0231_0321_1203_2301
Counting sequence:
1, 1, 2, 6, 20, 62, 177, 479, 1261, 3283, 8526, 22166, 57742, 150686, 393745, 1029717, 2694237, 7051399, 18457836, 48319338, 126496626, 331166056, 866995957, 2269814941, 5942440495, 15557496447, 40730036772, 106632599544, 279167744986, 730870615668, 1913444079051, 5009461594921, 13114940675147, 34335360395521, 89891140471520, 235338060973752, 616123042398532, 1613031066164164, 4222970156029211, 11055879401851023, 28944668049443051, 75778124746388261, 198389706189622062, 519390993822367676, 1359783275277359320, 3559958832009576382, 9320093220751222767, 24400320830243930759, 63880869269980393261, 167242286979697056653, 437845991669110567126, 1146295688027634416826, 3001041072413792435952, 7856827529213742622906, 20569441515227435142645, 53851497016468562491587, 140985049534178251993977, 369103651586066193126079, 966325905224020326992386, 2529874064085994787430058, 6623296287033964034846026, 17340014797015897316623866, 45396748104013727914507317, 118850229515025286426343961, 311153940441062131363932745, 814611591808161107664822867, 2132680834983421191629862912, 5583430913142102467224049374, 14617611904442886210041523086, 38269404800186556162899709988, 100190602496116782278656747001, 262302402688163790673069618881, 686716605568374589740551142907, 1797847414016959978548582786091, 4706825636482505345905196132120, 12322629495430556059167004464972, 32261062849809162831595816052822, 84460559053996932435620442416144, 221120614312181634475265509848111, 578901283882547970990176085707693, 1515583237335462278495262745778551, 3967848428123838864495612150052621, 10387962047036054314991573702721972, 27196037712984324080479108956370796, 71200151091916917926445753164559520, 186404415562766429698858150535385152, 488013095596382371170128698439578207, 1277634871226380683811527944781233139, 3344891518082759680264455135901902711, 8757039683021898356981837462922150673, 22926227530982935390681057252862115426, 60021642909926907815061334295661648336, 157138701198797788054502945634120165012, 411394460686466456348447502606696060826, 1077044680860601580990839562185965106195, 2819739581895338286624071183951196216907, 7382174064825413278881373989667620369817, 19326782612580901550020050785051661579609, 50598173772917291371178778365487360913386, 132467738706170972563516284311410417557678, 346805042345595626319370074568743888004876
Generating function in Maple syntax:
-(3*x^6-13*x^5+24*x^4-29*x^3+20*x^2-7*x+1)/(x^2-3*x+1)/(x-1)^5
Generating function in latex syntax:
-\frac{3 x^{6}-13 x^{5}+24 x^{4}-29 x^{3}+20 x^{2}-7 x +1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(-3*x**6 + 13*x**5 - 24*x**4 + 29*x**3 - 20*x**2 + 7*x - 1)/((x - 1)**5*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^5*F(x)+3*x^6-13*x^5+24*x^4-29*x^3+20*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{5} F \! \left(x \right)+3 x^{6}-13 x^{5}+24 x^{4}-29 x^{3}+20 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
1/120*(-24*5^(1/2)+120)*(3/2-1/2*5^(1/2))^(-n)+1/120*(24*5^(1/2)+120)*(3/2+1/2*5^(1/2))^(-n)+1/24*n^4-1/4*n^3-1/24*n^2+1/4*n-1
Explicit closed form in latex syntax:
\frac{\left(-24 \sqrt{5}+120\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(24 \sqrt{5}+120\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{n^{4}}{24}-\frac{n^{3}}{4}-\frac{n^{2}}{24}+\frac{n}{4}-1
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 62
a(6) = 177
a(n+2) = -a(n)+3*a(n+1)-1/24*n*(n^3-2*n^2-25*n+2), n >= 7
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) = 62
a \! \left(6\right) = 177
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-\frac{n \left(n^{3}-2 n^{2}-25 n +2\right)}{24}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15812/
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[16,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = x
F[11,x] = F[12,x]+F[13,x]+F[14,x]
F[12,x] = 0
F[13,x] = F[10,x]*F[6,x]
F[14,x] = F[10,x]*F[15,x]
F[15,x] = F[9,x]
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[12,x]+F[18,x]+F[41,x]
F[18,x] = F[10,x]*F[19,x]
F[19,x] = F[20,x]+F[26,x]
F[20,x] = F[21,x]+F[6,x]
F[21,x] = F[22,x]
F[22,x] = F[10,x]*F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[10,x]+F[25,x]
F[25,x] = F[22,x]
F[26,x] = F[17,x]+F[27,x]
F[27,x] = 2*F[12,x]+F[28,x]+F[32,x]
F[28,x] = F[10,x]*F[29,x]
F[29,x] = F[30,x]+F[31,x]
F[30,x] = F[21,x]
F[31,x] = F[27,x]
F[32,x] = F[10,x]*F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]+F[40,x]
F[35,x] = F[36,x]
F[36,x] = F[10,x]*F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[10,x]*F[39,x]
F[39,x] = F[1,x]+F[37,x]
F[40,x] = F[32,x]
F[41,x] = F[10,x]*F[42,x]
F[42,x] = F[43,x]+F[47,x]
F[43,x] = F[37,x]+F[44,x]
F[44,x] = F[12,x]+F[41,x]+F[45,x]
F[45,x] = F[10,x]*F[46,x]
F[46,x] = F[44,x]+F[6,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[12,x]+F[36,x]+F[49,x]
F[49,x] = F[10,x]*F[50,x]
F[50,x] = F[10,x]+F[48,x]
F[51,x] = F[12,x]+F[52,x]+F[55,x]+F[56,x]
F[52,x] = F[10,x]*F[53,x]
F[53,x] = F[25,x]+F[54,x]
F[54,x] = 2*F[12,x]+F[32,x]+F[52,x]
F[55,x] = F[10,x]*F[44,x]
F[56,x] = F[10,x]*F[57,x]
F[57,x] = F[58,x]
F[58,x] = F[35,x]+F[59,x]
F[59,x] = 2*F[12,x]+F[55,x]+F[56,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_{16}\! \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_{10}\! \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_{11}\! \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_{14}\! \left(x \right)
F_{12}\! \left(x \right) = 0
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{9}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)+F_{41}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{26}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{6}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = 2 F_{12}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)
F_{31}\! \left(x \right) = F_{27}\! \left(x \right)
F_{32}\! \left(x \right) = F_{10}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{32}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{47}\! \left(x \right)
F_{43}\! \left(x \right) = F_{37}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{44}\! \left(x \right)+F_{6}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right)+F_{36}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{10}\! \left(x \right)+F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right)+F_{52}\! \left(x \right)+F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{52}\! \left(x \right) = F_{10}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{25}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = 2 F_{12}\! \left(x \right)+F_{32}\! \left(x \right)+F_{52}\! \left(x \right)
F_{55}\! \left(x \right) = F_{10}\! \left(x \right) F_{44}\! \left(x \right)
F_{56}\! \left(x \right) = F_{10}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{35}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = 2 F_{12}\! \left(x \right)+F_{55}\! \left(x \right)+F_{56}\! \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_16(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_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_13(x) + F_14(x))
Eq(F_12(x), 0)
Eq(F_13(x), F_10(x)*F_6(x))
Eq(F_14(x), F_10(x)*F_15(x))
Eq(F_15(x), F_9(x))
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_12(x) + F_18(x) + F_41(x))
Eq(F_18(x), F_10(x)*F_19(x))
Eq(F_19(x), F_20(x) + F_26(x))
Eq(F_20(x), F_21(x) + F_6(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_10(x)*F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_10(x) + F_25(x))
Eq(F_25(x), F_22(x))
Eq(F_26(x), F_17(x) + F_27(x))
Eq(F_27(x), 2*F_12(x) + F_28(x) + F_32(x))
Eq(F_28(x), F_10(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_31(x))
Eq(F_30(x), F_21(x))
Eq(F_31(x), F_27(x))
Eq(F_32(x), F_10(x)*F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x) + F_40(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_10(x)*F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_10(x)*F_39(x))
Eq(F_39(x), F_1(x) + F_37(x))
Eq(F_40(x), F_32(x))
Eq(F_41(x), F_10(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_47(x))
Eq(F_43(x), F_37(x) + F_44(x))
Eq(F_44(x), F_12(x) + F_41(x) + F_45(x))
Eq(F_45(x), F_10(x)*F_46(x))
Eq(F_46(x), F_44(x) + F_6(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_12(x) + F_36(x) + F_49(x))
Eq(F_49(x), F_10(x)*F_50(x))
Eq(F_50(x), F_10(x) + F_48(x))
Eq(F_51(x), F_12(x) + F_52(x) + F_55(x) + F_56(x))
Eq(F_52(x), F_10(x)*F_53(x))
Eq(F_53(x), F_25(x) + F_54(x))
Eq(F_54(x), 2*F_12(x) + F_32(x) + F_52(x))
Eq(F_55(x), F_10(x)*F_44(x))
Eq(F_56(x), F_10(x)*F_57(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_35(x) + F_59(x))
Eq(F_59(x), 2*F_12(x) + F_55(x) + F_56(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": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "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, 2, 0, 3], "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, 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, 2, 0, 3], "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": 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