0123_0132_0231_0312_1023_1203
Counting sequence:
1, 1, 2, 6, 18, 55, 174, 567, 1891, 6426, 22177, 77530, 274013, 977499, 3515206, 12729871, 46383734, 169929331, 625567208, 2312939597, 8585251537, 31980115168, 119510658592, 447931978571, 1683412060428, 6342324239480, 23949886725524, 90632088788217, 343651565542291, 1305429852596077, 4967462007844261, 18932737172039517, 72268095210518990, 276245117340999575, 1057356258557271569, 4052227366112092626, 15548258553511741148, 59725168096426103643, 229665133963475082414, 884038475664977651166, 3406150188846305258456, 13135683151467388047649, 50701281957835079437491, 195859704691939086889063, 757207304773562393305167, 2929630119706654549626382, 11342919030811342448053373, 43947843441148794405132112, 170387797431617130541102281, 661021893970299571830696191, 2566011450159990501274001922, 9966813473139918671790785940, 38734617284365473372270365773, 150618446214396081327472129924, 585982008339265735797198333170, 2280916417774508080802682556792, 8882712754338248252711292117107, 34608726678655637004873324887129, 134903233660477196778697687012777, 526076826692748238715807252271205, 2052389228101744782177903675435826, 8010285677535187380764327923421771, 31275789299864180918614484852602962, 122161706844216851953225999731949526, 477334961745941560068644921022961434, 1865811547556935129514126526086128739, 7295649347526943407674656026619964746, 28536929898262670644788217765962196124, 111658898989065123603209514275597658882, 437036887493289766212446558940019155403, 1711108503348421575471790718216428820433, 6701436616978985921953654515535465729377, 26253399968288323182309703018535032615823, 102879059882683086207543804577595962136624, 403263454941226775783386993168127071178191, 1581131474761163168335827128382319723970186, 6200993213075335803211440428395316977467679, 24325720886814056256724816968210261615497851, 95450569666930843438312397445900034957478558, 374625150880382261500122856863243707352606032, 1470680382627037692824591301690049223905682527, 5774841775639276942142035539241895126059958438, 22680877518863057248907314558129309277772639162, 89099492691981702833715481980782136850061275836, 350093419924224357496850066444779364691301187269, 1375890300360260311208153560004787234634478799909, 5408448457906970955973603450266302236241483423221, 21264176161793868517205364918907389940254699463629, 83619859195172854402055840572140503614456574805792, 328892066282046067380668875281223917309690145589553, 1293834156460341141437636888933046262086873879074188, 5090767834486924795313606797661419628070042213926707, 20033909549608917904177019625788267460971797417519683, 78854084314241982272883454137567112060626749931274644, 310425312115599731445961310432548821508641470921525426, 1222258152544025315869961664607160236953885595606075611, 4813268891200463632912657430876122644643009712555062134, 18957767801096693192302597529112199430032265192276605166, 74679730799637363416980205009646223897100490806379778108, 294228930987262050017599602441172919175221168035985603557, 1159401208758975556627460037564433742567765726188713127977
Generating function in Maple syntax:
1/2*(x^2+x-1+(x^2-x+1)*(1-4*x)^(1/2))/x/(x^4-2*x^3+4*x^2-3*x+1)*(x-1)
Generating function in latex syntax:
\frac{\left(x^{2}+x -1+\left(x^{2}-x +1\right) \sqrt{1-4 x}\right) \left(x -1\right)}{2 x \left(x^{4}-2 x^{3}+4 x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(x - 1)*(x**2/2 + x/2 + sqrt(1 - 4*x)*(x**2 - x + 1)/2 - 1/2)/(x*(x**4 - 2*x**3 + 4*x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
x*(x^4-2*x^3+4*x^2-3*x+1)*F(x)^2-(x-1)*(x^2+x-1)*F(x)+(x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{4}-2 x^{3}+4 x^{2}-3 x +1\right) F \! \left(x \right)^{2}-\left(x -1\right) \left(x^{2}+x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 55
a(6) = 174
a(7) = 567
a(n+8) = -2*(3+2*n)/(9+n)*a(n)+(43+17*n)/(9+n)*a(1+n)-4*(35+11*n)/(9+n)*a(n+2)+2*(147+37*n)/(9+n)*a(n+3)-4*(103+21*n)/(9+n)*a(n+4)+(387+65*n)/(9+n)*a(n+5)-2*(111+16*n)/(9+n)*a(n+6)+(71+9*n)/(9+n)*a(n+7), 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) = 18
a \! \left(5\right) = 55
a \! \left(6\right) = 174
a \! \left(7\right) = 567
a \! \left(n +8\right) = -\frac{2 \left(3+2 n \right) a \! \left(n \right)}{9+n}+\frac{\left(43+17 n \right) a \! \left(1+n \right)}{9+n}-\frac{4 \left(35+11 n \right) a \! \left(n +2\right)}{9+n}+\frac{2 \left(147+37 n \right) a \! \left(n +3\right)}{9+n}-\frac{4 \left(103+21 n \right) a \! \left(n +4\right)}{9+n}+\frac{\left(387+65 n \right) a \! \left(n +5\right)}{9+n}-\frac{2 \left(111+16 n \right) a \! \left(n +6\right)}{9+n}+\frac{\left(71+9 n \right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22068/
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[14,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[8,x]
F[13,x] = F[6,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[5,x]*F[8,x]
F[16,x] = F[17,x]+F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[8,x]
F[19,x] = F[20,x,1]
F[20,x,k[0]] = (F[21,x,k[0]]*k[0]-F[21,x,1])/(-1+k[0])
F[21,x,k[0]] = F[22,x,k[0]]+F[26,x,k[0]]
F[22,x,k[0]] = F[0,x]+F[23,x,k[0]]
F[23,x,k[0]] = F[24,x,k[0]]
F[24,x,k[0]] = F[21,x,k[0]]*F[25,x,k[0]]
F[25,x,k[0]] = k[0]*x
F[26,x,k[0]] = F[27,x,k[0]]
F[27,x,k[0]] = F[20,x,k[0]]*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_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \left(x \right)+F_{5}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{4}\! \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_{20}\! \left(x , 1\right)
F_{20}\! \left(x , y\right) = \frac{y F_{21}\! \left(x , y\right)-F_{21}\! \left(x , 1\right)}{-1+y}
F_{21}\! \left(x , y\right) = F_{22}\! \left(x , y\right)+F_{26}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{0}\! \left(x \right)+F_{23}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = F_{24}\! \left(x , y\right)
F_{24}\! \left(x , y\right) = F_{21}\! \left(x , y\right) F_{25}\! \left(x , y\right)
F_{25}\! \left(x , y\right) = y x
F_{26}\! \left(x , y\right) = F_{27}\! \left(x , y\right)
F_{27}\! \left(x , y\right) = F_{20}\! \left(x , y\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_4(x)*F_8(x))
Eq(F_4(x), F_14(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_8(x))
Eq(F_13(x), F_6(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_5(x)*F_8(x))
Eq(F_16(x), F_17(x) + F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_8(x))
Eq(F_19(x), F_20(x, 1))
Eq(F_20(x, k_0), (-k_0*F_21(x, k_0) + F_21(x, 1))/(1 - k_0))
Eq(F_21(x, k_0), F_22(x, k_0) + F_26(x, k_0))
Eq(F_22(x, k_0), F_0(x) + F_23(x, k_0))
Eq(F_23(x, k_0), F_24(x, k_0))
Eq(F_24(x, k_0), F_21(x, k_0)*F_25(x, k_0))
Eq(F_25(x, k_0), k_0*x)
Eq(F_26(x, k_0), F_27(x, k_0))
Eq(F_27(x, k_0), F_20(x, k_0)*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.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": "point_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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "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_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, 0]]}, {"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, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0, 3], "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, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, 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