0123_0231_1032_2031
Counting sequence:
1, 1, 2, 6, 20, 64, 198, 606, 1854, 5680, 17414, 53396, 163720, 501970, 1539034, 4718658, 14467368, 44356874, 135997964, 416969088, 1278425110, 3919644846, 12017611038, 36845934016, 112969445478, 346363742836, 1061949466392, 3255931639586, 9982669775690, 30606814540834, 93840337042968, 287713994038826, 882129636084844, 2704604958332720, 8292304987170694, 25424164733707966, 77950359207333630, 238995403160533840, 732758685305614534, 2246634386227721428, 6888169552402760488, 21119092663010265074, 64751030228812062394, 198526327934342071938, 608680707377480195080, 1866211939688328306506, 5721796274504570350668, 17542998150789902522976, 53786742720976287564918, 164909878440717553482446, 505612844942297643852382, 1550206399931001999310816, 4752924903759699144982310, 14572443477065128130426804, 44679037264880466199746488, 136985700034337195971572546, 419997456584578748882913834, 1287710056548229183779360610, 3948112455775115367235007224, 12104892622513120216177402986, 37113538949033124931696653740, 113789920842390012132150374160, 348879316065726671985950447334, 1069662200987718896962071013598, 3279578844411444943984410800958, 10055171984930780223862838279024, 30829105944144920502201501187974, 94521881350183037534071533287188, 289803605403448395331816703283400, 888536373854930559454196081877266, 2724247983610057056350544513753498, 8352530402334613498373727770158786, 25608815548970228166580021907340008, 78516497663734771121455706807949706, 240731180776034210650553364144632972, 738080570608415254450433864070629696, 2262951259382005528564772311037052630, 6938197002147475539531352778016756974, 21272476568388171828990515721992505758, 65221304499218263019754294834946582016, 199968186445254258186437273222836057574, 613101438206947438498022911932521658100, 1879765877830454624478915406280560892504, 5763352579615036895675946058832590436706, 17670409569989671035661692881000444535050, 54177385481427347507257257954440382112930, 166107587148863718271766538323465365556184, 509285013723597434759054586031141701749418, 1561465250657089518100618043487067370018028, 4787444482575859445318504781541580030762224, 14678280329390035688557428585382422374800454, 45003532513495734637425321806926659814531838, 137980600809075224759695477735390845549555070, 423047817277988777628339804756134745793599376, 1297062446853032161148997597375268738523678278, 3976786836677785688761287249361148325487418196, 12192808127892442148493443145625753248703750248, 37383087439455172805163221305354191317697954866, 114616355137174762336899576912542431400224343162, 351413159392123952905123641640536602549489492226, 1077430951683622991169577255748571394272130899400
Generating function in Maple syntax:
(x-1)*(x^3-2*x^2+3*x-1)/(2*x^4-7*x^3+8*x^2-5*x+1)
Generating function in latex syntax:
\frac{\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}{2 x^{4}-7 x^{3}+8 x^{2}-5 x +1}
Generating function in sympy syntax:
(x - 1)*(x**3 - 2*x**2 + 3*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)-(x-1)*(x^3-2*x^2+3*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)-\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) = 0
Explicit closed form in Maple syntax:
-162/751*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))+208/751*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))+54/751*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))+24/751*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4-7*_Z^3+8*_Z^2-5*_Z+1))+piecewise(n = 0,1/2,0)
Explicit closed form in latex syntax:
-\frac{162 \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{208 \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{54 \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{24 \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}+\left(\left\{\begin{array}{cc}\frac{1}{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) = 20
a(n+4) = -2*a(n)+7*a(n+1)-8*a(n+2)+5*a(n+3), 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) = 20
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 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14983/
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[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[4,x]
F[18,x] = F[19,x]
F[19,x] = F[15,x]
F[20,x] = F[2,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]+F[60,x]
F[22,x] = 0
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[30,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[1,x]+F[4,x]
F[29,x] = F[15,x]
F[30,x] = F[31,x]+F[54,x]
F[31,x] = F[22,x]+F[23,x]+F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[2,x]+F[34,x]
F[34,x] = F[22,x]+F[35,x]+F[53,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[19,x]+F[4,x]
F[38,x] = F[34,x]+F[39,x]
F[39,x] = 2*F[22,x]+F[40,x]+F[49,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[45,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[4,x]
F[45,x] = F[46,x]
F[46,x] = 2*F[22,x]+F[40,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[34,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[34,x]
F[52,x] = F[39,x]
F[53,x] = F[2,x]*F[4,x]
F[54,x] = 2*F[22,x]+F[49,x]+F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[58,x]
F[57,x] = F[43,x]
F[58,x] = F[59,x]
F[59,x] = 2*F[22,x]+F[47,x]+F[55,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[62,x]+F[65,x]
F[62,x] = F[2,x]+F[63,x]
F[63,x] = F[22,x]+F[35,x]+F[64,x]
F[64,x] = F[4,x]*F[62,x]
F[65,x] = F[21,x]+F[66,x]
F[66,x] = 2*F[22,x]+F[40,x]+F[49,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_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \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_{60}\! \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_{30}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{54}\! \left(x \right)
F_{31}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{22}\! \left(x \right)+F_{35}\! \left(x \right)+F_{53}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{19}\! \left(x \right)+F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{40}\! \left(x \right)+F_{49}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{4}\! \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_{40}\! \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_{34}\! \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_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{34}\! \left(x \right)
F_{52}\! \left(x \right) = F_{39}\! \left(x \right)
F_{53}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{49}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{58}\! \left(x \right)
F_{57}\! \left(x \right) = F_{43}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{47}\! \left(x \right)+F_{55}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)+F_{65}\! \left(x \right)
F_{62}\! \left(x \right) = F_{2}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{22}\! \left(x \right)+F_{35}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{65}\! \left(x \right) = F_{21}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = 2 F_{22}\! \left(x \right)+F_{40}\! \left(x \right)+F_{49}\! \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_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_15(x))
Eq(F_20(x), F_2(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x) + F_60(x))
Eq(F_22(x), 0)
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_30(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_1(x) + F_4(x))
Eq(F_29(x), F_15(x))
Eq(F_30(x), F_31(x) + F_54(x))
Eq(F_31(x), F_22(x) + F_23(x) + F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_2(x) + F_34(x))
Eq(F_34(x), F_22(x) + F_35(x) + F_53(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_19(x) + F_4(x))
Eq(F_38(x), F_34(x) + F_39(x))
Eq(F_39(x), 2*F_22(x) + F_40(x) + F_49(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_45(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_4(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), 2*F_22(x) + F_40(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_34(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_34(x))
Eq(F_52(x), F_39(x))
Eq(F_53(x), F_2(x)*F_4(x))
Eq(F_54(x), 2*F_22(x) + F_49(x) + F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_58(x))
Eq(F_57(x), F_43(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), 2*F_22(x) + F_47(x) + F_55(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_62(x) + F_65(x))
Eq(F_62(x), F_2(x) + F_63(x))
Eq(F_63(x), F_22(x) + F_35(x) + F_64(x))
Eq(F_64(x), F_4(x)*F_62(x))
Eq(F_65(x), F_21(x) + F_66(x))
Eq(F_66(x), 2*F_22(x) + F_40(x) + F_49(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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