0132_1203_1230_1302_2103
Counting sequence:
1, 1, 2, 6, 19, 54, 141, 357, 885, 2159, 5207, 12451, 29582, 69942, 164752, 386969, 906891, 2121688, 4957014, 11569068, 26978340, 62870649, 146439074, 340949231, 793566652, 1846576733, 4296003717, 9992944552, 23241716927, 54050557173, 125689308184, 292260258852, 679547866827, 1579987398114, 3673448607879, 8540510939195, 19855711940393, 46161565424419, 107317467190829, 249491758337409, 580014368461270, 1348399977838634, 3134705114949234, 7287407301995263, 16941354280246791, 39384215682401940, 91557828290094756, 212847295244138130, 494812077125127884, 1150302471427501421, 2674136077275597916, 6216625522636930421, 14451925565576619048, 33596696088060339337, 78102925852322989283, 181567427183201454486, 422093339725782294559, 981248483889465359529, 2281126922655771456090, 5302978470188866421778, 12327932495901159098367, 28658971970268592283462, 66624037666510739314949, 154882116779258320477321, 360057274976989693701905, 837031680542611340322951, 1945862703183090861516703, 4523582197676633284553759, 10516053187675549226229738, 24446858460291263320799302, 56832052287338929586419656, 132118495123153427240797389, 307138242922691697908887815, 714009797554908401558858736, 1659871414349958096762411326, 3858732913683631290507123752, 8970465751880032278374310544, 20853802919825283515781531449, 48479210311396936750261155642, 112700491107594432330482112403, 261996856100188874473661449832, 609068797280320539279282629253, 1415913172373272469289851739709, 3291598781648379383089453291044, 7652038802976567941110586891691, 17788832028117980064959000221221, 41354017278647614604709024309168, 96136426616891299263942345771848, 223490077384414689733470713839895, 519551396313833404910925629739562, 1207810461002412342888192374081711, 2807818668155477477654403023752743, 6527386479495508249168060122206581, 15174332564502392009590497876485987, 35276043405107606548745224208606613, 82006851570293990388455426147525749, 190642800473409976004861396037420402, 443190741699906795766701765113667090, 1030293475751072940098748891789578682, 2395141744378091787095293776778135899, 5568029023426348615582868714931974291
Generating function in Maple syntax:
-(2*x^7-2*x^6-3*x^5+3*x^4-x^3+5*x^2-4*x+1)/(x-1)/(x^3-2*x^2+3*x-1)/(x^3+x^2+x-1)
Generating function in latex syntax:
-\frac{2 x^{7}-2 x^{6}-3 x^{5}+3 x^{4}-x^{3}+5 x^{2}-4 x +1}{\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(-2*x**7 + 2*x**6 + 3*x**5 - 3*x**4 + x**3 - 5*x**2 + 4*x - 1)/((x - 1)*(x**3 - 2*x**2 + 3*x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^3-2*x^2+3*x-1)*(x^3+x^2+x-1)*F(x)+2*x^7-2*x^6-3*x^5+3*x^4-x^3+5*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+2 x^{7}-2 x^{6}-3 x^{5}+3 x^{4}-x^{3}+5 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
29721/5566*Sum(_alpha^(-n+5),_alpha = RootOf(_Z^7-2*_Z^6+3*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))-43973/5566*Sum(_alpha^(-n+4),_alpha = RootOf(_Z^7-2*_Z^6+3*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))+32584/2783*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^7-2*_Z^6+3*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))-26938/2783*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^7-2*_Z^6+3*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))+5452/253*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^7-2*_Z^6+3*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))-171273/5566*Sum(_alpha^(-n),_alpha = RootOf(_Z^7-2*_Z^6+3*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))+28536/2783*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^7-2*_Z^6+3*_Z^5-3*_Z^4+5*_Z^3-8*_Z^2+5*_Z-1))-piecewise(n = 0,2,0)
Explicit closed form in latex syntax:
\frac{29721 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +5}\right)}{5566}-\frac{43973 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{5566}+\frac{32584 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{2783}-\frac{26938 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{2783}+\frac{5452 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{253}-\frac{171273 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{5566}+\frac{28536 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{7}-2 \textit{\_Z}^{6}+3 \textit{\_Z}^{5}-3 \textit{\_Z}^{4}+5 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{2783}-\left(\left\{\begin{array}{cc}2 & 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) = 141
a(7) = 357
a(n+6) = -a(n)+a(n+1)-2*a(n+2)+a(n+3)-4*a(n+4)+4*a(n+5)+1, n >= 8
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) = 141
a \! \left(7\right) = 357
a \! \left(n +6\right) = -a \! \left(n \right)+a \! \left(n +1\right)-2 a \! \left(n +2\right)+a \! \left(n +3\right)-4 a \! \left(n +4\right)+4 a \! \left(n +5\right)+1, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/19819/
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[24,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[16,x]+F[22,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[4,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[7,x]
F[24,x] = F[2,x]+F[25,x]
F[25,x] = F[15,x]+F[26,x]+F[35,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[33,x]
F[28,x] = F[29,x]+F[7,x]
F[29,x] = F[15,x]+F[16,x]+F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[23,x]+F[32,x]
F[32,x] = F[30,x]
F[33,x] = F[34,x]+F[39,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[2,x]
F[38,x] = F[34,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[43,x]
F[42,x] = F[34,x]
F[43,x] = F[40,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_{24}\! \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_{16}\! \left(x \right)+F_{22}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{2}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{35}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{33}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{7}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \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_{23}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{39}\! \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_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{43}\! \left(x \right)
F_{42}\! \left(x \right) = F_{34}\! \left(x \right)
F_{43}\! \left(x \right) = F_{40}\! \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_24(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) + F_16(x) + F_22(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_7(x))
Eq(F_24(x), F_2(x) + F_25(x))
Eq(F_25(x), F_15(x) + F_26(x) + F_35(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_33(x))
Eq(F_28(x), F_29(x) + F_7(x))
Eq(F_29(x), F_15(x) + F_16(x) + F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_23(x) + F_32(x))
Eq(F_32(x), F_30(x))
Eq(F_33(x), F_34(x) + F_39(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) + F_38(x))
Eq(F_37(x), F_2(x))
Eq(F_38(x), F_34(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_43(x))
Eq(F_42(x), F_34(x))
Eq(F_43(x), F_40(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": [1, 2, 0, 3], "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]]}, {"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": [1, 2, 0, 3], "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]]}, {"patt": [2, 1, 0, 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