0231_0321_2013
Counting sequence:
1, 1, 2, 6, 21, 74, 257, 881, 2995, 10132, 34182, 115143, 387538, 1303745, 4384933, 14746009, 49585430, 166730986, 560620697, 1885023706, 6338138505, 21311063589, 71655198707, 240929422288, 810087129234, 2723789200907, 9158306712498, 30793342721777, 103537688773817, 348128906036113, 1170527707580426, 3935711972895710, 13233201203278013, 44494519642786778, 149605695959332833, 503025191149069961, 1691341637866390099, 5686865361310721804, 19121173930882851438, 64291884763276937935, 216171165077215378978, 726840919069237224145, 2443886174364438201405, 8217175831084851066281, 27628937610540805255070, 92897877467620131765282, 312354233797591368619905, 1050241082259859424829258, 3531267424986640822771273, 11873321123506344384235549, 39922140561876390326282771, 134231803424136807489522824, 451332939489276037870500442, 1517534720325449715636558099, 5102467437894471164177525410, 17156229512290691974453358497, 57685073870819480657976549681, 193956821637180450804897463201, 652148747245117937949875376018, 2192745709810339166383389823222, 7372756243422564721215753859301, 24789711995208072349066966891562, 83351436086551977744439160890881, 280255853679684736700812852101217, 942315420218670214804950276127859, 3168384672517031507380875765243012, 10653185990218988967222440902264214, 35819631601752131606298737984951511, 120437774132850320395424684159078322, 404952725347569997518565724831348641, 1361588678860313111370330168293712789, 4578123850899767060074271586978615609, 15393208183634980068878000327920895142, 51757197031303008818796549092955320730, 174025285215400922740814530819508609513, 585132148404119376532408179563903269242, 1967413130064621231389253145863983552329, 6615111534903010437621392318954339517141, 22242253012599108200836694037143481895091, 74786013276755365729830410478781648054336, 251455991381039888001115050716146135314274, 845480496031127405392371205431489812026651, 2842792749709525963697456646656298569348306, 9558435298906668381253027515170533164034065, 32138707745304497164994226982424339484689769, 108061257228601219396703455248387924276608561, 363338669568379800359737542946133156200874138, 1221668081507190578866435012039048013815982414, 4107663252982155950905074661950222779007014029, 13811359775466668478870426618911995437830018362, 46438485118976561282453627303676900422683930977, 156141968293092342694972289111963127380071517433, 525002359572627726055757530465557366690284909587, 1765236345935190462717889378779529391143529117180, 5935324480345605559203168290649610767928761934078, 19956577921199912373009851787531255277925665782751, 67100796871973780869653396257314721134387745918594, 225615682139113321827529275146397637137596405928241, 758596594973643485602562859161001680228120730416813, 2550659548349903169508870979345922028461771796388425, 8576184199475045556746135446889121188027450263920174
Generating function in Maple syntax:
-(x^2-3*x+1)*(x^2+2*x-1)/(x-1)/(x^4-2*x^3-5*x^2+5*x-1)
Generating function in latex syntax:
-\frac{\left(x^{2}-3 x +1\right) \left(x^{2}+2 x -1\right)}{\left(x -1\right) \left(x^{4}-2 x^{3}-5 x^{2}+5 x -1\right)}
Generating function in sympy syntax:
(-x**2 + 3*x - 1)*(x**2 + 2*x - 1)/((x - 1)*(x**4 - 2*x**3 - 5*x**2 + 5*x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^4-2*x^3-5*x^2+5*x-1)*F(x)+(x^2-3*x+1)*(x^2+2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{4}-2 x^{3}-5 x^{2}+5 x -1\right) F \! \left(x \right)+\left(x^{2}-3 x +1\right) \left(x^{2}+2 x -1\right) = 0
Explicit closed form in Maple syntax:
-2222/2777*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^5-3*_Z^4-3*_Z^3+10*_Z^2-6*_Z+1))+4890/2777*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^5-3*_Z^4-3*_Z^3+10*_Z^2-6*_Z+1))+10457/2777*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^5-3*_Z^4-3*_Z^3+10*_Z^2-6*_Z+1))-13472/2777*Sum(_alpha^(-n),_alpha = RootOf(_Z^5-3*_Z^4-3*_Z^3+10*_Z^2-6*_Z+1))+3124/2777*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^5-3*_Z^4-3*_Z^3+10*_Z^2-6*_Z+1))
Explicit closed form in latex syntax:
-\frac{2222 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+10 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{2777}+\frac{4890 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+10 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{2777}+\frac{10457 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+10 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{2777}-\frac{13472 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+10 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{2777}+\frac{3124 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}-3 \textit{\_Z}^{3}+10 \textit{\_Z}^{2}-6 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{2777}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(n+4) = a(n)-2*a(n+1)-5*a(n+2)+5*a(n+3)+2, n >= 5
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) = 21
a \! \left(n +4\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-5 a \! \left(n +2\right)+5 a \! \left(n +3\right)+2, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14755/
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[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[7,x]
F[17,x] = F[18,x]+F[2,x]
F[18,x] = F[19,x]+F[20,x]+F[51,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[26,x]
F[22,x] = F[23,x]+F[7,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[11,x]
F[26,x] = F[18,x]+F[27,x]
F[27,x] = 2*F[19,x]+F[28,x]+F[32,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[23,x]
F[31,x] = F[27,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]
F[34,x] = F[19,x]+F[35,x]+F[49,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[37,x]+F[41,x]
F[37,x] = F[11,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[11,x]
F[41,x] = F[34,x]+F[42,x]
F[42,x] = 2*F[19,x]+F[43,x]+F[47,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[45,x]+F[46,x]
F[45,x] = F[38,x]
F[46,x] = F[42,x]
F[47,x] = F[4,x]*F[48,x]
F[48,x] = F[34,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[2,x]+F[34,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[17,x]+F[53,x]
F[53,x] = F[34,x]+F[54,x]
F[54,x] = F[55,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[18,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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \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_{1}\! \left(x \right)+F_{11}\! \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_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{2}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{51}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{7}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \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) = 2 F_{19}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \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_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{23}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{19}\! \left(x \right)+F_{35}\! \left(x \right)+F_{49}\! \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_{41}\! \left(x \right)
F_{37}\! \left(x \right) = F_{11}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{11}\! \left(x \right)
F_{41}\! \left(x \right) = F_{34}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = 2 F_{19}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{46}\! \left(x \right)
F_{45}\! \left(x \right) = F_{38}\! \left(x \right)
F_{46}\! \left(x \right) = F_{42}\! \left(x \right)
F_{47}\! \left(x \right) = F_{4}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{34}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{2}\! \left(x \right)+F_{34}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{17}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{34}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{18}\! \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_6(x))
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_7(x))
Eq(F_17(x), F_18(x) + F_2(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_51(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_26(x))
Eq(F_22(x), F_23(x) + F_7(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_11(x))
Eq(F_26(x), F_18(x) + F_27(x))
Eq(F_27(x), 2*F_19(x) + F_28(x) + F_32(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_23(x))
Eq(F_31(x), F_27(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_19(x) + F_35(x) + F_49(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_37(x) + F_41(x))
Eq(F_37(x), F_11(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_11(x))
Eq(F_41(x), F_34(x) + F_42(x))
Eq(F_42(x), 2*F_19(x) + F_43(x) + F_47(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_46(x))
Eq(F_45(x), F_38(x))
Eq(F_46(x), F_42(x))
Eq(F_47(x), F_4(x)*F_48(x))
Eq(F_48(x), F_34(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_2(x) + F_34(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_17(x) + F_53(x))
Eq(F_53(x), F_34(x) + F_54(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_18(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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": 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