0132_0321_1032_1230_1302
Counting sequence:
1, 1, 2, 6, 19, 55, 154, 428, 1187, 3289, 9110, 25230, 69871, 193495, 535846, 1483916, 4109399, 11380129, 31514906, 87273990, 241687195, 669302503, 1853494306, 5132867612, 14214411035, 39363859801, 109010035982, 301880659182, 835995801367, 2315116780903, 6411235200526, 17754584622044, 49167635446511, 136159556918017, 377065619929586, 1044204938317254, 2891708751940195, 8007986937432919, 22176456998675818, 61413092810534732, 170070808370361299, 470975789299224985, 1304269652337848390, 3611903976084681486, 10002418065215421055, 27699619871899430071, 76708345526567550646, 212427834773018642924, 588274778664387808199, 1629104846746712332321, 4511467596348767831690, 12493572720963978970950, 34598355413289881413483, 95812961115254433101191, 265334100653437396861042, 734787696258347638895420, 2034841949236859952926411, 5635072252105669616744857, 15605162503338496536203582, 43215257207146679944792302, 119675681370207212914918183, 331416949406813455336165831, 917790424057379833038371614, 2541630004135533256693867196, 7038516537756033392379395807, 19491710033189946753482691841, 53978243565949340124762546914, 149481537202414107013184344710, 413958078074482927668273179059, 1146370940586984016116397739383, 3174636280888883227694104383802, 8791495980154148594634188591084, 24346222600160546572324859128835, 67421808101216374350034493179225, 186710697683642725072412808999926, 517056507549872347439528792949390, 1431880633067190691596208063027471, 3965296089335342452421508205132951, 10981064142488709013107845345320774, 30409776971197975278498235893856844, 84213562860440559274808370541382711, 233211975752612411559034721075612065, 645832141368594650680794028579310330, 1788498011213612099758155735203701638, 4952873868024854060152707898107406651, 13715955734229544731380761987697828647, 37983491345877392233747733800189780930, 105187392171427486030015147310978920796, 291294640902634610587678470120444810107, 806680021882353737966767996850545290457, 2233932816915854089342640667742169982062, 6186413069767843140648832476496409465838, 17131986414270029070570624758597603705911, 47443478989958098160405682467038571633959, 131384863625372108551295255119014520661806, 363843098300428256564050707648602594057372, 1007587910265870759301262550293831833144271, 2790305496122556129891032198351907426037249, 7727171676402225886935884752410493039024850, 21398797443349992290217884609173403524181574, 59259526149770307113608801273461624959557123
Generating function in Maple syntax:
(2*x^4+2*x^2-3*x+1)/(x-1)/(x^3-x^2+3*x-1)
Generating function in latex syntax:
\frac{2 x^{4}+2 x^{2}-3 x +1}{\left(x -1\right) \left(x^{3}-x^{2}+3 x -1\right)}
Generating function in sympy syntax:
(2*x**4 + 2*x**2 - 3*x + 1)/((x - 1)*(x**3 - x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^3-x^2+3*x-1)*F(x)-2*x^4-2*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{3}-x^{2}+3 x -1\right) F \! \left(x \right)-2 x^{4}-2 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
-27/38*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^4-2*_Z^3+4*_Z^2-4*_Z+1))+51/38*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^4-2*_Z^3+4*_Z^2-4*_Z+1))-48/19*Sum(_alpha^(-n),_alpha = RootOf(_Z^4-2*_Z^3+4*_Z^2-4*_Z+1))+17/19*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^4-2*_Z^3+4*_Z^2-4*_Z+1))+piecewise(n = 0,2,0)
Explicit closed form in latex syntax:
-\frac{27 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{38}+\frac{51 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{38}-\frac{48 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{19}+\frac{17 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-2 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{19}+\left(\left\{\begin{array}{cc}2 & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+3) = a(n)-a(n+1)+3*a(n+2)+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) = a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+2, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19360/
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[12,x]*F[4,x]
F[4,x] = F[24,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[12,x]
F[17,x] = F[18,x]+F[19,x]+F[23,x]
F[18,x] = 0
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[10,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[10,x]*F[12,x]
F[23,x] = F[12,x]*F[13,x]
F[24,x] = F[25,x]+F[41,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[1,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[12,x]*F[16,x]
F[31,x] = F[25,x]+F[32,x]
F[32,x] = F[18,x]+F[33,x]+F[39,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[12,x]
F[36,x] = F[37,x]
F[37,x] = F[18,x]+F[33,x]+F[38,x]
F[38,x] = F[12,x]*F[25,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[25,x]+F[37,x]
F[41,x] = F[18,x]+F[42,x]+F[60,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[10,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[12,x]*F[20,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[18,x]+F[42,x]+F[49,x]
F[49,x] = F[12,x]*F[50,x]
F[50,x] = F[25,x]+F[48,x]
F[51,x] = 2*F[18,x]+F[52,x]+F[58,x]
F[52,x] = F[12,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[21,x]
F[55,x] = F[56,x]
F[56,x] = 2*F[18,x]+F[52,x]+F[57,x]
F[57,x] = F[12,x]*F[48,x]
F[58,x] = F[12,x]*F[59,x]
F[59,x] = F[48,x]+F[56,x]
F[60,x] = F[12,x]*F[61,x]
F[61,x] = F[50,x]+F[62,x]
F[62,x] = F[63,x]+F[65,x]
F[63,x] = F[18,x]+F[33,x]+F[64,x]
F[64,x] = F[12,x]*F[40,x]
F[65,x] = F[18,x]+F[52,x]+F[66,x]+F[67,x]
F[66,x] = F[12,x]*F[59,x]
F[67,x] = F[12,x]*F[62,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{24}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{23}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{41}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{39}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{25}\! \left(x \right)+F_{37}\! \left(x \right)
F_{41}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{60}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{10}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{25}\! \left(x \right)+F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{58}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{21}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{48}\! \left(x \right)+F_{56}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{50}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{65}\! \left(x \right)
F_{63}\! \left(x \right) = F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{65}\! \left(x \right) = F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)
F_{66}\! \left(x \right) = F_{12}\! \left(x \right) F_{59}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_24(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_12(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_23(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_10(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_10(x)*F_12(x))
Eq(F_23(x), F_12(x)*F_13(x))
Eq(F_24(x), F_25(x) + F_41(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_1(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_12(x)*F_16(x))
Eq(F_31(x), F_25(x) + F_32(x))
Eq(F_32(x), F_18(x) + F_33(x) + F_39(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_12(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_18(x) + F_33(x) + F_38(x))
Eq(F_38(x), F_12(x)*F_25(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_25(x) + F_37(x))
Eq(F_41(x), F_18(x) + F_42(x) + F_60(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_10(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_12(x)*F_20(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_18(x) + F_42(x) + F_49(x))
Eq(F_49(x), F_12(x)*F_50(x))
Eq(F_50(x), F_25(x) + F_48(x))
Eq(F_51(x), 2*F_18(x) + F_52(x) + F_58(x))
Eq(F_52(x), F_12(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_21(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), 2*F_18(x) + F_52(x) + F_57(x))
Eq(F_57(x), F_12(x)*F_48(x))
Eq(F_58(x), F_12(x)*F_59(x))
Eq(F_59(x), F_48(x) + F_56(x))
Eq(F_60(x), F_12(x)*F_61(x))
Eq(F_61(x), F_50(x) + F_62(x))
Eq(F_62(x), F_63(x) + F_65(x))
Eq(F_63(x), F_18(x) + F_33(x) + F_64(x))
Eq(F_64(x), F_12(x)*F_40(x))
Eq(F_65(x), F_18(x) + F_52(x) + F_66(x) + F_67(x))
Eq(F_66(x), F_12(x)*F_59(x))
Eq(F_67(x), F_12(x)*F_62(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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