0132_0231_0321_1032_2031_2301

Counting sequence:
1, 1, 2, 6, 18, 52, 146, 402, 1092, 2938, 7850, 20868, 55266, 145954, 384644, 1012074, 2659770, 6983620, 18323858, 48053490, 125967684, 330111706, 864891722, 2265612036, 5934041538, 15540706882, 40696467716, 106565473482, 279033507162, 730602156868, 1912907181170, 5008387822098, 13112793156036, 34331065387834, 89882550491114, 235320881052804, 616088682601890, 1612962346622050, 4222832717002628, 11055604523862570, 28944118293538554, 75777025234660036, 198387507166255442, 519386595775734066, 1359774479184202308, 3559941239823383962, 9320058036378971786, 24400250461499575812, 63880728532491844482, 167242005504720135298, 437845428719156916740, 1146294562127727325578, 3001038820613978481306, 7856823025614114960964, 20569432508028180086834, 53851479002070052670034, 140985013505381232664260, 369103579528472154804730, 966325761108832250713898, 2529873775855618635264900, 6623295710573211730936674, 17340013644094392709256866, 45396745798170718700257412, 118850224903339267998362346, 311153931217690094508523578, 814611573361417033954596292, 2132680798089933044210041106, 5583430839355126172385078642, 14617611756868933620364298052, 38269404505038650983546021978, 100190601905820971919950180810, 262302401507572169955657346308, 686716603207191348305727509826, 1797847409294593495678936486594, 4706825627037772380165904556804, 12322629476541090127688422397514, 32261062812030230968638653063130, 84460558978439068709706117646660, 221120614161065907023436861586418, 578901283580316516086518790531730, 1515583236730999368687948156847044, 3967848426914913044880982973685946, 10387962044618202675762315351563882, 27196037708148620802020592255711876, 71200151082245511369528719764984098, 186404415543423616585024083738065122, 488013095557696744942460564846860676, 1277634871149009431356191677597815722, 3344891517928017175353782601537184122, 8757039682712413347160492394194931908, 22926227530363965371038367115410002130, 60021642908688967775775954020759855538, 157138701196321907975932185084319126596, 411394460681514696191305981507096648474, 1077044680850698060676556519986769067274, 2819739581875531245995505099552807050244, 7382174064785799197624241820870845077250, 19326782612501673387505786447458114169090, 50598173772758835046150249690300269405188, 132467738705854059913459226961036237996810, 346805042344961801019255959867995532485914

Generating function in Maple syntax:
-(x^3+4*x^2-4*x+1)/(2*x-1)/(x^2-3*x+1)

Generating function in latex syntax:
-\frac{x^{3}+4 x^{2}-4 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right)}

Generating function in sympy syntax:
(-x**3 - 4*x**2 + 4*x - 1)/((2*x - 1)*(x**2 - 3*x + 1))

Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2-3*x+1)*F(x)+x^3+4*x^2-4*x+1 = 0

Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+x^{3}+4 x^{2}-4 x +1 = 0

Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/5*(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^(n-1))

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \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^{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(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2), n >= 4

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(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right), \quad n \geq 4

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22328/
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[38,x]+F[9,x]
F[9,x] = F[10,x]*F[18,x]
F[10,x] = F[11,x]+F[24,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[19,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[4,x]
F[18,x] = F[1,x]+F[17,x]
F[19,x] = F[20,x]+F[21,x]+F[23,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[13,x]+F[19,x]
F[23,x] = F[15,x]*F[4,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[31,x]
F[26,x] = F[12,x]+F[27,x]
F[27,x] = F[13,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[13,x]+F[28,x]
F[31,x] = F[15,x]+F[32,x]
F[32,x] = F[19,x]+F[33,x]
F[33,x] = 2*F[20,x]+F[34,x]+F[36,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[28,x]+F[33,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[19,x]+F[33,x]
F[38,x] = F[11,x]*F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[18,x]*F[4,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_{38}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \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_{19}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right)+F_{19}\! \left(x \right)
F_{23}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{13}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{19}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{28}\! \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_{19}\! \left(x \right)+F_{33}\! \left(x \right)
F_{38}\! \left(x \right) = F_{11}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \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_38(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_18(x))
Eq(F_10(x), F_11(x) + F_24(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_19(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_4(x))
Eq(F_18(x), F_1(x) + F_17(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_23(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_13(x) + F_19(x))
Eq(F_23(x), F_15(x)*F_4(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_31(x))
Eq(F_26(x), F_12(x) + F_27(x))
Eq(F_27(x), F_13(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_13(x) + F_28(x))
Eq(F_31(x), F_15(x) + F_32(x))
Eq(F_32(x), F_19(x) + F_33(x))
Eq(F_33(x), 2*F_20(x) + F_34(x) + F_36(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_28(x) + F_33(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_19(x) + F_33(x))
Eq(F_38(x), F_11(x)*F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_18(x)*F_4(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, 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": [0, 3, 2, 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, 0, 3, 1], "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, 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": [0, 3, 2, 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, 0, 3, 1], "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, 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": [0, 3, 2, 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, 0, 3, 1], "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, 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, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 0]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"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": [0, 3, 2, 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, 0, 3, 1], "pos": [[0, 0], 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