0231_0312_1203
Counting sequence:
1, 1, 2, 6, 21, 76, 276, 1002, 3641, 13261, 48451, 177651, 653753, 2414426, 8947576, 33266626, 124062001, 463991881, 1739969551, 6541168951, 24647883001, 93078189751, 352207870015, 1335293573131, 5071418015121, 19293438101001, 73514652074501, 280531912316293, 1072007803888561, 4101918749601366, 15715143261459776, 60278571820325426, 231469715790049633, 889789498870352081, 3423885943288280471, 13187694661947683151, 50841248199001927801, 196174587693577438555, 757584789873571409051, 2927962836538397607951, 11324801179267241666001, 43834108061009917917001, 169784111889645082065181, 658070974503343345650301, 2552275268894425863373201, 9904905476668774789544251, 38461905682129398208749751, 149437324210300088507062363, 580932820459478604937940641, 2259552942425451562414101361, 8793071591985837448073743501, 34235156288978872785357409501, 133355270606378794646090389297, 519693333980740890900205193581, 2026172796046370356019828345821, 7902996689849050282588725395701, 30838058646642621818312314538401, 120380567116984925534393707843537, 470104439502511389528340767749641, 1836520059924072508157405953466761, 7177204831887179916936988783663201, 28058765288219954961288090154776526, 109731355942546346940914346420679888, 429276914082241670223398151049604626, 1679905034983370387855455065831096001, 6576098841426498927249295390753380001, 25750430517651046071103898527379403751, 100862455535291635841585424078012249151, 395184524888594629545776437488754779001, 1548789101895474723505897455879435707251, 6071602724191365455162120163459673421251, 23808388120595283811840182121580411523751, 93383300605580587476840577190015968399441, 366368508862146226257360085083883997593801, 1437718845383270796979640333889787202678701, 5643311590150351653054963375910913940967501, 22156224361466108645384454225779558120610001, 87007493067477408650424751744636324739635471, 341754578033074032028471490770842484113772851, 1342661397321576566122626512521621580761205151, 5276074443599151659277049502401923542564884001, 20736976135918184686145871778427813417296158001, 81520712656671310739713447703078280628111896421, 320535790936868676594997320210639406604074769301, 1260576415829098344624228317329646648680112381201, 4958428316918126429498094134293640226804439856501, 19507396155323158713803898432028430491007515390501, 76759691381377480209532673281165988841870356638541, 302095386474115306728035565686163104565500509132001, 1189132589486941136153609261932565563773195782198001, 4681555722540230132804165496024836567054297177715001, 18434115301448752215495436262257968705701967203955001, 72598073734547647321656412844766119562385571893751201, 285954637958537608097321104165339076710358004677653251, 1126515155991746767174968706637904592204095250162211751, 4438573540889140488177872092974232379421665616998576251, 17491013205618585856700937609451004708591373469436040001, 68936685941554851752273316418217317715692442678170969441, 271736812777044805483899149495076926886733205358879378301, 1071292310408470758308567841641901213881535820520360726701, 4224037443457215789645432940097847068801775872469819390001
Generating function in Maple syntax:
-1/2*(-3*x+1+(x-1)*(-4*x+1)^(1/2))/x^3*(x^2-3*x+1)/(x-1)
Generating function in latex syntax:
-\frac{\left(-3 x +1+\left(x -1\right) \sqrt{-4 x +1}\right) \left(x^{2}-3 x +1\right)}{2 x^{3} \left(x -1\right)}
Generating function in sympy syntax:
(3*x/2 - sqrt(1 - 4*x)*(x - 1)/2 - 1/2)*(x**2 - 3*x + 1)/(x**3*(x - 1))
Implicit equation for the generating function in Maple syntax:
x^3*(x-1)^2*F(x)^2-(x-1)*(3*x-1)*(x^2-3*x+1)*F(x)+(x^2-3*x+1)^2 = 0
Implicit equation for the generating function in latex syntax:
x^{3} \left(x -1\right)^{2} F \! \left(x \right)^{2}-\left(x -1\right) \left(3 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+\left(x^{2}-3 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(n+3) = 2*(1+2*n)/(6+n)*a(n)-(32+13*n)/(6+n)*a(n+1)+(30+7*n)/(6+n)*a(n+2)+3*(n+2)/(6+n), 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 +3\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{6+n}-\frac{\left(32+13 n \right) a \! \left(n +1\right)}{6+n}+\frac{\left(30+7 n \right) a \! \left(n +2\right)}{6+n}+\frac{3 n +6}{6+n}, \quad n \geq 5
Specification 1
Strategy pack name: row_and_col_placements_tracked_fusion
Tree: http://permpal.com/tree/341/
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[4,x]
F[3,x] = x
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[7,x,1]
F[7,x,k[0]] = F[12,x,k[0]]*F[8,x,k[0]]
F[8,x,k[0]] = F[16,x,k[0]]+F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]+F[13,x,k[0]]
F[10,x,k[0]] = F[0,x]+F[11,x,k[0]]
F[11,x,k[0]] = F[12,x,k[0]]*F[9,x,k[0]]
F[12,x,k[0]] = k[0]*x
F[13,x,k[0]] = F[14,x,k[0]]
F[14,x,k[0]] = F[15,x,k[0]]*F[3,x]
F[15,x,k[0]] = 1/(-1+k[0])*(F[8,x,k[0]]*k[0]-F[8,x,1])
F[16,x,k[0]] = F[17,x,k[0]]
F[17,x,k[0]] = F[0,x]*F[18,x]*F[21,x]*F[24,x]*F[26,x,k[0]]*F[3,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[18,x]*F[21,x]*F[3,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[21,x]*F[24,x]*F[3,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[24,x]*F[3,x]
F[26,x,k[0]] = F[1,x]+F[27,x,k[0]]
F[27,x,k[0]] = F[12,x,k[0]]*F[26,x,k[0]]
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_{4}\! \left(x \right)
F_{3}\! \left(x \right) = x
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_{7}\! \left(x , 1\right)
F_{7}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = F_{16}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right)+F_{13}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{0}\! \left(x \right)+F_{11}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{9}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = y x
F_{13}\! \left(x , y\right) = F_{14}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{3}\! \left(x \right)
F_{15}\! \left(x , y\right) = \frac{y F_{8}\! \left(x , y\right)-F_{8}\! \left(x , 1\right)}{-1+y}
F_{16}\! \left(x , y\right) = F_{17}\! \left(x , y\right)
F_{17}\! \left(x , y\right) = F_{0}\! \left(x \right) F_{18}\! \left(x \right) F_{21}\! \left(x \right) F_{24}\! \left(x \right) F_{26}\! \left(x , y\right) F_{3}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right) F_{21}\! \left(x \right) F_{3}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{21}\! \left(x \right) F_{24}\! \left(x \right) F_{3}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{24}\! \left(x \right) F_{3}\! \left(x \right)
F_{26}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{27}\! \left(x , y\right)
F_{27}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{26}\! \left(x , y\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)*F_4(x))
Eq(F_3(x), x)
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_7(x, 1))
Eq(F_7(x, k_0), F_12(x, k_0)*F_8(x, k_0))
Eq(F_8(x, k_0), F_16(x, k_0) + F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0) + F_13(x, k_0))
Eq(F_10(x, k_0), F_0(x) + F_11(x, k_0))
Eq(F_11(x, k_0), F_12(x, k_0)*F_9(x, k_0))
Eq(F_12(x, k_0), k_0*x)
Eq(F_13(x, k_0), F_14(x, k_0))
Eq(F_14(x, k_0), F_15(x, k_0)*F_3(x))
Eq(F_15(x, k_0), (k_0*F_8(x, k_0) - F_8(x, 1))/(k_0 - 1))
Eq(F_16(x, k_0), F_17(x, k_0))
Eq(F_17(x, k_0), F_0(x)*F_18(x)*F_21(x)*F_24(x)*F_26(x, k_0)*F_3(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_18(x)*F_21(x)*F_3(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_21(x)*F_24(x)*F_3(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_24(x)*F_3(x))
Eq(F_26(x, k_0), F_1(x) + F_27(x, k_0))
Eq(F_27(x, k_0), F_12(x, k_0)*F_26(x, k_0))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "row_and_col_placements_tracked_fusion", "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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 2], [1, 0], [1, 2]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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, 1, 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_placement", "direction": 2, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "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], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[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, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}], "requirements": []}], "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": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 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