0123_0132_0321_1023_1032
Counting sequence:
1, 1, 2, 6, 19, 57, 172, 521, 1576, 4767, 14421, 43625, 131969, 399218, 1207670, 3653308, 11051579, 33432002, 101134756, 305941561, 925500219, 2799719830, 8469399537, 25620680952, 77504820687, 234458921719, 709258927205, 2145570840861, 6490541121976, 19634459629008, 59395972951756, 179678059368317, 543541984649146, 1644262466519202, 4974039053393403, 15046907053140965, 45518221597275488, 137696636927532317, 416544477262431908, 1260083800224895879, 3811864687354528013, 11531227043873675825, 34882979340393527221, 105524090630822886010, 319219685761408381950, 965667717849516793408, 2921230058454712535967, 8836978700523506690610, 26732640357266397215328, 80868596008793462213185, 244634638892144951464823, 740041369574199525692610, 2238690445316335738169749, 6772236142466298844332372, 20486611923180012693546895, 61973808836815829815960391, 187476240392706917290181437, 567132170371051831137256713, 1715624859961144998996304368, 5189916590679330241953626864, 15699955652786812892601814040, 47493751237186773097237051625, 143672788411946307704534916290, 434622862847305930523303004814, 1314772511882851608639988811123, 3977302865933340774537425432369, 12031692132586248865629272417716, 36396930395535040036776665138353, 110103926165925451737064140220368, 333074092386613070919106853402959, 1007578520424268414288301255079429, 3048013934515074365662785302764233, 9220511113204450224396557816116569, 27892859749098467493902696856402850, 84378362048575326947875166508410966, 255251990869476013922408185480741572, 772159796196602534446671352363630195, 2335851520026981545075687272070379282, 7066156967104171527759816971818654636, 21375748353721589762933430730349699993, 64663524997929069751060228572911477299, 195612869124640815156342456794048697774, 591746190273408710631365479896788104761, 1790084442143607850591529989830757736960, 5415163397205884637665878538410149254063, 16381347118643851923565255952966226994103, 49555020548403705384868224432141813811973, 149908309967856376380456126375277287111301, 453485865785657603939010348012147032082920, 1371834760270882928717102878024240306818464, 4149921202565053899838463055630748352948644, 12553877832996702845308619037721695702894277, 37976588217721819880384483168789756064224154, 114882530469403090079618696457939388899901258, 347529792075804982355770679422254125389882299, 1051308287576578307357240820304477992475133853, 3180300338930697669545263571996072158493960232, 9620689159711369990951939519325802067007736325, 29103433652089675225421899322979436553818676972, 88040454927970761748740709220903171709609841143, 266330144978185681590726299217738694886374205437
Generating function in Maple syntax:
(x^2+x+1)*(x^2+2*x-1)/(x^4+3*x^3+2*x^2+2*x-1)
Generating function in latex syntax:
\frac{\left(x^{2}+x +1\right) \left(x^{2}+2 x -1\right)}{x^{4}+3 x^{3}+2 x^{2}+2 x -1}
Generating function in sympy syntax:
(x**2 + x + 1)*(x**2 + 2*x - 1)/(x**4 + 3*x**3 + 2*x**2 + 2*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^4+3*x^3+2*x^2+2*x-1)*F(x)-(x^2+x+1)*(x^2+2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}+3 x^{3}+2 x^{2}+2 x -1\right) F \! \left(x \right)-\left(x^{2}+x +1\right) \left(x^{2}+2 x -1\right) = 0
Explicit closed form in Maple syntax:
187/2859*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^4+3*_Z^3+2*_Z^2+2*_Z-1))+605/2859*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^4+3*_Z^3+2*_Z^2+2*_Z-1))+60/953*Sum(_alpha^(-n),_alpha = RootOf(_Z^4+3*_Z^3+2*_Z^2+2*_Z-1))+80/2859*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^4+3*_Z^3+2*_Z^2+2*_Z-1))+piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
\frac{187 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+3 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{2859}+\frac{605 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+3 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{2859}+\frac{60 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+3 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{953}+\frac{80 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}+3 \textit{\_Z}^{3}+2 \textit{\_Z}^{2}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{2859}+\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) = 19
a(n+4) = a(n)+3*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) = 19
a \! \left(n +4\right) = a \! \left(n \right)+3 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/16211/
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[14,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]+F[16,x]+F[18,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[4,x]
F[18,x] = F[12,x]*F[4,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[15,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[4,x]
F[24,x] = F[15,x]+F[18,x]+F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = F[14,x]
F[28,x] = F[29,x]+F[31,x]
F[29,x] = F[15,x]+F[21,x]+F[30,x]
F[30,x] = F[2,x]*F[4,x]
F[31,x] = F[15,x]+F[32,x]+F[33,x]+F[39,x]
F[32,x] = 0
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[29,x]
F[36,x] = F[37,x]
F[37,x] = F[15,x]+F[32,x]+F[38,x]+F[39,x]
F[38,x] = F[35,x]*F[4,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[15,x]+F[21,x]+F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[2,x]+F[29,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[42,x]+F[45,x]
F[45,x] = F[37,x]+F[40,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_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \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_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \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_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{31}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15}\! \left(x \right)+F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{39}\! \left(x \right)
F_{32}\! \left(x \right) = 0
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_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{29}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right)+F_{32}\! \left(x \right)+F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{2}\! \left(x \right)+F_{29}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{42}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \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_14(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_18(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_12(x)*F_4(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_15(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_4(x))
Eq(F_24(x), F_15(x) + F_18(x) + F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), F_14(x))
Eq(F_28(x), F_29(x) + F_31(x))
Eq(F_29(x), F_15(x) + F_21(x) + F_30(x))
Eq(F_30(x), F_2(x)*F_4(x))
Eq(F_31(x), F_15(x) + F_32(x) + F_33(x) + F_39(x))
Eq(F_32(x), 0)
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_29(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_15(x) + F_32(x) + F_38(x) + F_39(x))
Eq(F_38(x), F_35(x)*F_4(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_15(x) + F_21(x) + F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_2(x) + F_29(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_42(x) + F_45(x))
Eq(F_45(x), F_37(x) + F_40(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, 1, 3, 2], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 1, 3, 2], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 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