0213_0231_0312_2301
Counting sequence:
1, 1, 2, 6, 20, 65, 201, 595, 1704, 4764, 13090, 35519, 95503, 255073, 677894, 1794978, 4739792, 12489917, 32861013, 86355247, 226728996, 594880296, 1560009022, 4089341051, 10716402715, 28076644285, 73547084546, 192631718190, 504482287724, 1321083580409, 3459305324385, 9057906134539, 23716560562848, 62096070521268, 162580240935514, 425661832154423, 1114439615266087, 2917725733120537, 7638875023048958, 19999174213933242, 52359197374564616, 137079517421388341, 358881553912855917, 939569542363690471, 2459835869271237660, 6439955657636066880, 16860066288009051766, 44140313575135266035, 115561015174885101619, 302543013424496749429, 792068588048558567930, 2073663876621085796934, 5428925293614512508068, 14213116507822079097713, 37210433237050979526009, 97418201217729368962243, 255044206444934146324632, 667714490174667107939532, 1748099408194255253349778, 4576584022638474803821487, 11981653236181921461538111, 31368376838828794187639761, 82123479586147470315075062, 215002066531299635184973266, 562882729231123472094620480, 1473646139608814854808439725, 3858055726488809239749801861, 10100521113644589159279172255, 26443507762018910827764127764, 69230002467560048503366036824, 181246500230957045041039634350, 474509499405902707337164169579, 1242282000347934318405275481163, 3252336506360266730748307487533, 8514727528177598839578937408754, 22291846097061995719467085593438, 58360810800787320181779481081052, 152790586380857828551785681068777, 400010948492901892925406208963473, 1047242259400079305128090239498107, 2741715830311798932266179096883856, 7177905232744243311285076225859556, 18791999870338782640818307930107082, 49198094383107807889628364263286311, 128802283288656047584983818257401175, 337208755502203147979157157304215945, 882823983256639022580155787245844206, 2311263194345085172216646471614511850, 6050965599933358998980456161960081784, 15841633605764476834546067082990514469, 41473935217979041524300435224461023645, 108580172049410587777640618865291680599, 284266580932728601887192181921212266508, 744219570753726978041077447997941615728, 1948392131338355852550323204271805574374, 5100956823281147620238458249215861094883, 13354478338544701089422183712172549685347, 34962478192432183810542357224895331911397, 91532956238910306667233416637700533949418, 239636390524615648841214950038580445738102, 627376215335570465156525548178789154867476
Generating function in Maple syntax:
-(4*x^4-11*x^3+13*x^2-6*x+1)/(2*x-1)/(x^2-3*x+1)/(x-1)^2
Generating function in latex syntax:
-\frac{4 x^{4}-11 x^{3}+13 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-4*x**4 + 11*x**3 - 13*x**2 + 6*x - 1)/((x - 1)**2*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x-1)^2*F(x)+4*x^4-11*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+4 x^{4}-11 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
n+1+(3/2+1/2*5^(1/2))^(-n)-2^(n+1)+(3/2-1/2*5^(1/2))^(-n)
Explicit closed form in latex syntax:
n +1+\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}-2^{n +1}+\left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)+n+1, 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) = 20
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+n +1, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21307/
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[14,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[15,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]*F[8,x]
F[8,x] = F[11,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[6,x]
F[11,x] = F[12,x]*F[6,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[12,x]*F[14,x]
F[14,x] = x
F[15,x] = F[16,x]
F[16,x] = F[14,x]*F[17,x]
F[17,x] = F[18,x]+F[5,x]
F[18,x] = F[19,x]*F[6,x]
F[19,x] = F[20,x]+F[28,x]
F[20,x] = F[12,x]+F[21,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[13,x]
F[23,x] = F[24,x]+F[25,x]+F[27,x]
F[24,x] = 0
F[25,x] = F[14,x]*F[26,x]
F[26,x] = F[22,x]+F[23,x]
F[27,x] = F[14,x]*F[21,x]
F[28,x] = F[22,x]+F[29,x]
F[29,x] = F[24,x]+F[30,x]+F[32,x]
F[30,x] = F[14,x]*F[31,x]
F[31,x] = F[22,x]+F[29,x]
F[32,x] = F[14,x]*F[28,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_{14}\! \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_{15}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{11}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{6}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{6}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = x
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{14}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{5}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{6}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{28}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{14}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right) F_{21}\! \left(x \right)
F_{28}\! \left(x \right) = F_{22}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{24}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{14}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{22}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{14}\! \left(x \right) F_{28}\! \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_14(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_15(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_14(x)*F_8(x))
Eq(F_8(x), F_11(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_6(x))
Eq(F_11(x), F_12(x)*F_6(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_12(x)*F_14(x))
Eq(F_14(x), x)
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_14(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_5(x))
Eq(F_18(x), F_19(x)*F_6(x))
Eq(F_19(x), F_20(x) + F_28(x))
Eq(F_20(x), F_12(x) + F_21(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_13(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_27(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_14(x)*F_26(x))
Eq(F_26(x), F_22(x) + F_23(x))
Eq(F_27(x), F_14(x)*F_21(x))
Eq(F_28(x), F_22(x) + F_29(x))
Eq(F_29(x), F_24(x) + F_30(x) + F_32(x))
Eq(F_30(x), F_14(x)*F_31(x))
Eq(F_31(x), F_22(x) + F_29(x))
Eq(F_32(x), F_14(x)*F_28(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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 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": [2, 3, 0, 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, 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": [2, 3, 0, 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, 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": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 2, 1, 3], "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": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 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": [1, 2, 0], "pos": [[0, 2], [0, 2], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "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}}, {"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": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"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": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 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