0123_0132_0213_1023_1302
Counting sequence:
1, 1, 2, 6, 19, 58, 173, 511, 1505, 4430, 13040, 38387, 113008, 332691, 979433, 2883425, 8488726, 24990578, 73571575, 216592690, 637642909, 1877203147, 5526434321, 16269670314, 47897460964, 141008804887, 415126034984, 1222119604939, 3597886431845, 10592078487353, 31182787118906, 91801265791598, 270260396186011, 795639157225050, 2342339748788381, 6895783658875559, 20300997024284065, 59765575686171302, 175948207510534216, 517986673277090011, 1524938488939154224, 4489376879787155251, 13216601794075136753, 38909311394554107057, 114547940294082171166, 337225978958165537242, 992783987144011801919, 2922728694196576638898, 8604432716984269954925, 25331212756119669175571, 74574392153617313802449, 219544955017519131005954, 646334296287073453350748, 1902790353454358444933823, 5601762354245928916281992, 16491433970368605138987659, 48550327057856422268872333, 142930824672982661632112009, 420784408256484632415407362, 1238777699889871332774396406, 3646927404232761616756996195, 10736453758350913159553612218, 31607823937328465719999878605, 93052562469808104450130474383, 273944179117361630714482641889, 806484112639395570106646490238, 2374266998610346215273007318112, 6989776602345452728957868108611, 20577709659146099961334769273648, 60580210056216486583405758431123, 178346468642302886151206872431161, 525047087946106890351191181374849, 1545724155119594616290407888322086, 4550569308110187881123662653594882, 13396750616420466557574649512900743, 39439664562123921751339584565245938, 116109285401363288684091542845330813, 341822535918881298687020093479874459, 1006316124142152340059615459844060945, 2962566932534850018995989185221781466, 8721715392596788334486273353565730196, 25676489720478213538271550585713264359, 75590878019871943810750624016622207272, 222537461390526046331693549326226868747, 655144152567203717025223321376459049205, 1928726327518321212371605114681959912025, 5678117147020891539700558193305480046282, 16716220375742450897325581206452035137342, 49212092039523215191879491773962658596907, 144879042538881258960583875512552968753370, 426519907955254722448772561908482824177981, 1255662852916336799516110291735213802129591, 3696636829339748202141298589250779084424865, 10882796935732490205489258670914098044639446, 32038654217904888860914573747985806771509560, 94320914940915010224081440973420581471954219, 277678173832892380512375381157312365697498480, 817476890162384582927321145302830107691900595, 2406629432645755957669906144261934365746171841, 7085050716144807721833448641928775937155395537, 20858194024145363192421505751941095275086162990
Generating function in Maple syntax:
-(2*x-1)*(x^2+x-1)/(2*x^4-x^3-3*x^2+4*x-1)
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x^{2}+x -1\right)}{2 x^{4}-x^{3}-3 x^{2}+4 x -1}
Generating function in sympy syntax:
(1 - 2*x)*(x**2 + x - 1)/(2*x**4 - x**3 - 3*x**2 + 4*x - 1)
Implicit equation for the generating function in Maple syntax:
(2*x^4-x^3-3*x^2+4*x-1)*F(x)+(2*x-1)*(x^2+x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(2 x^{4}-x^{3}-3 x^{2}+4 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x^{2}+x -1\right) = 0
Explicit closed form in Maple syntax:
-898/2191*Sum(_alpha^(-n+2),_alpha = RootOf(2*_Z^4-_Z^3-3*_Z^2+4*_Z-1))-229/2191*Sum(_alpha^(-n+1),_alpha = RootOf(2*_Z^4-_Z^3-3*_Z^2+4*_Z-1))+1584/2191*Sum(_alpha^(-n),_alpha = RootOf(2*_Z^4-_Z^3-3*_Z^2+4*_Z-1))-278/2191*Sum(_alpha^(-n-1),_alpha = RootOf(2*_Z^4-_Z^3-3*_Z^2+4*_Z-1))
Explicit closed form in latex syntax:
-\frac{898 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{2191}-\frac{229 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{2191}+\frac{1584 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{2191}-\frac{278 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(2 \textit{\_Z}^{4}-\textit{\_Z}^{3}-3 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{2191}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = 2*a(n)-a(n+1)-3*a(n+2)+4*a(n+3), n >= 4
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(n +4\right) = 2 a \! \left(n \right)-a \! \left(n +1\right)-3 a \! \left(n +2\right)+4 a \! \left(n +3\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15971/
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[10,x]*F[4,x]
F[4,x] = F[14,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[11,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = x
F[11,x] = F[12,x]+F[6,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[6,x]
F[14,x] = F[15,x]+F[2,x]
F[15,x] = F[16,x]+F[17,x]+F[41,x]
F[16,x] = 0
F[17,x] = F[10,x]*F[18,x]
F[18,x] = F[19,x]+F[26,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[21,x]
F[21,x] = F[10,x]*F[22,x]
F[22,x] = F[1,x]+F[20,x]
F[23,x] = F[24,x]
F[24,x] = F[10,x]*F[25,x]
F[25,x] = F[10,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[16,x]+F[17,x]+F[28,x]
F[28,x] = F[10,x]*F[29,x]
F[29,x] = F[2,x]+F[27,x]
F[30,x] = F[31,x]
F[31,x] = F[10,x]*F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[16,x]+F[34,x]+F[40,x]
F[34,x] = F[10,x]*F[35,x]
F[35,x] = F[36,x]+F[38,x]
F[36,x] = F[10,x]+F[37,x]
F[37,x] = F[24,x]
F[38,x] = F[33,x]+F[39,x]
F[39,x] = F[31,x]
F[40,x] = F[10,x]*F[2,x]
F[41,x] = F[10,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[2,x]+F[33,x]
F[44,x] = F[15,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[10,x]*F[15,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{11}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{6}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{41}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{2}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{10}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{16}\! \left(x \right)+F_{34}\! \left(x \right)+F_{40}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{24}\! \left(x \right)
F_{38}\! \left(x \right) = F_{33}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{31}\! \left(x \right)
F_{40}\! \left(x \right) = F_{10}\! \left(x \right) F_{2}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_14(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_10(x)*F_8(x))
Eq(F_8(x), F_11(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_6(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_6(x))
Eq(F_14(x), F_15(x) + F_2(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_41(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_10(x)*F_18(x))
Eq(F_18(x), F_19(x) + F_26(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_10(x)*F_22(x))
Eq(F_22(x), F_1(x) + F_20(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_10(x)*F_25(x))
Eq(F_25(x), F_10(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_16(x) + F_17(x) + F_28(x))
Eq(F_28(x), F_10(x)*F_29(x))
Eq(F_29(x), F_2(x) + F_27(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_10(x)*F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_16(x) + F_34(x) + F_40(x))
Eq(F_34(x), F_10(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_38(x))
Eq(F_36(x), F_10(x) + F_37(x))
Eq(F_37(x), F_24(x))
Eq(F_38(x), F_33(x) + F_39(x))
Eq(F_39(x), F_31(x))
Eq(F_40(x), F_10(x)*F_2(x))
Eq(F_41(x), F_10(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_2(x) + F_33(x))
Eq(F_44(x), F_15(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_10(x)*F_15(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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