0213_0231_2013_2301
Counting sequence:
1, 1, 2, 6, 20, 65, 202, 608, 1800, 5299, 15606, 46098, 136652, 406389, 1211650, 3619412, 10826512, 32415271, 97116270, 291088486, 872743316, 2617183881, 7849457402, 23544181256, 70624159000, 211855704155, 635533562982, 1906533585658, 5719466545500, 17158131208029, 51473856760946, 154420496549628, 463259342174752, 1389773731567439, 4169312604779230, 12507920634481102, 37523727543718692, 112571113911694321, 337713204296145770, 1013139338010548016, 3039417464275849256, 9118251293315940611, 27354751680924588502, 82064250644727278306, 246192743138088838380, 738578211822080498245, 2215734600281869435362, 6647203730476864159908, 19941611050693104158000, 59824832870604335799159, 179474498048863054014286, 538423493020689255240758, 1615270476810267952080052, 4845811425927204228915289, 14537434268774413432053210, 43612302788308841786728792, 130836908328897728341276472, 392510724914635590985958571, 1177532174599791584882080070, 3532596523511144378494591946, 10597789569956972383180419196, 31793368708717995644934480941, 95380106123848143925589822802, 286140318366932745758342158156, 858420955091574865238171780160, 2575262865256277851640805874335, 7725788595731940066774998609342, 23177365787122033224030157715358, 69532097361218525719500796831300, 208596292083360429253323037770689, 625788876249490991949610407767626, 1877366628747292384228113812111488, 5632099886239515969442906613844552, 16896299658713825541845850196441939, 50688898976132031892571811299025590, 152066696928377206211783955316354482, 456200090785093839703488908787491852, 1368600272355205961246740812039200021, 4105800817065466768012770607470911170, 12317402451196098072583408165119212276, 36952207353587689754840417180770445136, 110856622060761860338701636913136796871, 332569866182283163164465681481061152302, 997709598546844653790118585926484812806, 2993128795640524289963798840746056976340, 8979386386921553527078282688171375824425, 26938159160764621895608620396380537076922, 80814477482293788315573405852874430244008, 242443432446881210204215306886088928557400, 727330297340643321127636099313198061114619, 2181990892021929344412888655249456734012966, 6545972676065786795298626680368095303153818, 19637918028197357910015801470343736111460252, 58913754084592068778287247269510108738139389, 176741262253776196431341427525488127021688370, 530223786761328569486983654010379982679350172, 1590671360283985668846869704898971151266356832, 4772014080851956927312446600432575860255410671, 14316042242555870623481014772769052393678631070, 42948126727667611553530394261249806806860401006, 128844380183002834026765882669634719672229919076
Generating function in Maple syntax:
-(11*x^5-29*x^4+38*x^3-25*x^2+8*x-1)/(2*x-1)/(3*x-1)/(x-1)^4
Generating function in latex syntax:
-\frac{11 x^{5}-29 x^{4}+38 x^{3}-25 x^{2}+8 x -1}{\left(2 x -1\right) \left(3 x -1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-11*x**5 + 29*x**4 - 38*x**3 + 25*x**2 - 8*x + 1)/((x - 1)**4*(2*x - 1)*(3*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(3*x-1)*(x-1)^4*F(x)+11*x^5-29*x^4+38*x^3-25*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(3 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+11 x^{5}-29 x^{4}+38 x^{3}-25 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
-1/6*n^3-4/3*n-1/4+2^n+1/4*3^n
Explicit closed form in latex syntax:
-\frac{n^{3}}{6}-\frac{4 n}{3}-\frac{1}{4}+2^{n}+\frac{3^{n}}{4}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(n+2) = -1/3*n^3+3/2*n^2+5*a(n+1)-6*a(n)-13/6*n+3, n >= 6
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) = 20
a \! \left(5\right) = 65
a \! \left(n +2\right) = -\frac{n^{3}}{3}+\frac{3 n^{2}}{2}+5 a \! \left(n +1\right)-6 a \! \left(n \right)-\frac{13 n}{6}+3, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21351/
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[8,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[22,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = 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[8,x]
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[8,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]+F[19,x]+F[21,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[8,x]
F[20,x] = F[11,x]+F[17,x]
F[21,x] = F[13,x]*F[8,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[8,x]
F[24,x] = F[25,x]+F[5,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[8,x]
F[27,x] = F[28,x]+F[51,x]
F[28,x] = F[29,x]*F[31,x]
F[29,x] = F[30,x]+F[6,x]
F[30,x] = F[1,x]+F[7,x]
F[31,x] = F[32,x]+F[38,x]
F[32,x] = F[16,x]+F[33,x]
F[33,x] = F[11,x]+F[34,x]
F[34,x] = F[18,x]+F[35,x]+F[37,x]
F[35,x] = F[36,x]*F[8,x]
F[36,x] = F[14,x]+F[34,x]
F[37,x] = F[33,x]*F[8,x]
F[38,x] = F[39,x]+F[44,x]
F[39,x] = F[14,x]+F[40,x]
F[40,x] = F[18,x]+F[41,x]+F[43,x]
F[41,x] = F[42,x]*F[8,x]
F[42,x] = F[14,x]+F[40,x]
F[43,x] = F[39,x]*F[8,x]
F[44,x] = F[17,x]+F[45,x]
F[45,x] = F[18,x]+F[46,x]+F[48,x]+F[50,x]
F[46,x] = F[47,x]*F[8,x]
F[47,x] = F[34,x]+F[45,x]
F[48,x] = F[49,x]*F[8,x]
F[49,x] = F[40,x]+F[45,x]
F[50,x] = F[44,x]*F[8,x]
F[51,x] = F[16,x]*F[52,x]
F[52,x] = F[14,x]+F[53,x]
F[53,x] = F[18,x]+F[54,x]+F[56,x]
F[54,x] = F[55,x]*F[8,x]
F[55,x] = F[14,x]+F[53,x]
F[56,x] = F[52,x]*F[8,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{22}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \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) = x
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_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{11}\! \left(x \right)+F_{17}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right) F_{8}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{8}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{5}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{8}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{51}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{31}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{6}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{38}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{11}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{18}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{8}\! \left(x \right)
F_{36}\! \left(x \right) = F_{14}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{33}\! \left(x \right) F_{8}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{44}\! \left(x \right)
F_{39}\! \left(x \right) = F_{14}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right) F_{8}\! \left(x \right)
F_{42}\! \left(x \right) = F_{14}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right) F_{8}\! \left(x \right)
F_{44}\! \left(x \right) = F_{17}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right) F_{8}\! \left(x \right)
F_{47}\! \left(x \right) = F_{34}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right) F_{8}\! \left(x \right)
F_{49}\! \left(x \right) = F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{44}\! \left(x \right) F_{8}\! \left(x \right)
F_{51}\! \left(x \right) = F_{16}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{14}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{18}\! \left(x \right)+F_{54}\! \left(x \right)+F_{56}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right) F_{8}\! \left(x \right)
F_{55}\! \left(x \right) = F_{14}\! \left(x \right)+F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{52}\! \left(x \right) F_{8}\! \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_8(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_22(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_8(x)*F_9(x))
Eq(F_8(x), 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_8(x))
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_8(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_21(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_8(x))
Eq(F_20(x), F_11(x) + F_17(x))
Eq(F_21(x), F_13(x)*F_8(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_8(x))
Eq(F_24(x), F_25(x) + F_5(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_8(x))
Eq(F_27(x), F_28(x) + F_51(x))
Eq(F_28(x), F_29(x)*F_31(x))
Eq(F_29(x), F_30(x) + F_6(x))
Eq(F_30(x), F_1(x) + F_7(x))
Eq(F_31(x), F_32(x) + F_38(x))
Eq(F_32(x), F_16(x) + F_33(x))
Eq(F_33(x), F_11(x) + F_34(x))
Eq(F_34(x), F_18(x) + F_35(x) + F_37(x))
Eq(F_35(x), F_36(x)*F_8(x))
Eq(F_36(x), F_14(x) + F_34(x))
Eq(F_37(x), F_33(x)*F_8(x))
Eq(F_38(x), F_39(x) + F_44(x))
Eq(F_39(x), F_14(x) + F_40(x))
Eq(F_40(x), F_18(x) + F_41(x) + F_43(x))
Eq(F_41(x), F_42(x)*F_8(x))
Eq(F_42(x), F_14(x) + F_40(x))
Eq(F_43(x), F_39(x)*F_8(x))
Eq(F_44(x), F_17(x) + F_45(x))
Eq(F_45(x), F_18(x) + F_46(x) + F_48(x) + F_50(x))
Eq(F_46(x), F_47(x)*F_8(x))
Eq(F_47(x), F_34(x) + F_45(x))
Eq(F_48(x), F_49(x)*F_8(x))
Eq(F_49(x), F_40(x) + F_45(x))
Eq(F_50(x), F_44(x)*F_8(x))
Eq(F_51(x), F_16(x)*F_52(x))
Eq(F_52(x), F_14(x) + F_53(x))
Eq(F_53(x), F_18(x) + F_54(x) + F_56(x))
Eq(F_54(x), F_55(x)*F_8(x))
Eq(F_55(x), F_14(x) + F_53(x))
Eq(F_56(x), F_52(x)*F_8(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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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