0213_0231_0321_1230_1302
Counting sequence:
1, 1, 2, 6, 19, 57, 165, 468, 1309, 3622, 9936, 27067, 73313, 197633, 530654, 1420059, 3789338, 10087008, 26794945, 71049313, 188099987, 497311914, 1313277053, 3464467600, 9131149482, 24047596639, 63287388699, 166455606399, 437569660586, 1149717740005, 3019637230132, 7927916463296, 20807636721711, 54596176032595, 143216111757343, 375598980283680, 984852033794655, 2581912313495936, 6767754895522436, 17737323159468877, 46481342156813795, 121793014448795581, 319098350241580472, 835968488730251861, 2189891486345901214, 5736226825641025562, 15024649268238750635, 39351344471789530373, 103061054925283176029, 269905445930752971452, 706826441683462292481, 1850971775093860956522, 4847013879514521594108, 12692220218066939329963, 33234645742679427924749, 87023338200632669039889, 227862384849833525427958, 596626620907383228280863, 1562163480757293554654790, 4090203236893013925563268, 10709235275295858222230857, 28039336896944835157197469, 73413039664169766987054123, 192209695270931386185102130, 503239091485413187652346977, 1317561153093571843999641044, 3449568912021882674242630694, 9031435113172078630045104879, 23645409503549463577278838631, 61906358109468218613212221935, 162077302338920693525164804402, 424334005101561663089247655801, 1110944371232428603109281413056, 2908544808521246967347582381700, 7614796293768819948653982091463, 19936091049513354492924644624335, 52194051006006175894718747055975, 136647396708191194706127937263016, 357751242011883749653466777074131, 936613542679271967878483644818132, 2452106155034421233509910825993840, 6419743905639151485416408049627661, 16807216186863346235603148885913567, 44002115332469995833982153532871341, 115199619576358500831148144577390784, 301597881963983181141562384506879817, 789596673163619455522818771061362730, 2067198290702016934815337839281083278, 5412012503339177445712067058664862499, 14168872473003498606839872213272794657, 37094682221116917544175275497669316901, 97115353903704831221539986588046936852, 254251797272867092573790338234934535653, 665641009142235440637527651322038029918, 1742673487983474756041045209718409330344, 4562384703625286935770318167352779612219, 11944492824911748316857231296129520266953, 31271122137363484121696173947352603944593, 81868939530717655946025751406819167964510, 214335849754918649461562188406933474390723, 561138966113601411985510014720002048493602
Generating function in Maple syntax:
(x^5-2*x^4+4*x^3-8*x^2+5*x-1)/(x^2-3*x+1)/(x^3-2*x^2+3*x-1)
Generating function in latex syntax:
\frac{x^{5}-2 x^{4}+4 x^{3}-8 x^{2}+5 x -1}{\left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}
Generating function in sympy syntax:
(x**5 - 2*x**4 + 4*x**3 - 8*x**2 + 5*x - 1)/((x**2 - 3*x + 1)*(x**3 - 2*x**2 + 3*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x^3-2*x^2+3*x-1)*F(x)-x^5+2*x^4-4*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)-x^{5}+2 x^{4}-4 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
384/115*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^5-5*_Z^4+10*_Z^3-12*_Z^2+6*_Z-1))-1814/115*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^5-5*_Z^4+10*_Z^3-12*_Z^2+6*_Z-1))+3259/115*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^5-5*_Z^4+10*_Z^3-12*_Z^2+6*_Z-1))-3467/115*Sum(_alpha^(-n),_alpha = RootOf(_Z^5-5*_Z^4+10*_Z^3-12*_Z^2+6*_Z-1))+981/115*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^5-5*_Z^4+10*_Z^3-12*_Z^2+6*_Z-1))+piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
\frac{384 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+10 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{115}-\frac{1814 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+10 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{115}+\frac{3259 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+10 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{115}-\frac{3467 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+10 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{115}+\frac{981 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+10 \textit{\_Z}^{3}-12 \textit{\_Z}^{2}+6 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{115}+\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(5) = 57
a(n+5) = a(n)-5*a(n+1)+10*a(n+2)-12*a(n+3)+6*a(n+4), n >= 6
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(5\right) = 57
a \! \left(n +5\right) = a \! \left(n \right)-5 a \! \left(n +1\right)+10 a \! \left(n +2\right)-12 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20253/
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[10,x]*F[4,x]
F[4,x] = F[14,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = x
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]
F[13,x] = F[9,x]
F[14,x] = F[15,x]+F[37,x]
F[15,x] = F[16,x]
F[16,x] = F[10,x]*F[17,x]
F[17,x] = F[18,x]+F[22,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[10,x]*F[21,x]
F[21,x] = F[1,x]+F[10,x]
F[22,x] = F[15,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[29,x]
F[24,x] = 0
F[25,x] = F[10,x]*F[26,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[19,x]
F[28,x] = F[23,x]
F[29,x] = F[10,x]*F[30,x]
F[30,x] = F[15,x]+F[31,x]
F[31,x] = F[24,x]+F[32,x]+F[36,x]
F[32,x] = F[10,x]*F[33,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[10,x]
F[35,x] = F[31,x]
F[36,x] = F[10,x]*F[15,x]
F[37,x] = F[24,x]+F[38,x]+F[52,x]
F[38,x] = F[10,x]*F[39,x]
F[39,x] = F[40,x]+F[44,x]
F[40,x] = F[41,x]+F[6,x]
F[41,x] = F[42,x]
F[42,x] = F[10,x]*F[43,x]
F[43,x] = F[6,x]
F[44,x] = F[37,x]+F[45,x]
F[45,x] = 2*F[24,x]+F[46,x]+F[50,x]
F[46,x] = F[10,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[41,x]
F[49,x] = F[45,x]
F[50,x] = F[10,x]*F[51,x]
F[51,x] = F[37,x]
F[52,x] = F[10,x]*F[53,x]
F[53,x] = F[14,x]+F[54,x]
F[54,x] = F[31,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[10,x]*F[57,x]
F[57,x] = F[54,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \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) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{9}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{37}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \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_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{19}\! \left(x \right)
F_{28}\! \left(x \right) = F_{23}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{24}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)
F_{32}\! \left(x \right) = F_{10}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{37}\! \left(x \right) = F_{24}\! \left(x \right)+F_{38}\! \left(x \right)+F_{52}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{44}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{6}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{6}\! \left(x \right)
F_{44}\! \left(x \right) = F_{37}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{41}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{10}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{37}\! \left(x \right)
F_{52}\! \left(x \right) = F_{10}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{14}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{31}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{54}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_14(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_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_13(x))
Eq(F_13(x), F_9(x))
Eq(F_14(x), F_15(x) + F_37(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_10(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_22(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_10(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_10(x))
Eq(F_22(x), F_15(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_29(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_10(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_19(x))
Eq(F_28(x), F_23(x))
Eq(F_29(x), F_10(x)*F_30(x))
Eq(F_30(x), F_15(x) + F_31(x))
Eq(F_31(x), F_24(x) + F_32(x) + F_36(x))
Eq(F_32(x), F_10(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_10(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_10(x)*F_15(x))
Eq(F_37(x), F_24(x) + F_38(x) + F_52(x))
Eq(F_38(x), F_10(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_44(x))
Eq(F_40(x), F_41(x) + F_6(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_10(x)*F_43(x))
Eq(F_43(x), F_6(x))
Eq(F_44(x), F_37(x) + F_45(x))
Eq(F_45(x), 2*F_24(x) + F_46(x) + F_50(x))
Eq(F_46(x), F_10(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_41(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_10(x)*F_51(x))
Eq(F_51(x), F_37(x))
Eq(F_52(x), F_10(x)*F_53(x))
Eq(F_53(x), F_14(x) + F_54(x))
Eq(F_54(x), F_31(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_10(x)*F_57(x))
Eq(F_57(x), F_54(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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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