0132_0231_0321
Counting sequence:
1, 1, 2, 6, 21, 79, 311, 1265, 5275, 22431, 96900, 424068, 1876143, 8377299, 37704042, 170870106, 779058843, 3571051579, 16447100702, 76073821946, 353224531663, 1645807790529, 7692793487307, 36061795278341, 169498231169821, 798630930874009, 3771457603050116, 17847706407573180, 84625502414415405, 401983252480575399, 1912715749978934415, 9115523248724905737, 43506962747853388467, 207942410378389619811, 995174135193151905486, 4768637990515540843434, 22877004089340657326271, 109871768468179101721725, 528238490089334532793245, 2542187936649978961660155, 12246088719494802290837505, 59044480780490942389595325, 284927791983993824864529780, 1376089377586392463130571540, 6651179907936531441823381125, 32171842670674253932624202085, 155727139281064808573561774820, 754311339340515233886517218480, 3656138001037150263193410229365, 17732441651483971033653826787781, 86055297645519796258217673160170, 417868398378647487718289709643038, 2030227074808659828721582587555153, 9869240661169876791887212834148719, 48000777768179723225429924670681485, 233576992353654588696599325919826051, 1137159486478835403382899437436398707, 5538790083898908201944227711162943943, 26990050808779317695167376068124198964, 131577362522211780997425276513324515588, 641713339928021253862267081624847367991, 3130959277162925963491891514058694365451, 15282153677570751079870602250956622554122, 74620318548555904589087098053331681690026, 364493576802522448560823412355814744878963, 1781057719383880972834464601270434860031219, 8705961729350616939798325308187988375279622, 42569806782682080331943117346865820271360786, 208222898316056683274841717486189071758432143, 1018809723894112066551148150885440545720708109, 4986450809701191168937664594133159584862700317, 24412941114795738324605337486190836119954651771, 119557045322500868313492107710118051009383835473, 585670427334078528184029116132298678703625155549, 2869796854270774061046729657214609862004388770092, 14065836692191797555770724454767428354792864466156, 68959401814240990895597171629534435185133060227693, 338167573503049256153721158324730482446792311769129, 1658739445475723852425259422761297445961055267678902, 8138219526608677698666575522865163866262534080312566, 39937698029278924355069789033957645680018057427414113, 196036318038258395947674181335188850988178872746495489, 962470653551175275218457702054277296165177840916987642, 4726433658189839304980428749539435785965683842076016846, 23215203197099639578168763768724081294209716154772663453, 114051785099032128630479458452411563114553007426597936899, 560428557620613099566397011225614659686996198402689753977, 2754387064994333595666808980797718246896441940910195848471, 13539862822003262010540122110762144672403457466867555477623, 66571178419198128705135577718797387642615062160674497608299, 327370151363652222278247203093406558414754300223268954808812, 1610167274931863320673947156045029476630069131301740955867556, 7921004867777184423434418685702918429906506087791184337871103, 38973128839888361926405904209509816970335613079495334132387489, 191789297554919912951787510743305992723456512642767781444355507, 943965257201848988977947939070218917266487620319910191132725341, 4646850402350783479943018347876957531689237868951049078839638333, 22878679303211958700577794457643695315535025691013722739626110029, 112660421654550013278942792070877045703242369455154546573343437202, 554853823069796897398875430926014443034379114521384831568535766646, 2733073112795720153237297124938915907723365837935699807314396095313
Generating function in Maple syntax:
1/2*(-x-1+(5*x^2-6*x+1)^(1/2))/x/(x-2)
Generating function in latex syntax:
\frac{-x -1+\sqrt{5 x^{2}-6 x +1}}{2 x \left(x -2\right)}
Generating function in sympy syntax:
(-x/2 + sqrt(5*x**2 - 6*x + 1)/2 - 1/2)/(x*(x - 2))
Implicit equation for the generating function in Maple syntax:
x*(x-2)*F(x)^2+(x+1)*F(x)-1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -2\right) F \! \left(x \right)^{2}+\left(x +1\right) F \! \left(x \right)-1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(n+3) = 5/2*(1+n)/(n+4)*a(n)-1/2*(25+16*n)/(n+4)*a(1+n)+1/2*(34+13*n)/(n+4)*a(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 +3\right) = \frac{5 \left(1+n \right) a \! \left(n \right)}{2 \left(n +4\right)}-\frac{\left(25+16 n \right) a \! \left(1+n \right)}{2 \left(n +4\right)}+\frac{\left(34+13 n \right) a \! \left(n +2\right)}{2 n +8}, \quad n \geq 3
Specification 1
Strategy pack name: point_and_row_placements
Tree: http://permpal.com/tree/262/
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[9,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[4,x]*F[7,x]*F[9,x]
F[7,x] = F[1,x]+F[10,x]+F[8,x]
F[8,x] = F[7,x]*F[9,x]
F[9,x] = x
F[10,x] = F[11,x]*F[9,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[7,x]*F[9,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_{9}\! \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_{4}\! \left(x \right) F_{7}\! \left(x \right) F_{9}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{7}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{9}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right) F_{7}\! \left(x \right) F_{9}\! \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_9(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_4(x)*F_7(x)*F_9(x))
Eq(F_7(x), F_1(x) + F_10(x) + F_8(x))
Eq(F_8(x), F_7(x)*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_11(x)*F_9(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_7(x)*F_9(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"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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_and_row_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": [0, 3, 2, 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], "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": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 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], [0, 1], [2, 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": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 3, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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]]}]]}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"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": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [0, 1], "pos": [[0, 0], [0, 2]]}, {"patt": [0, 1], "pos": [[0, 0], [2, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[2, 0], [2, 0], [2, 2]]}, {"patt": [0, 1, 2], "pos": [[2, 0], [2, 2], [2, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 2], [2, 2]]}, {"patt": [0, 2, 1], "pos": [[2, 2], [2, 2], [2, 2]]}, {"patt": [1, 2, 0], "pos": [[2, 2], [2, 2], [2, 2]]}, {"patt": [2, 1, 0], "pos": [[2, 2], [2, 2], [2, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [2, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 2], [0, 2], [2, 2], [2, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [2, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 2], [0, 2], [2, 2], [2, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 2], [0, 2], [0, 2], [2, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 2], [0, 2], [2, 2], [2, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[2, 0], [2, 2], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}], "class_module": 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