0213_1032_1302_2301
Counting sequence:
1, 1, 2, 6, 20, 64, 192, 546, 1494, 3982, 10432, 27036, 69628, 178750, 458402, 1175946, 3020252, 7770088, 20027832, 51723546, 133832046, 346888966, 900543352, 2341109556, 6093425140, 15876251254, 41401110842, 108041868546, 282120514964, 737044607872, 1926328954032, 5036305109586, 13170775214598, 34451324472190, 90131658594352, 235836277128396, 617153834491372, 1615161369877678, 4227368202467282, 11064950372698746, 28963359747024716, 75816607653260056, 198468871026710952, 519553721543156106, 1360117526812068510, 3560644927265160694, 9321500595634613992, 24403205948755037796, 63886780244491123684, 167254390403695404262, 437870761467060875882, 1146346353523442086386, 3001144655205221688068, 7857039198396228744880, 20569873860791662395552, 53852379721995526769346, 140986850974029199826742, 369107326523362127057518, 966333399213800271075232, 2529889340295930827699196, 6623327415914588419228636, 17340078207698650692687646, 45396877231222243880812802, 118850492381128336786860906, 311154475396640268940296380, 814612680166061456527693384, 2132683048592710036775338392, 5583435414147656452353879546, 14617621054027946769978312654, 38269423394504582462126876902, 100190640275048645235817542232, 262302479426619137304803372436, 686716761406468524438842168788, 1797847730415514330814811018070, 4706826278724347016176944047002, 12322630798803705331189082232546, 32261065494334393238597134442612, 84460564418605256975537403824608, 221120625192514011006928080780432, 578901305945444178957158522596146, 1515583282065717604236542208328806, 3967848518793275335592800251355486, 10387962230792778896415208256315152, 27196038085333476521784894764038956, 71200151846286629365974358179287500, 186404417090848665691749427361970574, 488013098691232469383579385685269042, 1277634877493452132693765586455827546, 3344891530771645082939602953615598124, 8757039708709154172153478167076540792, 22926227582976417040667028798622781832, 60021643015151811154318657662084538986, 157138701411723474811588352916766741758, 411394461117269590019759838271588375510, 1077044681732111368647747275714945514952, 2819739583658164902566452695407545932996, 7382174068390680592023269181377094817732, 19326782619790664338818105506064157600198, 50598173787495273273803416482699444167882, 132467738735643849018822617896208763323346, 346805042405175204530096856439088934741476
Generating function in Maple syntax:
-(9*x^5-26*x^4+37*x^3-25*x^2+8*x-1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^2
Generating function in latex syntax:
-\frac{9 x^{5}-26 x^{4}+37 x^{3}-25 x^{2}+8 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-9*x**5 + 26*x**4 - 37*x**3 + 25*x**2 - 8*x + 1)/((x - 1)**2*(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*(x-1)^2*F(x)+9*x^5-26*x^4+37*x^3-25*x^2+8*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} \left(x -1\right)^{2} F \! \left(x \right)+9 x^{5}-26 x^{4}+37 x^{3}-25 x^{2}+8 x -1 = 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-3*2^n+2
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 -3 \,2^{n}+2
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 64
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)-2*n, n >= 6
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) = 64
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)-2 n, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21379/
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[11,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[14,x]*F[20,x]*F[43,x]
F[20,x] = F[21,x]+F[42,x]
F[21,x] = F[22,x]+F[35,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[1,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[14,x]*F[24,x]
F[27,x] = F[28,x]+F[30,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[14,x]
F[30,x] = F[31,x]+F[32,x]+F[34,x]
F[31,x] = 0
F[32,x] = F[14,x]*F[33,x]
F[33,x] = F[25,x]+F[30,x]
F[34,x] = F[14,x]*F[27,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]^2*F[14,x]*F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[14,x]*F[39,x]
F[39,x] = F[40,x]+F[42,x]
F[40,x] = F[1,x]+F[41,x]
F[41,x] = F[37,x]
F[42,x] = F[12,x]*F[37,x]
F[43,x] = F[12,x]+F[44,x]
F[44,x] = F[25,x]+F[45,x]
F[45,x] = F[31,x]+F[46,x]+F[48,x]
F[46,x] = F[14,x]*F[47,x]
F[47,x] = F[28,x]+F[45,x]
F[48,x] = F[14,x]*F[44,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_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right) F_{20}\! \left(x \right) F_{43}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{42}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{35}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right) F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = 0
F_{32}\! \left(x \right) = F_{14}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right) F_{27}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12} \left(x \right)^{2} F_{14}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{14}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{1}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{37}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{25}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{31}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{14}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{28}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{14}\! \left(x \right) F_{44}\! \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_11(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_14(x)*F_20(x)*F_43(x))
Eq(F_20(x), F_21(x) + F_42(x))
Eq(F_21(x), F_22(x) + F_35(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_1(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_14(x)*F_24(x))
Eq(F_27(x), F_28(x) + F_30(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_14(x))
Eq(F_30(x), F_31(x) + F_32(x) + F_34(x))
Eq(F_31(x), 0)
Eq(F_32(x), F_14(x)*F_33(x))
Eq(F_33(x), F_25(x) + F_30(x))
Eq(F_34(x), F_14(x)*F_27(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)**2*F_14(x)*F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_14(x)*F_39(x))
Eq(F_39(x), F_40(x) + F_42(x))
Eq(F_40(x), F_1(x) + F_41(x))
Eq(F_41(x), F_37(x))
Eq(F_42(x), F_12(x)*F_37(x))
Eq(F_43(x), F_12(x) + F_44(x))
Eq(F_44(x), F_25(x) + F_45(x))
Eq(F_45(x), F_31(x) + F_46(x) + F_48(x))
Eq(F_46(x), F_14(x)*F_47(x))
Eq(F_47(x), F_28(x) + F_45(x))
Eq(F_48(x), F_14(x)*F_44(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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 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