[67,x] = F[4,x]*F[68,x]
F[68,x] = F[54,x]+F[64,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_{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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{41}\! \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_{31}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{21}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{21}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{67}\! \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_{51}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \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_{17}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \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_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{55}\! \left(x \right)
F_{52}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{21}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{56}\! \left(x \right)+F_{63}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{61}\! \left(x \right)
F_{58}\! \left(x \right) = F_{52}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{65}\! \left(x \right)
F_{62}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{60}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{36}\! \left(x \right)+F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{67}\! \left(x \right) = F_{4}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{54}\! \left(x \right)+F_{64}\! \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_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_11(x) + F_14(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_11(x) + F_18(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_13(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_41(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_31(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_10(x) + F_28(x))
Eq(F_28(x), F_11(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_28(x)*F_4(x))
Eq(F_31(x), F_21(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_21(x) + F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_35(x)*F_4(x))
Eq(F_38(x), F_36(x) + F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_38(x)*F_4(x))
Eq(F_41(x), F_15(x) + F_42(x) + F_67(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_51(x))
Eq(F_44(x), F_11(x) + F_45(x))
Eq(F_45(x), F_15(x) + F_19(x) + F_46(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_17(x) + F_48(x))
Eq(F_48(x), F_14(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_48(x))
Eq(F_51(x), F_52(x) + F_55(x))
Eq(F_52(x), F_15(x) + F_42(x) + F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_21(x) + F_52(x))
Eq(F_55(x), 2*F_15(x) + F_56(x) + F_63(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), F_52(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_58(x))
Eq(F_61(x), F_62(x) + F_65(x))
Eq(F_62(x), 2*F_15(x) + F_60(x) + F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_36(x) + F_62(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_61(x))
Eq(F_67(x), F_4(x)*F_68(x))
Eq(F_68(x), F_54(x) + F_64(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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