0132_0213_1302_1320_2103
Counting sequence:
1, 1, 2, 6, 19, 55, 150, 396, 1027, 2634, 6701, 16939, 42601, 106708, 266436, 663591, 1649476, 4093593, 10146437, 25123555, 62157502, 153680920, 379765587, 938047758, 2316240865, 5717691447, 14111044769, 34819124500, 85903708624, 211911066559, 522700537128, 1289192822613, 3179471566537, 7840963288563, 19335952837598, 47681167703572, 117575312684755, 289918303119490, 714870216935973, 1762675419742851, 4346225762734297, 10716377121199428, 26422886966691932, 65149301621647783, 160633827162078364, 396061309730847553, 976531704559336173, 2407737144232130883, 5936504769649963054, 14636990043768361776, 36088770904822335395, 88979894000002043686, 219387184188410826745, 540916564630518582031, 1333671989904863300049, 3288270629681766719460, 8107480144393654022056, 19989599161479068187807, 49285839338190694392800, 121517865028943181885277, 299611187010046026514417, 738713193111619412557459, 1821350936886235328809134, 4490672351823298862983596, 11072076319809510107543299, 27299000366005518225647034, 67307644990894951201751261, 165951823233881808497257435, 409166101360871161648929865, 1008828288810364715698116724, 2487338247265956996006451188, 6132710107349034922415733351, 15120634576006282920592472884, 37281003542095496186045468841, 91918973676433011163936491989, 226632784855344084998235604387, 558779289929639117753275488094, 1377710172999417769557491654344, 3396842629972073124545812485939, 8375157590793251763830651128062, 20649547864537156261616083029073, 50912931658551108641648645900775, 125529460612803454631406385775873, 309501828809879836265031090554740, 763098809266775582153639426777728, 1881474479087353879834823523136447, 4638909364907657854668594453264600, 11437561508919150240565986923595813, 28200122682250296277635062889242521, 69529411346828253370135334082330675, 171429716638218674610902450220989246, 422672178136283704019515254274214596, 1042128363936124970321308638507271027, 2569441716060934919395073373857712562, 6335141581138361006781711546790606293, 15619742840389934478614236195542610227, 38511588612716317913731269094084416697, 94953065009661842832194353887584173668, 234113545530897906295954426363717920140, 577223622982756006964021038693555018855, 1423185959444169309272857675407084908812
Generating function in Maple syntax:
(x^6-x^5+2*x^4-6*x^3+9*x^2-5*x+1)/(2*x-1)/(3*x^3-5*x^2+4*x-1)
Generating function in latex syntax:
\frac{x^{6}-x^{5}+2 x^{4}-6 x^{3}+9 x^{2}-5 x +1}{\left(2 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right)}
Generating function in sympy syntax:
(x**6 - x**5 + 2*x**4 - 6*x**3 + 9*x**2 - 5*x + 1)/((2*x - 1)*(3*x**3 - 5*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
-(2*x-1)*(3*x^3-5*x^2+4*x-1)*F(x)+x^6-x^5+2*x^4-6*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)+x^{6}-x^{5}+2 x^{4}-6 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
-3637/186*Sum(_alpha^(2-n),_alpha = RootOf(6*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1))+6133/186*Sum(_alpha^(1-n),_alpha = RootOf(6*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1))-817/31*Sum(_alpha^(-n),_alpha = RootOf(6*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1))+1291/186*Sum(_alpha^(-n-1),_alpha = RootOf(6*_Z^4-13*_Z^3+13*_Z^2-6*_Z+1))+1/6*piecewise(n = 0,85/36,n = 1,7/6,n = 2,1,0)
Explicit closed form in latex syntax:
-\frac{3637 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(6 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{2-n}\right)}{186}+\frac{6133 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(6 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{186}-\frac{817 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(6 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{31}+\frac{1291 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(6 \textit{\_Z}^{4}-13 \textit{\_Z}^{3}+13 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{186}+\frac{\left(\left\{\begin{array}{cc}\frac{85}{36} & n =0 \\ \frac{7}{6} & n =1 \\ 1 & n =2 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)}{6}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 150
a(n+4) = -6*a(n)+13*a(n+1)-13*a(n+2)+6*a(n+3), n >= 7
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) = 55
a \! \left(6\right) = 150
a \! \left(n +4\right) = -6 a \! \left(n \right)+13 a \! \left(n +1\right)-13 a \! \left(n +2\right)+6 a \! \left(n +3\right), \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18508/
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[16,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[13,x]*F[4,x]
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[18,x]+F[19,x]+F[46,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[33,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[18,x]+F[23,x]+F[31,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[11,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[25,x]*F[4,x]
F[28,x] = F[14,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[28,x]*F[4,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[13,x]
F[33,x] = F[34,x]+F[40,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]
F[37,x] = F[2,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[37,x]*F[4,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[38,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[43,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[2,x]+F[49,x]
F[49,x] = F[18,x]+F[50,x]+F[56,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[52,x]+F[54,x]
F[52,x] = F[11,x]+F[53,x]
F[53,x] = F[15,x]+F[18,x]+F[23,x]
F[54,x] = F[38,x]+F[55,x]
F[55,x] = F[41,x]
F[56,x] = F[4,x]*F[48,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_{16}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{46}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{33}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{31}\! \left(x \right)
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_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{14}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{40}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \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_{37}\! \left(x \right) = F_{2}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \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_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{38}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{2}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{56}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{11}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{23}\! \left(x \right)
F_{54}\! \left(x \right) = F_{38}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{41}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \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_16(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_13(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_46(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_33(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_18(x) + F_23(x) + F_31(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_11(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_25(x)*F_4(x))
Eq(F_28(x), F_14(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_28(x)*F_4(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_13(x))
Eq(F_33(x), F_34(x) + F_40(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_2(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_37(x)*F_4(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_38(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_43(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_2(x) + F_49(x))
Eq(F_49(x), F_18(x) + F_50(x) + F_56(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_54(x))
Eq(F_52(x), F_11(x) + F_53(x))
Eq(F_53(x), F_15(x) + F_18(x) + F_23(x))
Eq(F_54(x), F_38(x) + F_55(x))
Eq(F_55(x), F_41(x))
Eq(F_56(x), F_4(x)*F_48(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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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