0132_0231_1203_1230_2031

Counting sequence:
1, 1, 2, 6, 19, 59, 178, 523, 1503, 4242, 11799, 32435, 88322, 238675, 641015, 1713042, 4559551, 12096683, 32009026, 84520219, 222796815, 586491666, 1542183207, 4051592291, 10636710914, 27908872099, 73194762983, 191893520658, 502938783823, 1317862354907, 3452594437954, 9043947490795, 23687569533567, 62035940979090, 162455686883895, 425404134116627, 1113907039321346, 2916626221492723, 7636607280316631, 19994501289515154, 52349576647821535, 137059726212088331, 358840871982628162, 939485979479979451, 2459664345457304559, 6439603813915178514, 16859345008381230663, 44138835831507535043, 115557989318885462018, 302536820975009114947, 792055921674606588359, 2073637980923228416530, 5428872376318890904687, 14213008421431022205755, 37210212560669238371650, 97417750857766631912587, 255043287710610162743391, 667712616677222121813138, 1748095589141771243169111, 4576576240418318707604339, 11981637383511233117392130, 31368344557026665195924371, 82123413869621707724797367, 215001932792405100790723986, 562882457141648384878734079, 1473645586206492643521891179, 3858054601237420743467153218, 10100518826248324019294771803, 26443503113439404252957120463, 69229993022827082764075609362, 181246481046343208383105953639, 474509460446379223662591156515, 1242281921248295730338718151682, 3252336345799806313180370221795, 8514727202334311521573417663655, 22291845435930688117716755675922, 58360809459635239046800240391311, 152790583660774734418870037979803, 400010942977177840933660599366466, 1047242248217515473692770373465899, 2741715807644439814491882071142975, 7177905186805062165929167587024786, 18791999777251494530491861477731447, 49198094194515380029746213009122771, 128802282906635488586760999050249474, 337208754728490623425794485492263219, 882823981689871160359596376826640023, 2311263191172863821547859517185505938, 6050965593511545045187545985920874783, 15841633592766106422049574196549710027, 41473935191672815689486104382854633794, 108580171996179166088369267044629338875, 284266580825027818469364098004988458927, 744219570535849531126850519615695752210, 1948392130897649198564727826407717350535, 5100956822389830791952984451288491653507, 13354478336742260392222670031919424815106, 34962478188787688334886197695592310195843, 91532956231542087553406833241500946568071, 239636390509720754288533254570994183074834, 627376215305463763401105099893242453739695

Generating function in Maple syntax:
-(x^4+7*x^3-12*x^2+6*x-1)/(x^2-3*x+1)/(2*x-1)^2

Generating function in latex syntax:
-\frac{x^{4}+7 x^{3}-12 x^{2}+6 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2}}

Generating function in sympy syntax:
(-x**4 - 7*x**3 + 12*x**2 - 6*x + 1)/((2*x - 1)**2*(x**2 - 3*x + 1))

Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(2*x-1)^2*F(x)+x^4+7*x^3-12*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 -1\right)^{2} F \! \left(x \right)+x^{4}+7 x^{3}-12 x^{2}+6 x -1 = 0

Explicit closed form in Maple syntax:
piecewise(n = 0,1,(3/2-1/2*5^(1/2))^(-n)+(3/2+1/2*5^(1/2))^(-n)+1/4*(-n-3)*2^n)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}+\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}+\frac{\left(-n -3\right) 2^{n}}{4} & \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(n+4) = -4*a(n)+16*a(n+1)-17*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) = 19
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 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/21712/
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[26,x]+F[9,x]
F[9,x] = F[10,x]+F[25,x]
F[10,x] = F[11,x]+F[20,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[1,x]+F[16,x]
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[12,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[17,x]*F[4,x]
F[20,x] = F[21,x]*F[23,x]
F[21,x] = F[22,x]
F[22,x] = F[13,x]*F[15,x]*F[17,x]*F[4,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[23,x]*F[4,x]
F[25,x] = F[12,x]*F[2,x]
F[26,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_{26}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{25}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{20}\! \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_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{23}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{2}\! \left(x \right)
F_{26}\! \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_26(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_25(x))
Eq(F_10(x), F_11(x) + F_20(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_1(x) + F_16(x))
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_12(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_17(x)*F_4(x))
Eq(F_20(x), F_21(x)*F_23(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_13(x)*F_15(x)*F_17(x)*F_4(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_23(x)*F_4(x))
Eq(F_25(x), F_12(x)*F_2(x))
Eq(F_26(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, 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, 2, 0, 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": [2, 0, 3, 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, 2, 0, 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": [2, 0, 3, 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, 2, 0, 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": [2, 0, 3, 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"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"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, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 1], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [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, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [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, 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, 2, 0, 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