0132_0231_0321_1203_3102
Counting sequence:
1, 1, 2, 6, 19, 52, 127, 295, 669, 1497, 3326, 7363, 16270, 35918, 79256, 174844, 385673, 850674, 1876269, 4138293, 9127347, 20131055, 44400500, 97928449, 215988060, 476376732, 1050682030, 2317352242, 5111081343, 11272844848, 24863042075, 54837165635, 120947176265, 266757394757, 588351955306, 1297651087039, 2862059569002, 6312471093482, 13922593274180, 30707246117544, 67726963328757, 149376519931886, 329460285981513, 726647535291985, 1602671590516063, 3534803467013851, 7796254469319904, 17195180529156093, 37925164525326264, 83646583519972664, 184488347569101658, 406901859663529822, 897450302847032555, 1979388953263167020, 4365679766189864119, 9628809835226761055, 21237008623716689397, 46839697013623243185, 103308203862473247702, 227853416348663185083, 502546529710949613638, 1108401263284372475270, 2444655942917408135920, 5391858415545765885780, 11892118094375904247137, 26228892131669216630506, 57849642678884199147109, 127591403452144302541677, 281411699035957821714187, 620673040750799842575815, 1368937484953743987693644, 3019286668943445797101817, 6659246378637691436779796, 14687430242229126861253588, 32394147153401699519609350, 71447540685441090475998858, 157582511613111307813251671, 347559170379624315146113064, 766565881444689720768225363, 1690714274502490749349702779, 3728987719384605813845519009, 8224541320213901348459263773, 18139796914930293446268230722, 40008581549245192706381980855, 88241704418704286761223225890, 194623205752338866968714682914, 429254993053922926643811347100, 946751690526550140048845920512, 2088126586805439147066406524365, 4605508166664801220776624396262, 10157768023856152581602094713473, 22403662634517744310270595951753, 49412833435700289841317816300215, 108983434895256732264237727314355, 240370532425031208838746050580920, 530153898285762707518809917462517, 1169291231466782147301857562239856, 2578952995358595503442461175061104, 5688059889002953714403732267585202, 12545411009472689576109322097410742, 27669775014303974655661105369883075
Generating function in Maple syntax:
(x^6-x^5-4*x^4-3*x^2+3*x-1)/(x^3+2*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{x^{6}-x^{5}-4 x^{4}-3 x^{2}+3 x -1}{\left(x^{3}+2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**6 - x**5 - 4*x**4 - 3*x**2 + 3*x - 1)/((x - 1)**2*(x**3 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
-(x^3+2*x-1)*(x-1)^2*F(x)+x^6-x^5-4*x^4-3*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{3}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{6}-x^{5}-4 x^{4}-3 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
1/4+15/236*Sum(_alpha^(1-n),_alpha = RootOf(_Z^3+2*_Z-1))-151/236*Sum(_alpha^(-n),_alpha = RootOf(_Z^3+2*_Z-1))+197/236*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^3+2*_Z-1))-5/2*n+piecewise(Or(n = 0,n = 1),1,0)
Explicit closed form in latex syntax:
\frac{1}{4}+\frac{15 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{236}-\frac{151 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{236}+\frac{197 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{3}+2 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{236}-\frac{5 n}{2}+\left(\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 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) = 52
a(6) = 127
a(n) = -2*a(n+2)+a(n+3)-5*n-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) = 52
a \! \left(6\right) = 127
a \! \left(n \right) = -2 a \! \left(n +2\right)+a \! \left(n +3\right)-5 n -2, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18670/
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[17,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[11,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[11,x]+F[14,x]
F[17,x] = F[18,x]+F[34,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[26,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[11,x]
F[26,x] = F[18,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[18,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[30,x]*F[4,x]
F[33,x] = F[31,x]
F[34,x] = F[35,x]+F[36,x]+F[57,x]
F[35,x] = 0
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[46,x]
F[38,x] = F[39,x]+F[7,x]
F[39,x] = F[35,x]+F[40,x]+F[43,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[16,x]+F[42,x]
F[42,x] = F[14,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[25,x]+F[45,x]
F[45,x] = F[14,x]
F[46,x] = F[34,x]+F[47,x]
F[47,x] = 2*F[35,x]+F[48,x]+F[54,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[31,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[50,x]
F[53,x] = F[51,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[33,x]+F[56,x]
F[56,x] = F[52,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[30,x]+F[59,x]
F[59,x] = F[60,x]+F[73,x]
F[60,x] = F[32,x]+F[35,x]+F[61,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[68,x]
F[63,x] = F[11,x]+F[64,x]
F[64,x] = F[35,x]+F[65,x]+F[67,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[16,x]
F[67,x] = F[25,x]*F[4,x]
F[68,x] = F[31,x]+F[69,x]
F[69,x] = 2*F[35,x]+F[70,x]+F[72,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[50,x]
F[72,x] = F[33,x]*F[4,x]
F[73,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{17}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
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_{26}\! \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_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{11}\! \left(x \right)
F_{26}\! \left(x \right) = F_{18}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{57}\! \left(x \right)
F_{35}\! \left(x \right) = 0
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{46}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{7}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{16}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{14}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{25}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{14}\! \left(x \right)
F_{46}\! \left(x \right) = F_{34}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{48}\! \left(x \right)+F_{54}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{31}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{33}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{52}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{30}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{73}\! \left(x \right)
F_{60}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{68}\! \left(x \right)
F_{63}\! \left(x \right) = F_{11}\! \left(x \right)+F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{35}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{16}\! \left(x \right)
F_{67}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{68}\! \left(x \right) = F_{31}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{35}\! \left(x \right)+F_{70}\! \left(x \right)+F_{72}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{50}\! \left(x \right)
F_{72}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{73}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_17(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_11(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_11(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_34(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_26(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_11(x))
Eq(F_26(x), F_18(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_18(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_30(x)*F_4(x))
Eq(F_33(x), F_31(x))
Eq(F_34(x), F_35(x) + F_36(x) + F_57(x))
Eq(F_35(x), 0)
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_46(x))
Eq(F_38(x), F_39(x) + F_7(x))
Eq(F_39(x), F_35(x) + F_40(x) + F_43(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_16(x) + F_42(x))
Eq(F_42(x), F_14(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_25(x) + F_45(x))
Eq(F_45(x), F_14(x))
Eq(F_46(x), F_34(x) + F_47(x))
Eq(F_47(x), 2*F_35(x) + F_48(x) + F_54(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_31(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_50(x))
Eq(F_53(x), F_51(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_33(x) + F_56(x))
Eq(F_56(x), F_52(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_30(x) + F_59(x))
Eq(F_59(x), F_60(x) + F_73(x))
Eq(F_60(x), F_32(x) + F_35(x) + F_61(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_68(x))
Eq(F_63(x), F_11(x) + F_64(x))
Eq(F_64(x), F_35(x) + F_65(x) + F_67(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_16(x))
Eq(F_67(x), F_25(x)*F_4(x))
Eq(F_68(x), F_31(x) + F_69(x))
Eq(F_69(x), 2*F_35(x) + F_70(x) + F_72(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_50(x))
Eq(F_72(x), F_33(x)*F_4(x))
Eq(F_73(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:
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