0213_0231_1203_1320
Counting sequence:
1, 1, 2, 6, 20, 66, 210, 644, 1914, 5546, 15748, 44002, 121378, 331396, 897386, 2414010, 6459332, 17209746, 45694194, 120986948, 319625946, 842871626, 2219474692, 5837572546, 15339380290, 40277037316, 105693058250, 277221567834, 726844060484, 1905122552946, 4992281694738, 13079507159492, 34262345911098, 89740816570346, 235028823276676, 615487387180450, 1611725396040802, 4220290096363396, 11050381843630634, 28933398055167738, 75755035002104516, 198342427189516626, 519294236799000882, 1359585363184224836, 3559554211730406810, 9319266388006973066, 24398631980383489540, 63877421201515494274, 167235250105279079554, 437831636445298094596, 1146266414630056259978, 3000981399718729507482, 7856705932023803328068, 20569193817247929450546, 53850992613310296656466, 140984022713463211155140, 369101561915839092822522, 966321653825972088821546, 2529865417174710235624324, 6623278704981018779943778, 17339979056449254503847586, 45396675469958937682591876, 118850081941072696749337322, 311153640681470933583085626, 814610983065606675248944580, 2132679599051568253089186066, 5583428404384908442724265330, 14617606813141521866204464964, 38269394470009874885549942874, 100190581540615514544605196746, 262302360186865444846261726468, 686716519385186277369524966722, 1797847239289400112371708793538, 4706825282305019130681803956996, 12322628777630850662980930770506, 32261061395320286107745088954330, 84460556107240247124961827719492, 221120608343110400128033958312946, 578901271793289774843884337146514, 1515583212854714431299021956400068, 3967848378557880260295815985438906, 10387961946695211286977352200363626, 27196037509884786385221407603899012, 71200150680882139257471833762533666, 186404414731025465803993278335514850, 488013093913557630266564887246461316, 1277634867822045575776732188806419370, 3344891511196718211739527356773196154, 8757039669095072915021309370304565444, 22926227502819799496938655998904488146, 60021642852981666007933841650299265458, 157138701083669364400962580068498822212, 411394460453733728962796010925657791258, 1077044680390184366062395057724294855946, 2819739580944600336452899132828665633796, 7382174062904130337910463803024176256770, 19326782608698721586821098242968004552962, 50598173765073703282266608943726506222596, 132467738690325340060663416792701623730954, 346805042313587448663607282180952128152858, 907947388265965725782954239918489374993476
Generating function in Maple syntax:
-(x-1)*(7*x^2-5*x+1)/(x^2-3*x+1)/(2*x-1)^2
Generating function in latex syntax:
-\frac{\left(x -1\right) \left(7 x^{2}-5 x +1\right)}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(1 - x)*(7*x**2 - 5*x + 1)/((2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x-1)^2*F(x)+(x-1)*(7*x^2-5*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+\left(x -1\right) \left(7 x^{2}-5 x +1\right) = 0
Explicit closed form in Maple syntax:
1/5*(5+5^(1/2))*(3/2-1/2*5^(1/2))^(-n)+1/5*(5-5^(1/2))*(3/2+1/2*5^(1/2))^(-n)-2^(-1+n)*n-2^n
Explicit closed form in latex syntax:
\frac{\left(5+\sqrt{5}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{5}+\frac{\left(5-\sqrt{5}\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{5}-2^{-1+n} n -2^{n}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3), n >= 4
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(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right), \quad n \geq 4
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21327/
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[7,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[7,x]*F[8,x]
F[7,x] = x
F[8,x] = F[55,x]+F[9,x]
F[9,x] = F[10,x]+F[4,x]
F[10,x] = F[11,x]*F[50,x]
F[11,x] = F[12,x]+F[28,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[15,x]*F[7,x]
F[15,x] = F[16,x]+F[19,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[16,x]*F[7,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[7,x]
F[22,x] = F[1,x]+F[20,x]
F[23,x] = F[24,x]+F[25,x]+F[27,x]
F[24,x] = 0
F[25,x] = F[26,x]*F[7,x]
F[26,x] = F[17,x]+F[23,x]
F[27,x] = F[19,x]*F[7,x]
F[28,x] = F[13,x]+F[29,x]
F[29,x] = F[24,x]+F[30,x]+F[40,x]
F[30,x] = F[31,x]*F[7,x]
F[31,x] = F[32,x]+F[35,x]
F[32,x] = F[17,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[32,x]*F[7,x]
F[35,x] = F[23,x]+F[36,x]
F[36,x] = 2*F[24,x]+F[37,x]+F[39,x]
F[37,x] = F[38,x]*F[7,x]
F[38,x] = F[33,x]+F[36,x]
F[39,x] = F[35,x]*F[7,x]
F[40,x] = F[41,x]*F[7,x]
F[41,x] = F[19,x]+F[42,x]
F[42,x] = F[43,x]+F[46,x]
F[43,x] = F[44,x]
F[44,x] = F[45,x]*F[7,x]
F[45,x] = F[20,x]+F[43,x]
F[46,x] = 2*F[24,x]+F[47,x]+F[49,x]
F[47,x] = F[48,x]*F[7,x]
F[48,x] = F[23,x]+F[46,x]
F[49,x] = F[42,x]*F[7,x]
F[50,x] = F[51,x]
F[51,x] = F[52,x]*F[7,x]
F[52,x] = F[53,x]+F[54,x]
F[53,x] = F[1,x]+F[50,x]
F[54,x] = F[50,x]
F[55,x] = F[56,x]
F[56,x] = F[12,x]*F[17,x]*F[57,x]
F[57,x] = F[2,x]+F[53,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_{7}\! \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_{7}\! \left(x \right) F_{8}\! \left(x \right)
F_{7}\! \left(x \right) = x
F_{8}\! \left(x \right) = F_{55}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{4}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{50}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{28}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{7}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{7}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \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_{26}\! \left(x \right) F_{7}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right)+F_{23}\! \left(x \right)
F_{27}\! \left(x \right) = F_{19}\! \left(x \right) F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{24}\! \left(x \right)+F_{30}\! \left(x \right)+F_{40}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{7}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{32}\! \left(x \right) = F_{17}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right) F_{7}\! \left(x \right)
F_{35}\! \left(x \right) = F_{23}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{7}\! \left(x \right)
F_{38}\! \left(x \right) = F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right) F_{7}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right) F_{7}\! \left(x \right)
F_{41}\! \left(x \right) = F_{19}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{46}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right) F_{7}\! \left(x \right)
F_{45}\! \left(x \right) = F_{20}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = 2 F_{24}\! \left(x \right)+F_{47}\! \left(x \right)+F_{49}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right) F_{7}\! \left(x \right)
F_{48}\! \left(x \right) = F_{23}\! \left(x \right)+F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{42}\! \left(x \right) F_{7}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right) F_{7}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{1}\! \left(x \right)+F_{50}\! \left(x \right)
F_{54}\! \left(x \right) = F_{50}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{2}\! \left(x \right)+F_{53}\! \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_7(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_7(x)*F_8(x))
Eq(F_7(x), x)
Eq(F_8(x), F_55(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_4(x))
Eq(F_10(x), F_11(x)*F_50(x))
Eq(F_11(x), F_12(x) + F_28(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_15(x)*F_7(x))
Eq(F_15(x), F_16(x) + F_19(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_16(x)*F_7(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_7(x))
Eq(F_22(x), F_1(x) + F_20(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_27(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_26(x)*F_7(x))
Eq(F_26(x), F_17(x) + F_23(x))
Eq(F_27(x), F_19(x)*F_7(x))
Eq(F_28(x), F_13(x) + F_29(x))
Eq(F_29(x), F_24(x) + F_30(x) + F_40(x))
Eq(F_30(x), F_31(x)*F_7(x))
Eq(F_31(x), F_32(x) + F_35(x))
Eq(F_32(x), F_17(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_32(x)*F_7(x))
Eq(F_35(x), F_23(x) + F_36(x))
Eq(F_36(x), 2*F_24(x) + F_37(x) + F_39(x))
Eq(F_37(x), F_38(x)*F_7(x))
Eq(F_38(x), F_33(x) + F_36(x))
Eq(F_39(x), F_35(x)*F_7(x))
Eq(F_40(x), F_41(x)*F_7(x))
Eq(F_41(x), F_19(x) + F_42(x))
Eq(F_42(x), F_43(x) + F_46(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_45(x)*F_7(x))
Eq(F_45(x), F_20(x) + F_43(x))
Eq(F_46(x), 2*F_24(x) + F_47(x) + F_49(x))
Eq(F_47(x), F_48(x)*F_7(x))
Eq(F_48(x), F_23(x) + F_46(x))
Eq(F_49(x), F_42(x)*F_7(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_52(x)*F_7(x))
Eq(F_52(x), F_53(x) + F_54(x))
Eq(F_53(x), F_1(x) + F_50(x))
Eq(F_54(x), F_50(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_17(x)*F_57(x))
Eq(F_57(x), F_2(x) + F_53(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, 2, 0, 3], "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, 2, 0, 3], "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": 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