0231_0321_1032_2031
Counting sequence:
1, 1, 2, 6, 20, 69, 245, 894, 3339, 12708, 49108, 192164, 759908, 3032102, 12192282, 49357361, 200997189, 822820399, 3384157870, 13977131659, 57946776025, 241063269204, 1005982794393, 4210095858768, 17665816236210, 74306374131205, 313248916563460, 1323285111019057, 5600854305683211, 23748448322225794, 100866054784862643, 429079013341716581, 1827973337122236024, 7798381250591461970, 33312414363249843665, 142476348289225089221, 610077595027529938251, 2615206471040551850676, 11222266439258693385152, 48204305838886447272118, 207252816524943518258394, 891876311556103644427622, 3841315905921905561611158, 16558062773037225598174747, 71429429542462153492047847, 308367145946906688713232721, 1332196693779275525750742162, 5759236092188141840918292667, 24914114254300403129198416545, 107844639167072328372199390128, 467103797261171992083612963931, 2024322848652044676456051410499, 8777848865435215438560160813097, 38082837770406811745445114200382, 165308363531047571546535512343254, 717920860696287409702392165777264, 3119369878405225384579864979203520, 13559961733035698173699937994238892, 58971808803983298733240609630169340, 256577331240083610316906747444012973, 1116795738176093178671272464616657761, 4863007347260453615138828719222625083, 21183913037459637837648063467681040176, 92314975797235131656096991906663937356, 402436845303903028108007940197067952822, 1755003679401502938933577101084567853677, 7656113903125949558357739808765160840523, 33410605546128055407790560073155136793664, 145848395839031612962011429512555926789244, 636877991926470184617744452335583742926658, 2781917211717876128561233319020873828305743, 12155191358630738633376090182878260946678073, 53125796785731089633167092065013255869634142, 232258563517172532173151163998318550388303880, 1015681029048272838205575370718298648988863387, 4442823973168784984161785518624177966217312040, 19439003245258658457641615490320690811189963094, 85074435195814407364528710097320943678312243891, 372418753424474333319507204365571709473701070148, 1630679794163931948282263216196212055068695689231, 7141803778184512648440495157376510113290708902373, 31285759718779835492350237359425305644077495650374, 137082699548627885381241809267017053848146319421637, 600777158278932616494783169643521553405321829394926, 2633520784754543302304289327381473059769960000656492, 11546503571535247585971643408889686021144691332852241, 50635202500482831206901038950204514689344009816258850, 222096083448205485435957409001643551318680834731493436, 974346856830026098732340151570948718793104794705513851, 4275321014966630621899756463911359953224456086454596115, 18763095926173585508737918123700339985270961473174817170, 82360529895506500893851758428029065216714363181274526104, 361585429807718147967134419007488239202122531261823048418, 1587735724228465745340288539568609931540585853218227221453, 6972995993324243619798805099699684104068101564889681267099, 30629010725888156713999940033789251920783524458711082572279, 134560433667281598043056350114187631709155785621451747235159, 591250042837476155994794208120824786195100697322367843124375, 2598322060090402206720826740910648693655903692239571109975837, 11420402122233910259925512864466054445029999829456124727129654, 50203628193875168522765715511041999708338825926474684999650466
Implicit equation for the generating function in Maple syntax:
(x-1)*x^2*F(x)^3+x*(x^2-2*x+2)*F(x)^2+(x-1)*(x+1)*F(x)-x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) x^{2} F \! \left(x \right)^{3}+x \left(x^{2}-2 x +2\right) F \! \left(x \right)^{2}+\left(x -1\right) \left(x +1\right) F \! \left(x \right)-x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 69
a(6) = 245
a(7) = 894
a(8) = 3339
a(9) = 12708
a(10) = 49108
a(n+11) = n*(n+1)/(n+12)/(n+11)*a(n)-1/5*(75*n+134)*(n+1)/(n+12)/(n+11)*a(n+1)+1/25*(2059*n^2+9747*n+11308)/(n+12)/(n+11)*a(n+2)-1/25*(6127*n^2+41355*n+69202)/(n+12)/(n+11)*a(n+3)+2/25*(5878*n^2+52109*n+114923)/(n+12)/(n+11)*a(n+4)-2/25*(7944*n^2+87762*n+241471)/(n+12)/(n+11)*a(n+5)+2/25*(7800*n^2+103044*n+339197)/(n+12)/(n+11)*a(n+6)-2/25*(5520*n^2+84417*n+321847)/(n+12)/(n+11)*a(n+7)+1/25*(5451*n^2+94221*n+406270)/(n+12)/(n+11)*a(n+8)-1/25*(1761*n^2+33831*n+162200)/(n+12)/(n+11)*a(n+9)+1/5*(65*n+658)/(n+12)*a(n+10), n >= 11
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) = 69
a \! \left(6\right) = 245
a \! \left(7\right) = 894
a \! \left(8\right) = 3339
a \! \left(9\right) = 12708
a \! \left(10\right) = 49108
a \! \left(n +11\right) = \frac{n \left(n +1\right) a \! \left(n \right)}{\left(n +12\right) \left(n +11\right)}-\frac{\left(75 n +134\right) \left(n +1\right) a \! \left(n +1\right)}{5 \left(n +12\right) \left(n +11\right)}+\frac{\left(2059 n^{2}+9747 n +11308\right) a \! \left(n +2\right)}{25 \left(n +12\right) \left(n +11\right)}-\frac{\left(6127 n^{2}+41355 n +69202\right) a \! \left(n +3\right)}{25 \left(n +12\right) \left(n +11\right)}+\frac{2 \left(5878 n^{2}+52109 n +114923\right) a \! \left(n +4\right)}{25 \left(n +12\right) \left(n +11\right)}-\frac{2 \left(7944 n^{2}+87762 n +241471\right) a \! \left(n +5\right)}{25 \left(n +12\right) \left(n +11\right)}+\frac{2 \left(7800 n^{2}+103044 n +339197\right) a \! \left(n +6\right)}{25 \left(n +12\right) \left(n +11\right)}-\frac{2 \left(5520 n^{2}+84417 n +321847\right) a \! \left(n +7\right)}{25 \left(n +12\right) \left(n +11\right)}+\frac{\left(5451 n^{2}+94221 n +406270\right) a \! \left(n +8\right)}{25 \left(n +12\right) \left(n +11\right)}-\frac{\left(1761 n^{2}+33831 n +162200\right) a \! \left(n +9\right)}{25 \left(n +12\right) \left(n +11\right)}+\frac{\left(65 n +658\right) a \! \left(n +10\right)}{5 n +60}, \quad n \geq 11
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21421/
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[5,x]
F[4,x] = x
F[5,x] = F[0,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[0,x]*F[4,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[14,x]*F[4,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[11,x]*F[4,x]
F[14,x] = F[0,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[0,x]*F[14,x]*F[4,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_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{0}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{0}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{0}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{0}\! \left(x \right) F_{14}\! \left(x \right) F_{4}\! \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_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_0(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_0(x)*F_4(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_14(x)*F_4(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_11(x)*F_4(x))
Eq(F_14(x), F_0(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_0(x)*F_14(x)*F_4(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, 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]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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, 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]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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, 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]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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"}}, {"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": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "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": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "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, 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]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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]]}, {"patt": [1, 0, 3, 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