0132_0231_1302
Counting sequence:
1, 1, 2, 6, 21, 77, 287, 1082, 4128, 15945, 62330, 246328, 982977, 3956136, 16041373, 65473465, 268790735, 1109203358, 4598545442, 19144360453, 80001660884, 335463135505, 1411066594333, 5952407116432, 25175573374242, 106737889289463, 453556262998468, 1931285171749227, 8239503725645011, 35215854610857967, 150767113659527948, 646486955254423492, 2776230289080192535, 11938654194674705020, 51407246277093222537, 221630880476383779905, 956629312597757195223, 4133685709017960652628, 17880784470631970725954, 77422629643336550430195, 335552210320555080284190, 1455604119325535467830175, 6319736725698844576106833, 27460602461721739231101839, 119415440566837220447103454, 519679160409795414382388974, 2263187413351482610142354401, 9862863714800912591526831129, 43010109339088337084966089439, 187677199942505807924227339650, 819436072484429199249256666616, 3579900934861940234027328858817, 15648390178684641954779477779218, 68438731559497207847107363978729, 299474099566815954270603904905791, 1311092952311633186038185161895517, 5742710166909484610485453030173862, 25165306085708656316128721651060238, 110327180205482345125754280875064191, 483895106671086670267459107085989954, 2123254767237667803806672670610662311, 9320285394629436852261363232041265037, 40928599646145859949456670134461187743, 179800024076080116120723852227026290250, 790155659684024297098524675299009814514, 3473687253212471664531563215458354778385, 15276337576541714110709505502180284620956, 67203792310091160975264083086788558505285, 295739983169801963524672443997781733325633, 1301858990574112072787823025834746577302615, 5732599152112292033472241555663460868239852, 25250453321804268434943711185118587833282096, 111253367477873385576075730052089060250897583, 490320538739380905725302538118904955705825068, 2161556699274044145390065557204868842549304681, 9531683825743642476001986506121849490850113967, 42042256953156603061405213819011098903118178499, 185486745508669057332762627249520422259919656422, 818554092853838865213584800404834738578046363808, 3613156867625243370291139457405272712816750213245, 15952493150315580783486591976888254425742876804098, 70448234997516795427416143073699116596152104845383, 311178112608014934748172025911441522574235094931039, 1374811023943189835252696080668027540542478234452959, 6075328322962988099223756254823724623378053024189488, 26852647372788975529184609943127934322501755442689516, 118711549433417755751044811010096829704059287067341735, 524910571993791315226121935057483719405970452591671584, 2321465261094731218345469999511503992670475806402686777, 10268847134622344711448930918219701943813534285885068211, 45432013940718045502322296669824801721078810960752866495, 201039464893305142292775426932654200408374446848646949702, 889770430036066912989000237912886703909420263150173932096, 3938676199608616733684171161958665251841511998275150133057, 17437998877406049702046954234974588718726990678000198545642, 77217460486818392286488889168137154922676330145222350924160, 341983679399294443676753321679689050023128881224087217656769, 1514833093482371901427107782005662840843669390682107917358352, 6711078378278742858383112130871406413927551090040784993600653, 29736276654755599848221864237222712502294149573619253864931121, 131779023894151760527514689786689594064297163811992252268080695
Generating function in Maple syntax:
-1/2*(2*x-1+(4*x^4-20*x^3+20*x^2-8*x+1)^(1/2))/x/(x^3-5*x^2+4*x-1)*(2*x-1)
Generating function in latex syntax:
-\frac{\left(2 x -1+\sqrt{4 x^{4}-20 x^{3}+20 x^{2}-8 x +1}\right) \left(2 x -1\right)}{2 x \left(x^{3}-5 x^{2}+4 x -1\right)}
Generating function in sympy syntax:
(2*x - 1)*(-x - sqrt(4*x**4 - 20*x**3 + 20*x**2 - 8*x + 1)/2 + 1/2)/(x*(x**3 - 5*x**2 + 4*x - 1))
Implicit equation for the generating function in Maple syntax:
x*(x^3-5*x^2+4*x-1)*F(x)^2+(2*x-1)^2*F(x)-(2*x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)^{2}+\left(2 x -1\right)^{2} F \! \left(x \right)-\left(2 x -1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 77
a(6) = 287
a(7) = 1082
a(n+8) = -8*(1+n)/(9+n)*a(n)+4*(21*n+38)/(9+n)*a(1+n)-2*(156*n+421)/(9+n)*a(n+2)+2*(260*n+957)/(9+n)*a(n+3)-6*(79*n+371)/(9+n)*a(n+4)+3*(85*n+487)/(9+n)*a(n+5)-9*(9*n+61)/(9+n)*a(n+6)+2*(7*n+55)/(9+n)*a(n+7), 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) = 21
a \! \left(5\right) = 77
a \! \left(6\right) = 287
a \! \left(7\right) = 1082
a \! \left(n +8\right) = -\frac{8 \left(1+n \right) a \! \left(n \right)}{9+n}+\frac{4 \left(21 n +38\right) a \! \left(1+n \right)}{9+n}-\frac{2 \left(156 n +421\right) a \! \left(n +2\right)}{9+n}+\frac{2 \left(260 n +957\right) a \! \left(n +3\right)}{9+n}-\frac{6 \left(79 n +371\right) a \! \left(n +4\right)}{9+n}+\frac{3 \left(85 n +487\right) a \! \left(n +5\right)}{9+n}-\frac{9 \left(9 n +61\right) a \! \left(n +6\right)}{9+n}+\frac{2 \left(7 n +55\right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/267/
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[19,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[19,x]*F[7,x]
F[7,x] = F[8,x]+F[9,x]
F[8,x] = F[0,x]*F[4,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[16,x]*F[20,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[19,x]
F[13,x] = F[14,x]+F[15,x]
F[14,x] = F[0,x]^2
F[15,x] = F[11,x]*F[16,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[16,x]*F[19,x]
F[19,x] = x
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[19,x]*F[23,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[19,x]*F[24,x]
F[27,x] = F[17,x]+F[28,x]
F[28,x] = F[29,x]+F[30,x]+F[32,x]
F[29,x] = 0
F[30,x] = F[19,x]*F[31,x]
F[31,x] = F[25,x]+F[28,x]
F[32,x] = F[19,x]*F[27,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_{19}\! \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_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{19}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)+F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = F_{0}\! \left(x \right) F_{4}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{0} \left(x \right)^{2}
F_{15}\! \left(x \right) = F_{11}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = x
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{19}\! \left(x \right) F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{17}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = 0
F_{30}\! \left(x \right) = F_{19}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{32}\! \left(x \right) = F_{19}\! \left(x \right) F_{27}\! \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_19(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_19(x)*F_7(x))
Eq(F_7(x), F_8(x) + F_9(x))
Eq(F_8(x), F_0(x)*F_4(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_16(x)*F_20(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_19(x))
Eq(F_13(x), F_14(x) + F_15(x))
Eq(F_14(x), F_0(x)**2)
Eq(F_15(x), F_11(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_16(x)*F_19(x))
Eq(F_19(x), x)
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_19(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_19(x)*F_24(x))
Eq(F_27(x), F_17(x) + F_28(x))
Eq(F_28(x), F_29(x) + F_30(x) + F_32(x))
Eq(F_29(x), 0)
Eq(F_30(x), F_19(x)*F_31(x))
Eq(F_31(x), F_25(x) + F_28(x))
Eq(F_32(x), F_19(x)*F_27(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": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "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": [0, 2, 3, 1], "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, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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": [{"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[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": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 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