0213_0231_1032_1320
Counting sequence:
1, 1, 2, 6, 20, 66, 215, 701, 2310, 7727, 26259, 90573, 316559, 1119121, 3995385, 14384390, 52163792, 190359898, 698503927, 2575538142, 9537605769, 35455811060, 132265558289, 494966966584, 1857608397685, 6989965624818, 26366136220109, 99675222201492, 377594090503163, 1433165339749858, 5449330657441525, 20754593030026229, 79170421492600454, 302445576496798359, 1156987801290568363, 4431723859823125953, 16996013701683513059, 65256335309204274373, 250826052527815466657, 965099751372667155291, 3717050870455199689439, 14329478901068474024355, 55290214348016941536017, 213517718322560103561011, 825221219745480078771109, 3191845599685486821621655, 12354740604352175007875037, 47855500308811245636820101, 185491341660594900823198319, 719443772986052033907981017, 2792158876717695140091934705, 10842832411108948258420932205, 42130308301741036640792143857, 163789592522104145375459783925, 637101716624595031796183903789, 2479439892873637424689922440217, 9654120558945889092247681510897, 37607857229150381934162296431177, 146569636098715671311645075929341, 571481476268607204696452771803785, 2229187291687559771561994607905173, 8699034572945406203310552440106401, 33960165577491917310179547270576517, 132628608240016688421751359878616166, 518164895862674808959841693855964416, 2025149375789658905188014065114187930, 7917710295743253979918750004425185047, 30966420680203306267255407407552521670, 121150960669038551454774211865784951505, 474136022008432648482595202832145911964, 1856159299958455881926341197581276771121, 7268752539534388372990804089894142868124, 28472996037484003080505546911439441031369, 111565920205842653862669914084215310111798, 437271966634955469693322045911717996168085, 1714313146202509619582447028333162223680088, 6722703805106397933546902574983435139693963, 26369997545574788361683279639529050638497342, 103463126395431974386705100896047736015305269, 406038970188717348742462309768631295542032322, 1593873208240275474823063297777921961914956117, 6258080956916753028389952354466386519245206376, 24576919829564822267592383773949280624986999733, 96540646714009916318606645871959647885015305966, 379303737344325468050156398845525030901624423391, 1490582372551647583280397646333424154814851372880, 5858880404914256482868780715503474622765082854237, 23033556702467234334673220683664113030737537940790, 90571819462202907375201405368048741726913810347075, 356212341435903402534293593554981133443164415740624, 1401221343732077189826115136880081223371113321896641, 5512957315049902408431454730679070102044883519063194, 21694061706055616200921348612766056636713601937943219, 85383465072270218824023040156986898563210676774179788, 336110210179526140522550920406783263577537999146740045, 1323314511085234613627843172777677743601867481304874528, 5210943301633012377642011216232979417748547089592036813, 20522961755849574424673324581909744179405104152858964602, 80841189745405673487856342613314626081219527051135061749, 318487895307731228158196834782429339400464585945878916452, 1254929656218371432745956003176740531537037808768094030667
Generating function in Maple syntax:
1/4*((x^3+2*x^2-3*x+1)*(1-4*x)^(1/2)+2*x^4-5*x^3+3*x-1)/x/(x-1/2)/(x-1)^2
Generating function in latex syntax:
\frac{\left(x^{3}+2 x^{2}-3 x +1\right) \sqrt{1-4 x}+2 x^{4}-5 x^{3}+3 x -1}{4 x \left(x -\frac{1}{2}\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(x**4/2 - 5*x**3/4 + 3*x/4 + sqrt(1 - 4*x)*(x**3 + 2*x**2 - 3*x + 1)/4 - 1/4)/(x*(x - 1)**2*(x - 1/2))
Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(x-1)^4*F(x)^2-(x^3-2*x^2-x+1)*(x-1)^2*(2*x-1)^2*F(x)+x^7-4*x^6+10*x^5-18*x^3+18*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)^{2}-\left(x^{3}-2 x^{2}-x +1\right) \left(x -1\right)^{2} \left(2 x -1\right)^{2} F \! \left(x \right)+x^{7}-4 x^{6}+10 x^{5}-18 x^{3}+18 x^{2}-7 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(6) = 215
a(7) = 701
a(n+6) = 4*(1+2*n)/(7+n)*a(n)+2*(19+n)/(7+n)*a(1+n)-(166+45*n)/(7+n)*a(n+2)+(257+63*n)/(7+n)*a(n+3)-(182+37*n)/(7+n)*a(n+4)+(59+10*n)/(7+n)*a(n+5)+1/(7+n), n >= 8
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) = 66
a \! \left(6\right) = 215
a \! \left(7\right) = 701
a \! \left(n +6\right) = \frac{4 \left(1+2 n \right) a \! \left(n \right)}{7+n}+\frac{2 \left(19+n \right) a \! \left(1+n \right)}{7+n}-\frac{\left(166+45 n \right) a \! \left(n +2\right)}{7+n}+\frac{\left(257+63 n \right) a \! \left(n +3\right)}{7+n}-\frac{\left(182+37 n \right) a \! \left(n +4\right)}{7+n}+\frac{\left(59+10 n \right) a \! \left(n +5\right)}{7+n}+\frac{1}{7+n}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21319/
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[10,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[10,x]*F[11,x]*F[7,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[7,x]^2*F[10,x]
F[10,x] = x
F[11,x] = F[12,x]+F[13,x]
F[12,x] = F[0,x]*F[7,x]
F[13,x] = F[14,x]*F[17,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[16,x]
F[16,x] = F[10,x]*F[14,x]
F[17,x] = F[18,x]+F[22,x]
F[18,x] = F[19,x]
F[19,x] = F[10,x]*F[14,x]*F[20,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[18,x]
F[22,x] = F[15,x]*F[8,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_{10}\! \left(x \right) F_{4}\! \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_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{7} \left(x \right)^{2} F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{0}\! \left(x \right) F_{7}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right) F_{8}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_10(x)*F_11(x)*F_7(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_7(x)**2)
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_13(x))
Eq(F_12(x), F_0(x)*F_7(x))
Eq(F_13(x), F_14(x)*F_17(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_10(x)*F_14(x))
Eq(F_17(x), F_18(x) + F_22(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_10(x)*F_14(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_18(x))
Eq(F_22(x), F_15(x)*F_8(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, 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, 3, 2, 0], "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, 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, 3, 2, 0], "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, 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, 3, 2, 0], "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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 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": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[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, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": 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