0213_1230_1302_1320
Counting sequence:
1, 1, 2, 6, 20, 66, 212, 662, 2016, 6010, 17600, 50778, 144680, 407910, 1139828, 3160850, 8708196, 23856250, 65035628, 176542446, 477448712, 1287006114, 3459224696, 9273966962, 24806502096, 66219635854, 176450715884, 469415812074, 1246986797132, 3308258364834, 8766506608484, 23205613194950, 61368157409328, 162149664939658, 428101884268592, 1129452108716106, 2977874172889592, 7846718877724278, 20665024896640100, 54396440692288898, 143123312455984308, 376419818265702538, 989628213113818460, 2600896108844902494, 6833416408999325144, 17948500295613978162, 47130741725412622568, 123730199747364354530, 324751490886010450848, 852194067292625615902, 2235862038678229908956, 5865124709602982286042, 15383009161751381877404, 40340630424178341509682, 105775836168308524170836, 277317517222023892732598, 726974954593240800061440, 1905534477400729343361946, 4994281005412084312102304, 13088540779328024121456954, 34298458449446517248262344, 89872301259425868657613446, 235476496258879864216706132, 616938756932416744466179826, 1616260966028207442531581700, 4234068097005847264122277786, 11091312435103983812813506892, 29052831398224102836550629582, 76098475277189087594845737896, 199318143713842916627362337730, 522038348053640402986257448280, 1367237234327540459293332726098, 3580736415321927907285631579760, 9377538467190043009847324909230, 24558074959515634980291876744908, 64311644816985304256535369325770, 168412972283048146022668360650476, 441014456035170412052293971103170, 1154840876648427713746434070314308, 3024016319621721652446709276192166, 7918434854579256509314187833750992, 20734249934779368921374286382648234, 54291465103901358794641515645372176, 142157407376779739952805353254442282, 372222431182103823313539517613140376, 974610496484345464436442126800442774, 2551851953063479556060497465762538564, 6681531762620508020158475953747812194, 17494159029448435342977243465169475412, 45804363114998022941652808464784304554, 119927181588858531064985607445691189564, 313997101987708704109558427729872043070, 822112216320306974073934946578685420408, 2152455651202350592618455402739462301778, 5635535037692917855663022584353366819272, 14754826166921601272757860127305944281026, 38630577173575883251502356955898138353472, 101140849479874096638391481397339911054398, 264801493228716210826791276834955726670780, 693286618257741613446748697215342130917818, 1815113859610231431129878097131924837788988
Generating function in Maple syntax:
-(x^5+4*x^4+3*x^3-11*x^2+6*x-1)/(2*x-1)/(x^2-3*x+1)/(x^2+2*x-1)
Generating function in latex syntax:
-\frac{x^{5}+4 x^{4}+3 x^{3}-11 x^{2}+6 x -1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+2 x -1\right)}
Generating function in sympy syntax:
(-x**5 - 4*x**4 - 3*x**3 + 11*x**2 - 6*x + 1)/((2*x - 1)*(x**2 - 3*x + 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*(x^2+2*x-1)*F(x)+x^5+4*x^4+3*x^3-11*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+x^{5}+4 x^{4}+3 x^{3}-11 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/10*(4*5^(1/2)+20)*(3/2-1/2*5^(1/2))^(-n)+1/10*(-4*5^(1/2)+20)*(3/2+1/2*5^(1/2))^(-n)+1/10*(15*2^(1/2)-20)*(-1-2^(1/2))^(-n)+1/10*(-15*2^(1/2)-20)*(2^(1/2)-1)^(-n)+3*2^(n-1))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(4 \sqrt{5}+20\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(-4 \sqrt{5}+20\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\\\frac{\left(15 \sqrt{2}-20\right) \left(-1-\sqrt{2}\right)^{-n}}{10}+\frac{\left(-15 \sqrt{2}-20\right) \left(\sqrt{2}-1\right)^{-n}}{10}+3 \,2^{n -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) = 20
a(5) = 66
a(n+5) = -2*a(n)+3*a(n+1)+11*a(n+2)-16*a(n+3)+7*a(n+4), 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) = 66
a \! \left(n +5\right) = -2 a \! \left(n \right)+3 a \! \left(n +1\right)+11 a \! \left(n +2\right)-16 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21385/
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[19,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[19,x]*F[7,x]
F[7,x] = F[26,x]+F[8,x]
F[8,x] = F[4,x]+F[9,x]
F[9,x] = F[10,x]*F[20,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[13,x]*F[19,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,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[19,x]
F[19,x] = x
F[20,x] = F[21,x]
F[21,x] = F[19,x]*F[22,x]
F[22,x] = F[23,x]+F[24,x]
F[23,x] = F[1,x]+F[20,x]
F[24,x] = F[20,x]*F[25,x]
F[25,x] = F[1,x]+F[19,x]
F[26,x] = F[27,x]
F[27,x] = F[28,x]+F[30,x]
F[28,x] = F[11,x]*F[29,x]
F[29,x] = F[20,x]+F[23,x]
F[30,x] = F[11,x]*F[17,x]*F[23,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_{19}\! \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_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{19}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{26}\! \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_{20}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{19}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \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_{19}\! \left(x \right)
F_{19}\! \left(x \right) = x
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{19}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{24}\! \left(x \right) = F_{20}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{30}\! \left(x \right) = F_{11}\! \left(x \right) F_{17}\! \left(x \right) F_{23}\! \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_19(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_19(x)*F_7(x))
Eq(F_7(x), F_26(x) + F_8(x))
Eq(F_8(x), F_4(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_20(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_13(x)*F_19(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(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_19(x))
Eq(F_19(x), x)
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_19(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_24(x))
Eq(F_23(x), F_1(x) + F_20(x))
Eq(F_24(x), F_20(x)*F_25(x))
Eq(F_25(x), F_1(x) + F_19(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_28(x) + F_30(x))
Eq(F_28(x), F_11(x)*F_29(x))
Eq(F_29(x), F_20(x) + F_23(x))
Eq(F_30(x), F_11(x)*F_17(x)*F_23(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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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": "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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[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], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [2, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": 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