0123_0132_1032_2103
Counting sequence:
1, 1, 2, 6, 20, 63, 196, 616, 1936, 6077, 19078, 59902, 188076, 590499, 1853992, 5820992, 18276192, 57381833, 180161994, 565655398, 1775990692, 5576085655, 17507260268, 54967620808, 172582076848, 541856693349, 1701269804430, 5341484165374, 16770680943548, 52654979515611, 165321066993104, 519059269287744, 1629692633454784, 5116753012004305, 16065091569048466, 50439637503787910, 158365548094054324, 497220996507891247, 1561126913926188180, 4901476925755739880, 15389188309677584080, 48317501116095074381, 151702667296346840982, 476301520840861386686, 1495445945667829505100, 4695258105551054370643, 14741722187690253821176, 46284648931676518564096, 145320112497941250768416, 456262185926649717061657, 1432528359138958771380058, 4497715486918011535649382, 14121496773314363234610756, 44337324514800533777225479, 139206089601318716480410172, 437065961786249988060662472, 1372257891154282633894879472, 4308484037830741711164401013, 13527365973919004791802441374, 42471929473475373505695190078, 133349300719574571341337505372, 418677376395268133994768359435, 1314523170045343919070399814048, 4127214084179948512012771938688, 12958231916190381112169429459328, 40685016810107839326630307130529, 127738923299452266682861760376610, 401062450136421582003700168667654, 1259217509860750198016153374988628, 3953570663622476144751045050210591, 12413042917410495813597782806198884, 38973284551918209454773369718776424, 122364590122729565709794945375758864, 384188633009804962017780896564497885, 1206238713224973380715038511174215846, 3787232906616229576669138706136792190, 11890791542089814459968862857007284204, 37333569649341396482189499860678509827, 117216370149002260333138395793441404360, 368024744484895260200538658854679341760, 1155488882491430070376018697309977384160, 3627893443496683891932236795085114598761, 11390512740362815746418080807589225545962, 35762842131139405338806325971152627851366, 112284750164465467949305816829754097342180, 352540915883151061237693319217425696367223, 1106874238840879126968477818401737775462156, 3475257836499340464225984520470801216581576, 10911282064706439831187615405104043061582896, 34258199505424530282414191082408916747279237, 107560617202781939465012984512114337810437038, 337708535184736720212282766370875085011508862, 1060304948990854759977811112310926641817757436, 3329044035678584309087482268659287555142118459, 10452213961686170837374756780960271747397098352, 32816861396247139493472570847595450427383989952, 103035241705553522177978017668681374495583922752, 323500194157381759619662650962372599500950574193, 1015694959195917634081295890995370564944796248754, 3188981857717554294594246867132998178155981306758, 10012460135573131462363670173112272007692109176052
Generating function in Maple syntax:
(3*x^4+4*x^3+2*x^2+x-1)/(3*x^4+4*x^3+2*x^2+2*x-1)
Generating function in latex syntax:
\frac{3 x^{4}+4 x^{3}+2 x^{2}+x -1}{3 x^{4}+4 x^{3}+2 x^{2}+2 x -1}
Generating function in sympy syntax:
(3*x**4 + 4*x**3 + 2*x**2 + x - 1)/(3*x**4 + 4*x**3 + 2*x**2 + 2*x - 1)
Implicit equation for the generating function in Maple syntax:
(3*x^4+4*x^3+2*x^2+2*x-1)*F(x)-3*x^4-4*x^3-2*x^2-x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(3 x^{4}+4 x^{3}+2 x^{2}+2 x -1\right) F \! \left(x \right)-3 x^{4}-4 x^{3}-2 x^{2}-x +1 = 0
Explicit closed form in Maple syntax:
3/35*Sum(_alpha^(-n+2),_alpha = RootOf(3*_Z^4+4*_Z^3+2*_Z^2+2*_Z-1))+1/5*Sum(_alpha^(-n+1),_alpha = RootOf(3*_Z^4+4*_Z^3+2*_Z^2+2*_Z-1))+3/70*Sum(_alpha^(-n),_alpha = RootOf(3*_Z^4+4*_Z^3+2*_Z^2+2*_Z-1))+1/35*Sum(_alpha^(-n-1),_alpha = RootOf(3*_Z^4+4*_Z^3+2*_Z^2+2*_Z-1))+piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
\frac{3 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{35}+\frac{\left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{5}+\frac{3 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{70}+\frac{\left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}+4 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{35}+\left(\left\{\begin{array}{cc}1 & 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) = 3*a(n)+4*a(n+1)+2*a(n+2)+2*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) = 3 a \! \left(n \right)+4 a \! \left(n +1\right)+2 a \! \left(n +2\right)+2 a \! \left(n +3\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14828/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[43,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[28,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[25,x]
F[25,x] = F[10,x]*F[4,x]
F[26,x] = F[13,x]+F[14,x]+F[27,x]
F[27,x] = F[11,x]*F[4,x]
F[28,x] = F[29,x]+F[34,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[2,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[2,x]*F[4,x]
F[34,x] = 2*F[13,x]+F[35,x]+F[40,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[32,x]
F[38,x] = F[39,x]
F[39,x] = F[20,x]*F[4,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[20,x]+F[42,x]
F[42,x] = 2*F[13,x]+F[35,x]+F[39,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[51,x]
F[45,x] = F[2,x]+F[46,x]
F[46,x] = F[13,x]+F[33,x]+F[47,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[49,x]+F[50,x]
F[49,x] = F[12,x]+F[4,x]
F[50,x] = F[32,x]+F[42,x]
F[51,x] = F[20,x]+F[52,x]
F[52,x] = F[13,x]+F[35,x]+F[39,x]+F[53,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[55,x]+F[60,x]
F[55,x] = F[56,x]+F[58,x]
F[56,x] = F[57,x]
F[57,x] = F[16,x]*F[4,x]
F[58,x] = F[59,x]
F[59,x] = F[17,x]*F[4,x]
F[60,x] = F[61,x]+F[64,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[32,x]
F[64,x] = F[65,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[39,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{43}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{34}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{35}\! \left(x \right)+F_{40}\! \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_{32}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{20}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = 2 F_{13}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{51}\! \left(x \right)
F_{45}\! \left(x \right) = F_{2}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{13}\! \left(x \right)+F_{33}\! \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_{49}\! \left(x \right)+F_{50}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{50}\! \left(x \right) = F_{32}\! \left(x \right)+F_{42}\! \left(x \right)
F_{51}\! \left(x \right) = F_{20}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{13}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \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_{55}\! \left(x \right)+F_{60}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{58}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)+F_{64}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{32}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{39}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_43(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_28(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_10(x)*F_4(x))
Eq(F_26(x), F_13(x) + F_14(x) + F_27(x))
Eq(F_27(x), F_11(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_34(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_2(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_2(x)*F_4(x))
Eq(F_34(x), 2*F_13(x) + F_35(x) + F_40(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_32(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_20(x)*F_4(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_20(x) + F_42(x))
Eq(F_42(x), 2*F_13(x) + F_35(x) + F_39(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_51(x))
Eq(F_45(x), F_2(x) + F_46(x))
Eq(F_46(x), F_13(x) + F_33(x) + F_47(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_49(x) + F_50(x))
Eq(F_49(x), F_12(x) + F_4(x))
Eq(F_50(x), F_32(x) + F_42(x))
Eq(F_51(x), F_20(x) + F_52(x))
Eq(F_52(x), F_13(x) + F_35(x) + F_39(x) + F_53(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_60(x))
Eq(F_55(x), F_56(x) + F_58(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_16(x)*F_4(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_17(x)*F_4(x))
Eq(F_60(x), F_61(x) + F_64(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_32(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_39(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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