0213_0321_1230_2031_2130
Counting sequence:
1, 1, 2, 6, 19, 54, 147, 387, 994, 2511, 6268, 15512, 38149, 93388, 227829, 554389, 1346436, 3265341, 7910374, 19147258, 46317399, 111988994, 270676007, 654041079, 1580045798, 3816501003, 9217403840, 22259320684, 53750781577, 129787988248, 313376610857, 756632903529, 1826811068680, 4410565238009, 10648512086154, 25708638799662, 62067719813403, 149847628485198, 361769506359803, 873398650967467, 2108588897692138, 5090617075087815, 12289897775763908, 29670550072645248, 71631250723716237, 172933516496665316, 417499138942325917, 1007933367385845629, 2433368766920412300, 5874676222662872885, 14182730999893284334, 34240156224739165602, 82663076560744668511, 199566370247538405914, 481795929070794160655, 1163158434416782363807, 2808113176848297108462, 6779385485099943117827, 16366885429006343737992, 39513158701001295744980, 95393207167842327310097, 230299581013366167992368, 555992383865976938155345, 1342284375730235928872913, 3240561184959447172958736, 7823406836938446811278321, 18887375026743571593630738, 45598157199255135710200662, 110083689993279936060296227, 265765538230577877386833510, 641614768376054199147928675, 1548995078517093746598960851, 3739604931911030541132126194, 9028204954295969656879724479, 21796014862494981002610632204, 52620234719735546486622833000, 127036484376364514776113710613, 306693203609304642811348567804, 740422891846661922796088416325, 1787538987765555118373558696549, 4315500868229226978683014989204, 10418540725790078647246708720941, 25152582322689835293793394952950, 60723705376467724646097390374474, 146599993085369780589376150017383, 353923691565130208260118518994226, 854447376248595590959533882653943, 2062818444122954202467763781850695, 4980084264606025124468828467137014, 12022986973540123786117685929103099, 29026058212063545972278810056651856, 70075103398361129469535948007475516, 169176265010062111260498222980958617, 408427633420832845354115162575125832, 986031531856045515420319475611367801, 2380490697140865389359075326793107769, 5747012926152383014117965038278720280, 13874516549472497364190412452907089129, 33496046025146791922238013115725822618, 80866608599856968054980563814629336190, 195229263225027895004847896096444561707
Generating function in Maple syntax:
(x^6+x^5-4*x^4-3*x^3-x^2+3*x-1)/(x-1)/(x^2+2*x-1)/(x^3+x^2+x-1)
Generating function in latex syntax:
\frac{x^{6}+x^{5}-4 x^{4}-3 x^{3}-x^{2}+3 x -1}{\left(x -1\right) \left(x^{2}+2 x -1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(x**6 + x**5 - 4*x**4 - 3*x**3 - x**2 + 3*x - 1)/((x - 1)*(x**2 + 2*x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2+2*x-1)*(x^3+x^2+x-1)*F(x)-x^6-x^5+4*x^4+3*x^3+x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}+2 x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)-x^{6}-x^{5}+4 x^{4}+3 x^{3}+x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
225/88*Sum(_alpha^(-n+4),_alpha = RootOf(_Z^6+2*_Z^5-_Z^4-2*_Z^3-3*_Z^2+4*_Z-1))+1145/176*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^6+2*_Z^5-_Z^4-2*_Z^3-3*_Z^2+4*_Z-1))+49/44*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^6+2*_Z^5-_Z^4-2*_Z^3-3*_Z^2+4*_Z-1))-185/44*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^6+2*_Z^5-_Z^4-2*_Z^3-3*_Z^2+4*_Z-1))-437/44*Sum(_alpha^(-n),_alpha = RootOf(_Z^6+2*_Z^5-_Z^4-2*_Z^3-3*_Z^2+4*_Z-1))+873/176*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^6+2*_Z^5-_Z^4-2*_Z^3-3*_Z^2+4*_Z-1))+piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
\frac{225 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{88}+\frac{1145 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{176}+\frac{49 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{44}-\frac{185 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{44}-\frac{437 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{44}+\frac{873 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{6}+2 \textit{\_Z}^{5}-\textit{\_Z}^{4}-2 \textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{176}+\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) = 54
a(6) = 147
a(n+2) = -1/2*a(n)-3/2*a(n+1)+3/2*a(n+4)-1/2*a(n+5)+2, 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) = 54
a \! \left(6\right) = 147
a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{2}-\frac{3 a \! \left(n +1\right)}{2}+\frac{3 a \! \left(n +4\right)}{2}-\frac{a \! \left(n +5\right)}{2}+2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20519/
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[12,x]*F[4,x]
F[4,x] = F[17,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[13,x]
F[12,x] = x
F[13,x] = F[1,x]+F[12,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[9,x]
F[17,x] = F[18,x]+F[30,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[12,x]*F[13,x]
F[24,x] = F[18,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[18,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[18,x]
F[30,x] = F[31,x]+F[32,x]+F[50,x]
F[31,x] = 0
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[42,x]
F[34,x] = F[35,x]+F[6,x]
F[35,x] = F[31,x]+F[36,x]+F[39,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]+F[6,x]
F[38,x] = F[39,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[12,x]+F[38,x]
F[42,x] = F[30,x]+F[43,x]
F[43,x] = 2*F[31,x]+F[44,x]+F[47,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[30,x]+F[46,x]
F[46,x] = F[47,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[49,x]
F[49,x] = F[28,x]+F[46,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[17,x]+F[52,x]
F[52,x] = F[53,x]+F[55,x]
F[53,x] = F[54,x]
F[54,x] = F[12,x]*F[27,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[52,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{17}\! \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_{12}\! \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_{14}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{24}\! \left(x \right) = F_{18}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{50}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{42}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{6}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{6}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right)+F_{38}\! \left(x \right)
F_{42}\! \left(x \right) = F_{30}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{30}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{28}\! \left(x \right)+F_{46}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{17}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{55}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{52}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_17(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_12(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_13(x))
Eq(F_12(x), x)
Eq(F_13(x), F_1(x) + F_12(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_9(x))
Eq(F_17(x), F_18(x) + F_30(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_12(x)*F_13(x))
Eq(F_24(x), F_18(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_18(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_18(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_50(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_42(x))
Eq(F_34(x), F_35(x) + F_6(x))
Eq(F_35(x), F_31(x) + F_36(x) + F_39(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_6(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_12(x) + F_38(x))
Eq(F_42(x), F_30(x) + F_43(x))
Eq(F_43(x), 2*F_31(x) + F_44(x) + F_47(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_30(x) + F_46(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_28(x) + F_46(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_17(x) + F_52(x))
Eq(F_52(x), F_53(x) + F_55(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_12(x)*F_27(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_52(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, 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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "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, 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": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 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