0231_1203_1302_3102
Counting sequence:
1, 1, 2, 6, 20, 65, 202, 606, 1775, 5117, 14591, 41276, 116047, 324631, 904280, 2509680, 6942662, 19150451, 52687120, 144613932, 396085181, 1082729951, 2954439017, 8048501606, 21892474765, 59465234845, 161310757022, 437053305546, 1182800440040, 3197615586197, 8635897752646, 23301364402026, 62816204209787, 169199987765681, 455393968945475, 1224759809106560, 3291608918622907, 8840459437471891, 23728223405672660, 63649180324475412, 170635772190660986, 457202530571299751, 1224388547871706732, 3277268903765432856, 8767918807240959785, 23446683658684937075, 62672219947179954317, 167450043377225742986, 447218023989243622105, 1193944302810373347577, 3186295593296734744826, 8500244329424537634126, 22668662243156140073852, 60433115092236961587881, 161058576261954719773570, 429098920686634043982390, 1142879213548567975837127, 3043105496217924728268581, 8100496576131150179522375, 21556995358994502877218692, 57352263580283345682338791, 152546506493329518793491151, 405645615154696177339008272, 1078418705624257928522459256, 2866337242423583387484270830, 7616744877109510656399129755, 20235626214588498069863544712, 53749168388243613595174004676, 142737253989221683353455541845, 378979684075071828336601336007, 1006027694683865636632777536881, 2670063998824490194619609769326, 7085220201885628451423258182117, 18797803708272500559183905512981, 49863756327156165920532474151190, 132247954384428769886091823585170, 350688008755604169094373900741072, 929785288559573040783244284401021, 2464767605025208838057371265611390, 6532847554145145824408692046115522, 17312665392195411803426160160629971, 45873489599167546739622331765773401, 121533936000701416708441009838789387, 321938375212392811110948678526890056, 852685227469451151507148793368664275, 2258119363885426940333481023683154635, 5979274996422254425379500476366168332, 15830509965398144306540655679367401116, 41907065787587177650563166184906815010, 110924315720791578143375306470913329487, 293571955457257126117921444391343246436, 776876144574960318727238636598086339376, 2055604185957095816275984603924315483457, 5438494942440762570220435980844089312731, 14387008521109026228532295617094240344085, 38055355730973383677697646900228266181650, 100650406122328493157377260894473173693617, 266177079877477133290562787186049466910961, 703853137783980650849880439061367247577330, 1861018029054633014169657893788270125418902, 4920135731846545232255261995276404656930276
Generating function in Maple syntax:
-(2*x-1)*(x^5-6*x^4+12*x^3-13*x^2+6*x-1)/(x^2-3*x+1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{\left(2 x -1\right) \left(x^{5}-6 x^{4}+12 x^{3}-13 x^{2}+6 x -1\right)}{\left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(1 - 2*x)*(x**5 - 6*x**4 + 12*x**3 - 13*x**2 + 6*x - 1)/((x - 1)**3*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)^2*(x-1)^3*F(x)+(2*x-1)*(x^5-6*x^4+12*x^3-13*x^2+6*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+\left(2 x -1\right) \left(x^{5}-6 x^{4}+12 x^{3}-13 x^{2}+6 x -1\right) = 0
Explicit closed form in Maple syntax:
1/50*((-5*n-29)*5^(1/2)+15*n+75)*(3/2-1/2*5^(1/2))^(-n)+1/50*((5*n+29)*5^(1/2)+15*n+75)*(3/2+1/2*5^(1/2))^(-n)-1/2*n^2+3/2*n-2
Explicit closed form in latex syntax:
\frac{\left(\left(-5 n -29\right) \sqrt{5}+15 n +75\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\frac{\left(\left(5 n +29\right) \sqrt{5}+15 n +75\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}-\frac{n^{2}}{2}+\frac{3 n}{2}-2
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(6) = 202
a(n+4) = -a(n)+6*a(n+1)-11*a(n+2)+6*a(n+3)-1/2*(n+2)*(n-1), 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) = 65
a \! \left(6\right) = 202
a \! \left(n +4\right) = -a \! \left(n \right)+6 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right)-\frac{\left(n +2\right) \left(n -1\right)}{2}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15892/
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[21,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[6,x]
F[10,x] = F[11,x]+F[16,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[14,x]+F[15,x]
F[14,x] = F[1,x]+F[11,x]
F[15,x] = F[11,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[19,x]+F[20,x]
F[19,x] = F[16,x]+F[7,x]
F[20,x] = F[16,x]
F[21,x] = F[22,x]+F[30,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[14,x]+F[25,x]
F[25,x] = F[22,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[25,x]+F[29,x]
F[29,x] = F[26,x]
F[30,x] = F[31,x]+F[32,x]+F[52,x]
F[31,x] = 0
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[46,x]
F[34,x] = F[35,x]+F[7,x]
F[35,x] = F[31,x]+F[36,x]+F[39,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[34,x]+F[38,x]
F[38,x] = F[16,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[11,x]
F[42,x] = F[43,x]
F[43,x] = F[4,x]*F[44,x]
F[44,x] = F[15,x]+F[45,x]
F[45,x] = F[43,x]
F[46,x] = F[30,x]+F[47,x]
F[47,x] = 2*F[31,x]+F[48,x]+F[62,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[69,x]
F[50,x] = F[51,x]+F[61,x]
F[51,x] = F[52,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[22,x]+F[51,x]
F[55,x] = F[26,x]+F[56,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]+F[60,x]
F[59,x] = F[51,x]+F[56,x]
F[60,x] = F[56,x]
F[61,x] = 2*F[31,x]+F[48,x]+F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[65,x]
F[64,x] = F[26,x]
F[65,x] = F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[29,x]+F[68,x]
F[68,x] = F[66,x]
F[69,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{21}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{16}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{15}\! \left(x \right) = F_{11}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right)+F_{7}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{30}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{14}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{26}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{52}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{46}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{7}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)+F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{34}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{16}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{11}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{30}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{48}\! \left(x \right)+F_{62}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{69}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{61}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{22}\! \left(x \right)+F_{51}\! \left(x \right)
F_{55}\! \left(x \right) = F_{26}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{59}\! \left(x \right) = F_{51}\! \left(x \right)+F_{56}\! \left(x \right)
F_{60}\! \left(x \right) = F_{56}\! \left(x \right)
F_{61}\! \left(x \right) = 2 F_{31}\! \left(x \right)+F_{48}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{65}\! \left(x \right)
F_{64}\! \left(x \right) = F_{26}\! \left(x \right)
F_{65}\! \left(x \right) = F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{29}\! \left(x \right)+F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{66}\! \left(x \right)
F_{69}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_21(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_6(x))
Eq(F_10(x), F_11(x) + F_16(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_15(x))
Eq(F_14(x), F_1(x) + F_11(x))
Eq(F_15(x), F_11(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_19(x) + F_20(x))
Eq(F_19(x), F_16(x) + F_7(x))
Eq(F_20(x), F_16(x))
Eq(F_21(x), F_22(x) + F_30(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_14(x) + F_25(x))
Eq(F_25(x), F_22(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_25(x) + F_29(x))
Eq(F_29(x), F_26(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_52(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_46(x))
Eq(F_34(x), F_35(x) + F_7(x))
Eq(F_35(x), F_31(x) + F_36(x) + F_39(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_34(x) + F_38(x))
Eq(F_38(x), F_16(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_11(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_4(x)*F_44(x))
Eq(F_44(x), F_15(x) + F_45(x))
Eq(F_45(x), F_43(x))
Eq(F_46(x), F_30(x) + F_47(x))
Eq(F_47(x), 2*F_31(x) + F_48(x) + F_62(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_69(x))
Eq(F_50(x), F_51(x) + F_61(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_22(x) + F_51(x))
Eq(F_55(x), F_26(x) + F_56(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x) + F_60(x))
Eq(F_59(x), F_51(x) + F_56(x))
Eq(F_60(x), F_56(x))
Eq(F_61(x), 2*F_31(x) + F_48(x) + F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_65(x))
Eq(F_64(x), F_26(x))
Eq(F_65(x), F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_29(x) + F_68(x))
Eq(F_68(x), F_66(x))
Eq(F_69(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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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