0231_1032_2031_2103_3012
Counting sequence:
1, 1, 2, 6, 19, 51, 121, 276, 620, 1380, 3057, 6757, 14919, 32922, 72630, 160210, 353375, 779415, 1719077, 3791568, 8362592, 18444304, 40680221, 89723081, 197890515, 436461302, 962645738, 2123182046, 4682825451, 10328296699, 22779775505, 50242376524, 110813049812, 244405875196, 539054126985, 1188921303853, 2622248482975, 5783551093010, 12756023489950, 28134295462954, 62052142018999, 136860307528031, 301854910519101, 665761963057288, 1468384233642696, 3238623377804584, 7143008718666549, 15754401670975889, 34747426719756459, 76637862158179566, 169030125987335122, 372807678694426806, 822253219547033283, 1813536565081401795, 3999880808857230505, 8822014837261494404, 19457566239604390716, 42915013288066012052, 94652041413393518625, 208761649066391428085, 460438311420848868343, 1015528664255091255434, 2239818977576573939078, 4940076266573996746626, 10895681197403084748815, 24031181372382743436839, 53002439011339483620437, 116900559220082051989824, 257832299812546847416624, 568667038636433178453824, 1254234636492948408897613, 2766301572798443665211993, 6101270184233320508877955, 13456775004959589426653670, 29679851582717622518519482, 65460973349668565545917070, 144378721704296720518487963, 318437294991311063555495563, 702335563332290692656908353, 1549049848368878105832304828, 3416536991729067275220105380, 7535409546790425243097119276, 16619868941949728592026543545, 36656274875628524459273192637, 80847959298047474161643504719, 178315787538044676915313553154, 393287849951717878289900299118, 867423659201483230741444103130, 1913163105941011138398201759591, 4219614061833740155086303818479, 9306651782868963540914051740269, 20526466671678938220226305240312, 45272547405191616595538914299288, 99851746593252196731991880339032, 220229959858183331684210065918565, 485732467121558279963959046136609, 1071316680836368756659909972612443, 2362863321530920845004030011143646, 5211459110183399969972019068424098, 11494234901203168696603948109460838, 25351333123937258238211926230065523
Generating function in Maple syntax:
(3*x^6-4*x^4-3*x^2+3*x-1)/(x^3+2*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{3 x^{6}-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:
(3*x**6 - 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)+3*x^6-4*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)+3 x^{6}-4 x^{4}-3 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
-3+6/59*Sum(_alpha^(1-n),_alpha = RootOf(_Z^3+2*_Z-1))-84/59*Sum(_alpha^(-n),_alpha = RootOf(_Z^3+2*_Z-1))+67/59*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^3+2*_Z-1))-n+3*piecewise(n = 0,2,n = 1,1,0)
Explicit closed form in latex syntax:
-3+\frac{6 \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)}{59}-\frac{84 \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)}{59}+\frac{67 \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)}{59}-n +3 \left(\left\{\begin{array}{cc}2 & n =0 \\ 1 & 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) = 51
a(6) = 121
a(n) = -2*a(n+2)+a(n+3)-2*n-7, 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) = 51
a \! \left(6\right) = 121
a \! \left(n \right) = -2 a \! \left(n +2\right)+a \! \left(n +3\right)-2 n -7, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21030/
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[32,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[6,x]
F[10,x] = F[11,x]+F[22,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[4,x]
F[17,x] = F[11,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[4,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]+F[29,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[1,x]+F[26,x]
F[29,x] = F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[26,x]
F[32,x] = F[33,x]+F[44,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[14,x]+F[36,x]
F[36,x] = F[33,x]+F[37,x]
F[37,x] = F[38,x]+F[39,x]+F[43,x]
F[38,x] = 0
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[21,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[33,x]*F[4,x]
F[43,x] = F[36,x]*F[4,x]
F[44,x] = F[38,x]+F[45,x]+F[54,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[25,x]+F[47,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[49,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[33,x]+F[48,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[48,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[32,x]+F[56,x]
F[56,x] = F[57,x]+F[64,x]
F[57,x] = F[38,x]+F[39,x]+F[58,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[36,x]+F[60,x]
F[60,x] = F[57,x]+F[61,x]
F[61,x] = F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[41,x]
F[64,x] = 2*F[38,x]+F[65,x]+F[71,x]
F[65,x] = F[4,x]*F[66,x]
F[66,x] = F[67,x]+F[69,x]
F[67,x] = F[68,x]
F[68,x] = F[26,x]*F[4,x]
F[69,x] = F[70,x]
F[70,x] = F[4,x]*F[48,x]
F[71,x] = F[4,x]*F[72,x]
F[72,x] = F[47,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_{32}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{22}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \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_{21}\! \left(x \right) = F_{4}\! \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_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{26}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{44}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{14}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{43}\! \left(x \right)
F_{38}\! \left(x \right) = 0
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{21}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{44}\! \left(x \right) = F_{38}\! \left(x \right)+F_{45}\! \left(x \right)+F_{54}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{25}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{33}\! \left(x \right)+F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{48}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{32}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{64}\! \left(x \right)
F_{57}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{36}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{41}\! \left(x \right)
F_{64}\! \left(x \right) = 2 F_{38}\! \left(x \right)+F_{65}\! \left(x \right)+F_{71}\! \left(x \right)
F_{65}\! \left(x \right) = F_{4}\! \left(x \right) F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{67}\! \left(x \right)+F_{69}\! \left(x \right)
F_{67}\! \left(x \right) = F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{71}\! \left(x \right) = F_{4}\! \left(x \right) F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{47}\! \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_32(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_6(x))
Eq(F_10(x), F_11(x) + F_22(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_4(x))
Eq(F_17(x), F_11(x) + F_18(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))
Eq(F_21(x), F_4(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x) + F_29(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_1(x) + F_26(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_26(x))
Eq(F_32(x), F_33(x) + F_44(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_14(x) + F_36(x))
Eq(F_36(x), F_33(x) + F_37(x))
Eq(F_37(x), F_38(x) + F_39(x) + F_43(x))
Eq(F_38(x), 0)
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_21(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_33(x)*F_4(x))
Eq(F_43(x), F_36(x)*F_4(x))
Eq(F_44(x), F_38(x) + F_45(x) + F_54(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_25(x) + F_47(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_33(x) + F_48(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_48(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_32(x) + F_56(x))
Eq(F_56(x), F_57(x) + F_64(x))
Eq(F_57(x), F_38(x) + F_39(x) + F_58(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_36(x) + F_60(x))
Eq(F_60(x), F_57(x) + F_61(x))
Eq(F_61(x), F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_41(x))
Eq(F_64(x), 2*F_38(x) + F_65(x) + F_71(x))
Eq(F_65(x), F_4(x)*F_66(x))
Eq(F_66(x), F_67(x) + F_69(x))
Eq(F_67(x), F_68(x))
Eq(F_68(x), F_26(x)*F_4(x))
Eq(F_69(x), F_70(x))
Eq(F_70(x), F_4(x)*F_48(x))
Eq(F_71(x), F_4(x)*F_72(x))
Eq(F_72(x), F_47(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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "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, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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