0132_0213_0231_0312_1032_1320_3021
Counting sequence:
1, 1, 2, 6, 17, 48, 147, 472, 1563, 5292, 18227, 63649, 224809, 801687, 2882453, 10437746, 38032111, 139339636, 512996371, 1896907981, 7041759121, 26233530211, 98046684061, 367525880671, 1381386710965, 5205006015103, 19657257219477, 74395497317457, 282115304822513, 1071775767567373, 4078734453842665, 15546878255878278, 59349051379140503, 226880766451470020, 868478107026778963, 3328621625319544373, 12772742428010184257, 49067089819529043627, 188693661172867154893, 726376176054597542755, 2798860905063498810221, 10794343963367873934411, 41666556954180936068061, 160967397796723249794721, 622344549504901137047281, 2407971333282333424914421, 9323628831126170083134361, 36125891611268294959683211, 140068226953702538868038221, 543421018936808184941038063, 2109589555998330019101766357, 8194337915693933523285424729, 31847428603528786603766191969, 123842657126296750635784662037, 481828884973726097397751926937, 1875572488064601305243033721853, 7304416645872826968997267973737, 28460367473383670775237700788637, 110940917216967177531047977276393, 432645947984351837634888664279777, 1687939424885242346720271097850977, 6588072953712432636217724290583449, 25723588707641746868398231325312209, 100477978517356835438422465793596462, 392618907618862286304720501259820263, 1534714423709691648660425279190245476, 6001160754436129394472924748562692051, 23474143595107437132231042508300815301, 91851605583520598461179556369550253601, 359519227532172493020142346069115574651, 1407641032419242556216127817853930653101, 5513055427156321534975723987450124341051, 21598312498129682133418940745444870570301, 84639059289723483400627177826344334600851, 331773768700812721000465290087063775052941, 1300859143795456189669968959640311399675521, 5101905808674407232465922499913673595051761, 20014538381240263387040122001095507324041061, 78535652541895762371018538063042193700360121, 308243223902074784029495207308578312457642271, 1210104821558868730391323028859314785906949541, 4751740131163585381917525143535623950764046971, 18662944593564113106666773394920323093063848221, 73316731182768515190351293332174780326484860841, 288084036055011894863377603807084222057067514721, 1132208144055054030680924710078145105866088610621, 4450637524595919852543338291460076024849422594761, 17498668537587700633042181618806362602090693799461, 68813340659319439105011333314317088921110286575321, 270659473138666223321022585739604533229914962421541, 1064767861386117696299895888022688045825142625211801, 4189536391888005481941807013542191156860579052050401, 16487492423292244189057726214314885096376392217786401, 64896188768924420436880417153592998401743760587461801, 255480703340568894576044103587255066238294758616065201, 1005934674512768623604068434035769245729200487142342051, 3961436549924031390732786318002723382715787524866923901, 15602903154107003644869483717111972358550185709134659151, 61464801274007312167751571149422102993730238322135305301, 242166760150358925098836418242345886717754637552440802441, 954263305159732854469357770682143488754858253163502520321
Generating function in Maple syntax:
1/2*((-x^3+x^2-x+1)*(1-4*x)^(1/2)-2*x^5-2*x^4+3*x^3-x^2+x-1)/(x-1)/x
Generating function in latex syntax:
\frac{\left(-x^{3}+x^{2}-x +1\right) \sqrt{1-4 x}-2 x^{5}-2 x^{4}+3 x^{3}-x^{2}+x -1}{2 \left(x -1\right) x}
Generating function in sympy syntax:
(-x**5 - x**4 + 3*x**3/2 - x**2/2 + x/2 + sqrt(1 - 4*x)*(-x**3 + x**2 - x + 1)/2 - 1/2)/(x*(x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)^2*x*F(x)^2+(x-1)*(2*x^5+2*x^4-3*x^3+x^2-x+1)*F(x)+x^9+2*x^8-2*x^7-x^6+2*x^4-2*x^3+2*x^2-2*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} x F \! \left(x \right)^{2}+\left(x -1\right) \left(2 x^{5}+2 x^{4}-3 x^{3}+x^{2}-x +1\right) F \! \left(x \right)+x^{9}+2 x^{8}-2 x^{7}-x^{6}+2 x^{4}-2 x^{3}+2 x^{2}-2 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 17
a(5) = 48
a(6) = 147
a(7) = 472
a(8) = 1563
a(n+3) = 2*(-3+2*n)/(n+4)*a(n)-n/(n+4)*a(n+1)+2*(5+2*n)/(n+4)*a(n+2)-6*n/(n+4), n >= 9
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) = 17
a \! \left(5\right) = 48
a \! \left(6\right) = 147
a \! \left(7\right) = 472
a \! \left(8\right) = 1563
a \! \left(n +3\right) = \frac{2 \left(-3+2 n \right) a \! \left(n \right)}{n +4}-\frac{n a \! \left(n +1\right)}{n +4}+\frac{2 \left(5+2 n \right) a \! \left(n +2\right)}{n +4}-\frac{6 n}{n +4}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/22782/
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[5,x]+F[9,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[5,x]^2*F[8,x]
F[8,x] = x
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[8,x]
F[11,x] = F[12,x]+F[15,x]
F[12,x] = F[5,x]^2*F[13,x]
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[1,x]+F[8,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[8,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[8,x]
F[19,x] = F[1,x]+F[18,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_{5}\! \left(x \right)+F_{9}\! \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_{5} \left(x \right)^{2} F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{15}\! \left(x \right)
F_{12}\! \left(x \right) = F_{5} \left(x \right)^{2} F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{8}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \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_5(x) + F_9(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_5(x)**2*F_8(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_8(x))
Eq(F_11(x), F_12(x) + F_15(x))
Eq(F_12(x), F_13(x)*F_5(x)**2)
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_1(x) + F_8(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_8(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_8(x))
Eq(F_19(x), F_1(x) + F_18(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, 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": [0, 3, 1, 2], "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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 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, 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": [0, 3, 1, 2], "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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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": [0, 3, 1, 2], "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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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"}}, {"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, 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, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [0, 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, 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, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 2], [0, 0]]}], "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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"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, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [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, 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, 0], [0, 0]]}, 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