0132_0213_0231_1032_1203
Counting sequence:
1, 1, 2, 6, 19, 59, 180, 544, 1637, 4917, 14758, 44282, 132855, 398575, 1195736, 3587220, 10761673, 32285033, 96855114, 290565358, 871696091, 2615088291, 7845264892, 23535794696, 70607384109, 211822152349, 635466457070, 1906399371234, 5719198113727, 17157594341207, 51472783023648, 154418349070972, 463255047212945, 1389765141638865, 4169295424916626, 12507886274749910, 37523658824249763, 112570976472749323, 337712929418248004, 1013138788254744048, 3039416364764232181, 9118249094292696581, 27354747282878089782, 82064241848634269386, 246192725545902808199, 738578176637708424639, 2215734529913125273960, 6647203589739375821924, 19941610769218127465817, 59824832307654382397497, 179474496922963147192538, 538423490768889441577662, 1615270472306668324733035, 4845811416920004974199155, 14537434250760014922597516, 43612302752280044767792600, 130836908256840134303377853, 392510724770520402910133613, 1177532174311561208730400894, 3532596522934683626191202738, 10597789568804050878573608271, 31793368706412152635720824871, 95380106119236457907162474672, 286140318357709373721487424076, 858420955073128121164462272289, 2575262865219384363493386816929, 7725788595658153090480160450850, 23177365786974459271440481352614, 69532097360923377814321444057907, 208596292082770133442964332173787, 625788876248310400328892996521428, 1877366628744931200986678989564352, 5632099886234793602960036968693125, 16896299658704380808880110906079445, 50688898976113142426640332718238406, 152066696928339427279920998154715290, 456200090785018281839762994464145943, 1368600272355054845519288983392437903, 4105800817065164536557866950177313784, 12317402451195493609673600850531941428, 36952207353586480829020802551595824361, 110856622060759442487062407654787473161, 332569866182278327461187222964362419562, 997709598546834982383561668893087258766, 2993128795640504947150685006679261776379, 8979386386921514841452055020037785329219, 26938159160764544524356165060113355987740, 80814477482293633573068495180340067963304, 242443432446880900719205485541020203889997, 727330297340642702157616456623060611670077, 2181990892021928106472849369869181835010318, 6545972676065784319418548109607545505031042, 19637918028197352958255644328822636515093215, 58913754084592058874766932986467909545279735, 176741262253776176624300798959403728635839296, 530223786761328529872902396878211185907517980, 1590671360283985589618707190634633557722554033, 4772014080851956768856121571903900673167662193, 14316042242555870306568364715711702019502986674, 42948126727667610919705094147135106058508960118, 128844380183002832759115282441405318175526880451
Generating function in Maple syntax:
-(2*x-1)^2/(3*x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{\left(2 x -1\right)^{2}}{\left(3 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
-(2*x - 1)**2/((x - 1)**2*(3*x - 1))
Implicit equation for the generating function in Maple syntax:
(3*x-1)*(x-1)^2*F(x)+(2*x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(3 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(2 x -1\right)^{2} = 0
Explicit closed form in Maple syntax:
3/4-1/2*n+1/4*3^n
Explicit closed form in latex syntax:
\frac{3}{4}-\frac{n}{2}+\frac{3^{n}}{4}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(n+1) = 3*a(n)+n-2, n >= 3
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(n +1\right) = 3 a \! \left(n \right)+n -2, \quad n \geq 3
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18146/
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[12,x]*F[4,x]
F[4,x] = F[19,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[13,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[47,x]
F[21,x] = 0
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[18,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[21,x]+F[22,x]+F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[2,x]+F[27,x]
F[30,x] = F[31,x]
F[31,x] = F[12,x]*F[32,x]
F[32,x] = F[33,x]+F[46,x]
F[33,x] = F[21,x]+F[34,x]+F[44,x]
F[34,x] = F[12,x]*F[35,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[12,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[12,x]*F[14,x]
F[39,x] = F[40,x]+F[42,x]
F[40,x] = F[21,x]+F[34,x]+F[41,x]
F[41,x] = F[12,x]*F[2,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[33,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[2,x]+F[33,x]
F[46,x] = F[31,x]
F[47,x] = F[12,x]*F[48,x]
F[48,x] = F[29,x]+F[32,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{19}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \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) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
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_{12}\! \left(x \right) F_{16}\! \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_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{47}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{18}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{12}\! \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_{12}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{46}\! \left(x \right)
F_{33}\! \left(x \right) = F_{21}\! \left(x \right)+F_{34}\! \left(x \right)+F_{44}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{21}\! \left(x \right)+F_{34}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{2}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{2}\! \left(x \right)+F_{33}\! \left(x \right)
F_{46}\! \left(x \right) = F_{31}\! \left(x \right)
F_{47}\! \left(x \right) = F_{12}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_19(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_13(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_47(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_18(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_21(x) + F_22(x) + F_28(x))
Eq(F_28(x), F_12(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_12(x)*F_32(x))
Eq(F_32(x), F_33(x) + F_46(x))
Eq(F_33(x), F_21(x) + F_34(x) + F_44(x))
Eq(F_34(x), F_12(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_12(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_12(x)*F_14(x))
Eq(F_39(x), F_40(x) + F_42(x))
Eq(F_40(x), F_21(x) + F_34(x) + F_41(x))
Eq(F_41(x), F_12(x)*F_2(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_33(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_2(x) + F_33(x))
Eq(F_46(x), F_31(x))
Eq(F_47(x), F_12(x)*F_48(x))
Eq(F_48(x), F_29(x) + F_32(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, 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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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, 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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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": [{"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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 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