0231_1032_2130
Counting sequence:
1, 1, 2, 6, 21, 75, 266, 935, 3263, 11326, 39155, 134955, 464094, 1593231, 5462447, 18709694, 64035275, 219039523, 748919582, 2559777591, 8746974087, 29883271518, 102078226067, 348648695387, 1190706243710, 4066228992799, 13885339795679, 47413708724094, 161896741574043, 552792500128083, 1887462784935518, 6444491990919623, 22003633680537335, 75127098749066014, 256505180374727267, 875777134210634763, 2990125953983119806, 10209022270971457839, 34856027568410300879, 119006564185710167102, 406315506575564995307, 1387252314385462553731, 4736387188785044015774, 16171067543097599423319, 55211557100611031273319, 188504253816892742613726, 643594321262489635397939, 2197369877503951422466235, 7502293747558020789885182, 25614442775337985060954687, 87453203346510118532271551, 298583979516073358865499902, 1019429646673125543104202555, 3480550981885203633947725491, 11883345561567255642661496030, 40572282710391843702725615591, 138522446074739856532425718487, 472945235519203489344815304862, 1614736093494110503169254520579, 5513054016997336059931358024811, 18822744179612250052363989976638, 64264869466228407520580474060367, 219413991552400241451578539979951, 749126232635503405756119852670910, 2557676961465579793620237629647499, 8732455417318053068474490069147619, 29814467842492908149675907486070046, 101792960787064352145304137956409207, 347542908122306213869494778835994951, 1186585712640471190266711477228231774, 4051257038834362828978248222649650131, 13831856741882405383251404912571205403, 47224912920821494723014236262864013694, 161235938280577068221577639423520052575, 550493927492872384880387484702092916383, 1879503833965900807302687354365347352190, 6417027482332347570682747131735510217947, 21909102265205490597599639174842250301459, 74802354106126483926222365822112386530142, 255391211920194702611784069740777358664199, 871960139536855870223783898338707195275959, 2977058134485924410456750622130729351667230, 10164312259338326878105891845598046346112931, 34703132769607591286904254431130901383213259, 118483906562963768200861341758575493616919230, 404529360721043928062610993054783939328030575, 1381149629780250232538187585624968092182332303, 4715539797736515206262953112276510688758633278, 16099859931536364699992245248591741844621915691, 54968360131067239275258073926134645624141172547, 187673720662229856024475994705581562402881140638, 640758162389491019629857400308415649526752286615, 2187685208240588960394458130339349083196216790823, 7469224508201921510007589706952755172252762297822, 25501527616375066648385878007252043328287844807283, 87067661449223551453272166949249635247132142518651, 297267590564476897626403979344815650361742514917886, 1014935039360331835049588951833579946763588390122623, 3465204976314654762187012886140817137733726742662015, 11830949826543927682922751385031677995805422210936318, 40393389353562036902897147962755657071544453208014459
Generating function in Maple syntax:
(x^4-3*x^3+10*x^2-6*x+1)/(x^2-3*x+1)/(2*x^2-4*x+1)
Generating function in latex syntax:
\frac{x^{4}-3 x^{3}+10 x^{2}-6 x +1}{\left(x^{2}-3 x +1\right) \left(2 x^{2}-4 x +1\right)}
Generating function in sympy syntax:
(x**4 - 3*x**3 + 10*x**2 - 6*x + 1)/((x**2 - 3*x + 1)*(2*x**2 - 4*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x^2-4*x+1)*F(x)-x^4+3*x^3-10*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(2 x^{2}-4 x +1\right) F \! \left(x \right)-x^{4}+3 x^{3}-10 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/20*(8*5^(1/2)-20)*(3/2-1/2*5^(1/2))^(-n)+1/20*(-8*5^(1/2)-20)*(3/2+1/2*5^(1/2))^(-n)+1/20*(-15*2^(1/2)+25)*(1-1/2*2^(1/2))^(-n)+1/20*(15*2^(1/2)+25)*(1+1/2*2^(1/2))^(-n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(8 \sqrt{5}-20\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\frac{\left(-8 \sqrt{5}-20\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{20}+\\\frac{\left(-15 \sqrt{2}+25\right) \left(1-\frac{\sqrt{2}}{2}\right)^{-n}}{20}+\frac{\left(15 \sqrt{2}+25\right) \left(1+\frac{\sqrt{2}}{2}\right)^{-n}}{20} & \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(n+4) = -2*a(n)+10*a(n+1)-15*a(n+2)+7*a(n+3), n >= 5
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(n +4\right) = -2 a \! \left(n \right)+10 a \! \left(n +1\right)-15 a \! \left(n +2\right)+7 a \! \left(n +3\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/356/
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[0,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[4,x]*F[8,x]
F[8,x] = F[23,x]+F[9,x]
F[9,x] = F[10,x]+F[5,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[15,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[4,x]
F[15,x] = F[16,x]+F[2,x]
F[16,x] = F[13,x]*F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[20,x]+F[22,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[17,x]
F[22,x] = F[12,x]*F[17,x]
F[23,x] = F[12,x]*F[6,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_{0}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{23}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{5}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{2}\! \left(x \right)
F_{16}\! \left(x \right) = F_{13}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{17}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{6}\! \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_0(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_4(x)*F_8(x))
Eq(F_8(x), F_23(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_5(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_15(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_2(x))
Eq(F_16(x), F_13(x)*F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_20(x) + F_22(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_17(x))
Eq(F_22(x), F_12(x)*F_17(x))
Eq(F_23(x), F_12(x)*F_6(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, 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, 1, 3, 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, 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, 1, 3, 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, 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, 1, 3, 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, 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, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [0, 1], [0, 1], [0, 0]]}], "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, 2, 1], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 2], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"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, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 2], [1, 2], [1, 2], [1, 0]]}], "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": [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, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[0, 1], [0, 1], [0, 1], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 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