02341_02413_02431_03241_03412_03421_04123_04132_04213_04231_04312_04321_20341_20413_20431_24013_24031_40123_40132_40213_40231_40312_40321_42013_42031
Counting sequence:
1, 1, 2, 6, 24, 95, 359, 1340, 5018, 18846, 70805, 265925, 998586, 3749854, 14081688, 52880811, 198582339, 745731356, 2800426018, 10516370838, 39491871929, 148302866585, 556918140946, 2091381115158, 7853712516504, 29492855178343, 110753800171663, 415911046143292, 1561860613603402, 5865217091380590, 22025506776797101, 82711848720608701, 310605789374285706, 1166410349726960942, 4380192354723485976, 16448829581176665939, 61769888780987290635, 231962957679484204444, 871084840806376495250, 3271163669723201994918, 12284121193305745057201, 46130260888044215574385, 173231844273783303930530, 650533322222635514151846, 2442931927995786638357784, 9173882721996490112526383, 34450458169741880417563511, 129370965824473415531201084, 485823634504146539107440506, 1824401652555099220464340158, 6851131055496989187932863493, 25727885454312096832849699829, 96615301121573306293790075610, 362817085274599657109933910526, 1362478156555297023800853458328, 5116481010494135179045990898619, 19213796422933420336319595403923, 72153101364931228985373787943708, 270954782802014122160244836520322, 1017509891251458655751438034732342, 3821030092504642064167520814108905, 14349021167616195601034367568157321, 53884529429009597335601889618095026, 202351259913024937392525124405621686, 759884753978098623958554182000750232, 2853576694192788738464477678159097271, 10715967002905468163809633346822734879, 40241409681067682328031285500601123900, 151117584878757134515710738996677600554, 567488182958268321018156918922425652110, 2131074540766745336269281357793972436893, 8002772277353594962532673721982379900781, 30052615663146415213426100636659245714090, 112855855058199351034807162430642895544910, 423804841604387458006577889836647204396056, 1591503991305505974747541544176686584691939, 5976536145157465411791500853266029306877403, 22443540506030073006749107303192522385563036, 84281680594193273465926326550853938837504754, 316500940743867539013763847767139614665787206, 1188548267969097972456285417855187173629592673, 4463326339480126477371667986560451492639432673, 16761020607717561157322290311169349840944594754, 62942252133204934658102580562853133704247491910, 236365505199353785513145597097269095894750597656, 887617620194312308897538866360337240619882465023, 3333248813167211896661276086875311385333440973959, 12517268019136852455983816163166764971676535501628, 47005791480063235997854852407852411230732782277978, 176519702964668807558441494812612166579309249179102, 662880137821965846014416576589719527641172148490357, 2489297623658579848687702274767817408255614503198245, 9347998688741094353365713938616707435772917936892666, 35104311615528436632585961464203678903266528249309726, 131826365731559775481544713647022079612581574068593816, 495044337924111090311225561158467195253742819526636747, 1859027935350651858525222749502397821674268915459921891, 6981162291252991203148851280761713779205112478981186844, 26216188584395833043387182494070966821976783788357617890, 98449013963439982431309732689385950855262809008532441814, 369703182412355349822652824765786012713648301919529643545
Generating function in Maple syntax:
-(6*x^3-4*x^2+4*x-1)/(x^4-9*x^3+7*x^2-5*x+1)
Generating function in latex syntax:
-\frac{6 x^{3}-4 x^{2}+4 x -1}{x^{4}-9 x^{3}+7 x^{2}-5 x +1}
Generating function in sympy syntax:
(-6*x**3 + 4*x**2 - 4*x + 1)/(x**4 - 9*x**3 + 7*x**2 - 5*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^4-9*x^3+7*x^2-5*x+1)*F(x)+6*x^3-4*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}-9 x^{3}+7 x^{2}-5 x +1\right) F \! \left(x \right)+6 x^{3}-4 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
-1306/61651*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^4-9*_Z^3+7*_Z^2-5*_Z+1))+16439/61651*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^4-9*_Z^3+7*_Z^2-5*_Z+1))-1317/61651*Sum(_alpha^(-n),_alpha = RootOf(_Z^4-9*_Z^3+7*_Z^2-5*_Z+1))+1294/61651*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^4-9*_Z^3+7*_Z^2-5*_Z+1))
Explicit closed form in latex syntax:
-\frac{1306 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-9 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{61651}+\frac{16439 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-9 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{61651}-\frac{1317 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-9 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{61651}+\frac{1294 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-9 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{61651}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = -a(n)+9*a(n+1)-7*a(n+2)+5*a(n+3), n >= 4
Recurrence in latex format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a{\left(n + 4 \right)} = - a{\left(n \right)} + 9 a{\left(n + 1 \right)} - 7 a{\left(n + 2 \right)} + 5 a{\left(n + 3 \right)}, \quad n \geq 4
Specification 1
Strategy pack name: regular_insertion_encoding_left
Tree: http://permpal.com/tree/25067/
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[16,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[2,x]+F[6,x]
F[6,x] = F[25,x]+F[7,x]+F[8,x]
F[7,x] = 0
F[8,x] = F[16,x]*F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = F[2,x]+F[6,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]+F[17,x]+F[7,x]
F[13,x] = F[14,x]*F[16,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]+F[2,x]
F[16,x] = x
F[17,x] = F[16,x]*F[18,x]
F[18,x] = F[19,x]+F[20,x]
F[19,x] = F[2,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[16,x]*F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[2,x]+F[21,x]
F[25,x] = F[16,x]*F[26,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[12,x]+F[2,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]+F[40,x]+F[7,x]
F[30,x] = F[16,x]*F[31,x]
F[31,x] = F[32,x]+F[33,x]
F[32,x] = F[2,x]+F[29,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]+F[38,x]+F[7,x]
F[35,x] = F[16,x]*F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[2,x]+F[34,x]
F[38,x] = F[16,x]*F[39,x]
F[39,x] = F[19,x]
F[40,x] = F[16,x]*F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[2,x]+F[21,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_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{2}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{25}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)
F_{7}\! \left(x \right) = 0
F_{8}\! \left(x \right) = F_{16}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{2}\! \left(x \right)+F_{6}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{7}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right)+F_{2}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{16}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right)+F_{2}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{40}\! \left(x \right)+F_{7}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{33}\! \left(x \right)
F_{32}\! \left(x \right) = F_{2}\! \left(x \right)+F_{29}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)+F_{7}\! \left(x \right)
F_{35}\! \left(x \right) = F_{16}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{38}\! \left(x \right) = F_{16}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{19}\! \left(x \right)
F_{40}\! \left(x \right) = F_{16}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \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_16(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_2(x) + F_6(x))
Eq(F_6(x), F_25(x) + F_7(x) + F_8(x))
Eq(F_7(x), 0)
Eq(F_8(x), F_16(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), F_2(x) + F_6(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x) + F_17(x) + F_7(x))
Eq(F_13(x), F_14(x)*F_16(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x) + F_2(x))
Eq(F_16(x), x)
Eq(F_17(x), F_16(x)*F_18(x))
Eq(F_18(x), F_19(x) + F_20(x))
Eq(F_19(x), F_2(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_16(x)*F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_2(x) + F_21(x))
Eq(F_25(x), F_16(x)*F_26(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_12(x) + F_2(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x) + F_40(x) + F_7(x))
Eq(F_30(x), F_16(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_33(x))
Eq(F_32(x), F_2(x) + F_29(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x) + F_38(x) + F_7(x))
Eq(F_35(x), F_16(x)*F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_2(x) + F_34(x))
Eq(F_38(x), F_16(x)*F_39(x))
Eq(F_39(x), F_19(x))
Eq(F_40(x), F_16(x)*F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_2(x) + F_21(x))
Pack JSON:
{"name": "regular_insertion_encoding_left", "initial_strats": [{"class_module": "tilings.strategies.factor", "strategy_class": "FactorFactory", "interleaving": null, "unions": false, "ignore_parent": true, "workable": true, "tracked": false}, {"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "CellInsertionFactory", "ignore_parent": true, "maxreqlen": 1, "extra_basis": [], "one_cell_only": true}], "inferral_strats": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}], "expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "strategy_class": "RowAndColumnPlacementFactory", "partial": false, "ignore_parent": false, "dirs": [0], "place_row": false, "place_col": true}]], "symmetries": [], "iterative": false}
Specification JSON:
{"root": {"class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 4, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 4, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}], 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