0231_1320_2013
Counting sequence:
1, 1, 2, 6, 21, 72, 236, 745, 2286, 6866, 20285, 59156, 170712, 488401, 1387226, 3916062, 10996581, 30737760, 85573316, 237387961, 656451270, 1810142186, 4978643597, 13661617196, 37409025456, 102238082977, 278920277426, 759695287350, 2066068144821, 5611044289656, 15218767591580, 41227831944841, 111560745754782, 301559884395266, 814339327146461, 2197013877114692, 5922149224693896, 15950218778385265, 42925415134220426, 115435965714133326, 310215391253992965, 833098996695428496, 2235915055529227892, 5997257751335478361, 16076859486109941366, 43073712988449325082, 115344455035970237741, 308719786508203020380, 825895132098131516256, 2208446158225927974721, 5902804760289369054050, 15770571827331592679526, 42117360418063932814677, 112436254811246363762856, 300047190472475108638796, 800417930592192654649897, 2134488656805343982635086, 5690181592341548982731570, 15164174722271190906664061, 40399564828109977871097716, 107598067920920717167035960, 286488061157600142877393681, 762582834061861644746888762, 2029317174334982621961120126, 5398822170354097779648270501, 14359453047653347561921240512, 38182994623973043882416299556, 101507600067441044169292654585, 269790555656508769901055804966, 716898247893386049276692186186, 1904555980919393047881878890445, 5058690892560723320484701469836, 13433628496954823921966638336272, 35666608801183959244482695005921, 94677328715045639200289478265106, 251274355566418503723742353073302, 666761541843157922684199389222901, 1768948703317433871101284626115800, 4692284064313331460223679115417596, 12444563544880745211609397787495881, 32999187239899270511119424822853630, 87489680128269980629253963341056866, 231922118335699047962642814604508957, 614696788497739378709170442025054116, 1628976322823467357930356046724213736, 4316236293351595288927864342355684017, 11434936821702168020625649377713568746, 30290182851788524714420354338689116590, 80225233509293404435276812884695333701, 212452774322948067587863011506517448176, 562545237624489937005029603961877149140, 1489352126398482780460925020169053850521, 3942606556949902376802125733588083817942, 10435584602740095230184893791933655998586, 27618403010758051982046500199185459948141, 73085274649708195840596998865201992761532, 193380115839401272245377728018185749313216, 511617507351425695887793487995071679192705, 1353417014762631303689141412762413329331906, 3579904928096804605000789477872604007989446, 9468167334461036192505564526801321751128725
Generating function in Maple syntax:
(x^5-3*x^4+8*x^3-12*x^2+6*x-1)/(x-1)/(x^2-3*x+1)^2
Generating function in latex syntax:
\frac{x^{5}-3 x^{4}+8 x^{3}-12 x^{2}+6 x -1}{\left(x -1\right) \left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(x**5 - 3*x**4 + 8*x**3 - 12*x**2 + 6*x - 1)/((x - 1)*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^2-3*x+1)^2*F(x)-x^5+3*x^4-8*x^3+12*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)-x^{5}+3 x^{4}-8 x^{3}+12 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/50*((-10*n+7)*5^(1/2)+30*n-25)*(3/2-1/2*5^(1/2))^(-n)+1+1/50*((10*n-7)*5^(1/2)+30*n-25)*(3/2+1/2*5^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-10 n +7\right) \sqrt{5}+30 n -25\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+1+\\\frac{\left(\left(10 n -7\right) \sqrt{5}+30 n -25\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 72
a(n+4) = -a(n)+6*a(n+1)-11*a(n+2)+6*a(n+3)+1, 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) = 21
a \! \left(5\right) = 72
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)+1, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14770/
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[16,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[7,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[7,x]
F[15,x] = F[11,x]
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[18,x]+F[19,x]+F[36,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[25,x]
F[21,x] = F[22,x]+F[7,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[7,x]
F[25,x] = F[17,x]+F[26,x]
F[26,x] = 2*F[18,x]+F[27,x]+F[31,x]
F[27,x] = F[28,x]*F[4,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[22,x]
F[30,x] = F[26,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[14,x]+F[35,x]
F[35,x] = F[33,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[16,x]+F[38,x]
F[38,x] = F[33,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_{16}\! \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_{7}\! \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_{7}\! \left(x \right)
F_{15}\! \left(x \right) = F_{11}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{36}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{7}\! \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_{7}\! \left(x \right)
F_{25}\! \left(x \right) = F_{17}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{22}\! \left(x \right)
F_{30}\! \left(x \right) = F_{26}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{14}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{16}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{33}\! \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_16(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_7(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_7(x))
Eq(F_15(x), F_11(x))
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_36(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_25(x))
Eq(F_21(x), F_22(x) + F_7(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_7(x))
Eq(F_25(x), F_17(x) + F_26(x))
Eq(F_26(x), 2*F_18(x) + F_27(x) + F_31(x))
Eq(F_27(x), F_28(x)*F_4(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_22(x))
Eq(F_30(x), F_26(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_14(x) + F_35(x))
Eq(F_35(x), F_33(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_16(x) + F_38(x))
Eq(F_38(x), F_33(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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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": true, "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": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 0], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 2], [1, 0], [1, 2], [1, 2]]}, {"patt": [2, 0, 1, 3], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], 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