0132_0231_1032_2301_3021
Counting sequence:
1, 1, 2, 6, 19, 55, 153, 417, 1123, 3001, 7977, 21123, 55777, 146977, 386691, 1016169, 2667961, 7000003, 18356625, 48119025, 126098755, 330373849, 865416009, 2266660611, 5936138689, 15544901185, 40704856323, 106582250697, 279067061593, 730669265731, 1913041398897, 5008656257553, 13113330026947, 34332139129657, 89884697974761, 235325176020099, 616097272536481, 1612979526491233, 4222867076740995, 11055673243339305, 28944255732492025, 75777300112566979, 198388056922069329, 519387695287361841, 1359776678207457859, 3559945637869895065, 9320066832471993993, 24400268053685620227, 63880763716863933313, 167242075873464312961, 437845569456645272067, 1146294843602704036233, 3001039383563931902617, 7856824151514021803587, 20569434759827993772081, 53851483505669680040529, 140985022512580487405251, 369103597542870664286713, 966325797137629269677865, 2529873847913212673192835, 6623295854688399806792545, 17340013932324768860968609, 45396746374631471003680899, 118850226056260772605209321, 311153933523533103722217529, 814611577973103052381984195, 2132680807313305081064816913, 5583430857801870246094630257, 14617611793762421767783401283, 38269404578825627278384228441, 100190602053394924509626593737, 262302401802720075135010172163, 686716603797487158664433161537, 1797847410475185116396347790017, 4706825629398955621600727163651, 12322629481263456610558067611209, 32261062821474963934377943490521, 84460558997328534641184698501443, 221120614198844838886394023295985, 578901283655874379812433113950865, 1515583236882115096139776803685315, 3967848427217144499784640267362489, 10387962045222665585569629938916969, 27196037709357546621635221430418051, 71200151084663363008757978114396449, 186404415548259319863482600436889825, 488013095567368151499377598244510083, 1277634871168352244470025744393114537, 3344891517966702801581450735127781753, 8757039682789784599615828661376127171, 22926227530518707875949039649772392657, 60021642908998452785597299089484636593, 157138701196940877995574875221768688707, 411394460682752636230591361781995772697, 1077044680853173940755127280536567315721, 2819739581880483006152646620652403547139, 7382174064795702717938524863070038071041, 19326782612521480428134352531856500156673, 50598173772798449127407381859097041380355, 132467738705933288075973491298629781947145, 346805042345120257344284488543182620386585
Generating function in Maple syntax:
(x^6-3*x^5-3*x^3+8*x^2-5*x+1)/(x-1)/(2*x-1)/(x^2-3*x+1)
Generating function in latex syntax:
\frac{x^{6}-3 x^{5}-3 x^{3}+8 x^{2}-5 x +1}{\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right)}
Generating function in sympy syntax:
(x**6 - 3*x**5 - 3*x**3 + 8*x**2 - 5*x + 1)/((x - 1)*(2*x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x-1)*(2*x-1)*(x^2-3*x+1)*F(x)+x^6-3*x^5-3*x^3+8*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+x^{6}-3 x^{5}-3 x^{3}+8 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/40*(-8*5^(1/2)+40)*(3/2-1/2*5^(1/2))^(-n)+1/40*(8*5^(1/2)+40)*(3/2+1/2*5^(1/2))^(-n)-3/8*2^n-1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-8 \sqrt{5}+40\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{40}+\frac{\left(8 \sqrt{5}+40\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{40}-\frac{3 \,2^{n}}{8}-1 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 153
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2)-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) = 19
a \! \left(5\right) = 55
a \! \left(6\right) = 153
a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)-1, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21708/
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[14,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[4,x]
F[10,x] = F[11,x]+F[6,x]
F[11,x] = F[12,x]*F[8,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[12,x]*F[4,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[21,x]
F[17,x] = F[12,x]+F[18,x]+F[19,x]
F[18,x] = F[17,x]*F[4,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[17,x]*F[4,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[44,x]
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[40,x]
F[26,x] = F[27,x]*F[28,x]
F[27,x] = F[4,x]+F[6,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[1,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[29,x]*F[4,x]
F[32,x] = F[33,x]+F[35,x]
F[33,x] = F[34,x]
F[34,x] = F[12,x]*F[4,x]
F[35,x] = F[36,x]+F[37,x]+F[39,x]
F[36,x] = 0
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[30,x]+F[35,x]
F[39,x] = F[32,x]*F[4,x]
F[40,x] = F[36,x]+F[41,x]+F[42,x]
F[41,x] = F[17,x]*F[4,x]
F[42,x] = F[43,x]
F[43,x] = F[12,x]*F[4,x]*F[40,x]
F[44,x] = F[28,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[17,x]*F[33,x]*F[4,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_{14}\! \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_{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_{6}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \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_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{21}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{44}\! \left(x \right)
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{40}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{4}\! \left(x \right)+F_{6}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{35}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = 0
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{30}\! \left(x \right)+F_{35}\! \left(x \right)
F_{39}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{40}\! \left(x \right) = F_{36}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{44}\! \left(x \right) = F_{28}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{17}\! \left(x \right) F_{33}\! \left(x \right) F_{4}\! \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_14(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_9(x))
Eq(F_9(x), F_10(x)*F_4(x))
Eq(F_10(x), F_11(x) + F_6(x))
Eq(F_11(x), F_12(x)*F_8(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_12(x)*F_4(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_21(x))
Eq(F_17(x), F_12(x) + F_18(x) + F_19(x))
Eq(F_18(x), F_17(x)*F_4(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_17(x)*F_4(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_44(x))
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_40(x))
Eq(F_26(x), F_27(x)*F_28(x))
Eq(F_27(x), F_4(x) + F_6(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_1(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_29(x)*F_4(x))
Eq(F_32(x), F_33(x) + F_35(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_12(x)*F_4(x))
Eq(F_35(x), F_36(x) + F_37(x) + F_39(x))
Eq(F_36(x), 0)
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_30(x) + F_35(x))
Eq(F_39(x), F_32(x)*F_4(x))
Eq(F_40(x), F_36(x) + F_41(x) + F_42(x))
Eq(F_41(x), F_17(x)*F_4(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_12(x)*F_4(x)*F_40(x))
Eq(F_44(x), F_28(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_17(x)*F_33(x)*F_4(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, 1, 3, 2], "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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 1, 3, 2], "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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 1, 3, 2], "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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": 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