0132_0231_0312_0321_1032_1302_1320_2031_2301

Counting sequence:
1, 1, 2, 6, 15, 38, 97, 249, 642, 1661, 4309, 11202, 29169, 76049, 198466, 518325, 1354461, 3540962, 9260233, 24223353, 63377058, 165842285, 434018725, 1135951746, 2973312225, 7782936353, 20373399682, 53333068389, 139617416877, 365502405026, 956856243769, 2504999217417, 6558007190754, 17168753919389, 44947717696501, 117673325428290, 308070111104721, 806532712918577, 2111519437716418, 5528008420361493, 14472471463629693, 37889337251050850, 99195402850569385, 259696596422750361, 679893836661867810, 1779983814051225293, 4660055406468552517, 12200178007307921154, 31940469819362188737, 83621213858592600641, 218923136572043524354, 573148125488793794757, 1500521099156849504589, 3928414890506778008354, 10284723009413531099161, 26925753011833908446505, 70492533774288380555106, 184551843807431605848317, 483162988640807182248853, 1264937104100591431416258, 3311648287632170093035953, 8670007686738324809763665, 22698374628467616260399170, 59425115910434147819722101, 155576972526374074895343645, 407305800515766572259461858, 1066340426715082632669347977, 2791715475017795307321194169, 7308805989114931252439458722, 19134702473880254376287630381, 50095301395632343729004329189, 131151201639229800515887150722, 343358303374483105228980710049, 898923708189071609991702153569, 2353412820602435914387420098946, 6161314752437644512453146839845, 16130531434349314381537197813741, 42230279545887932149288801387682, 110560307193869749100589915921913, 289450642016831849221002365523273, 757791618818846866699460018938338, 1983924214364150887151463367872605, 5193981024122490067303101437841205, 13598018857701087859854183651974466, 35600075548376310602452134930729105, 93202207786218918127887591965506673, 244006547807862592141981382616378562, 638817435632533155019598039184804309, 1672445759080065466359895701540384957, 4378519841588320430946254998641051746, 11463113765646210200251201160792172649, 30010821455272938917352012216554270937, 78569350600017864046894162954508249634, 205697230344471168213509131578245696909, 538522340432776670573990541642779278981, 1409869790952620903469177113075193015810, 3691087032422610159754970037033001520001, 9663391306310257815638591476924215047297, 25299086886498259766846521351540450628098, 66233869353164714444272406493298750849413, 173402521172956269484713565959559029944973

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

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

Generating function in sympy syntax:
(x - 1)**2*(-x**3 + x**2 + 2*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-x^2-2*x+1)*(x-1)^2 = 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)+\left(x^{3}-x^{2}-2 x +1\right) \left(x -1\right)^{2} = 0

Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/40*(-4*5^(1/2)+20)*(3/2-1/2*5^(1/2))^(-n)+1/40*(4*5^(1/2)+20)*(3/2+1/2*5^(1/2))^(-n)+1/8*2^n)

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-4 \sqrt{5}+20\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{40}+\frac{\left(4 \sqrt{5}+20\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{40}+\frac{2^{n}}{8} & \mathit{\text{otherwise}} \end{array}\right.

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 15
a(5) = 38
a(n+3) = 2*a(n)-7*a(n+1)+5*a(n+2), 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) = 15
a \! \left(5\right) = 38
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 6

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/23663/
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[11,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[11,x]*F[7,x]
F[7,x] = F[20,x]+F[8,x]
F[8,x] = F[12,x]*F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]*F[9,x]
F[11,x] = x
F[12,x] = F[13,x]+F[14,x]
F[13,x] = F[0,x]+F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[11,x]*F[16,x]
F[16,x] = F[17,x]+F[19,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[14,x]
F[19,x] = F[14,x]*F[9,x]
F[20,x] = F[21,x]
F[21,x] = F[9,x]^2*F[11,x]*F[14,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_{11}\! \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_{11}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{20}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{9}\! \left(x \right)
F_{11}\! \left(x \right) = x
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = F_{0}\! \left(x \right)+F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{11}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{19}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right) F_{9}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{11}\! \left(x \right) F_{14}\! \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_11(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_11(x)*F_7(x))
Eq(F_7(x), F_20(x) + F_8(x))
Eq(F_8(x), F_12(x)*F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x)*F_9(x))
Eq(F_11(x), x)
Eq(F_12(x), F_13(x) + F_14(x))
Eq(F_13(x), F_0(x) + F_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_11(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_19(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_14(x)*F_9(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_11(x)*F_14(x)*F_9(x)**2)
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, 1, 2], "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": [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]]}, {"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, 1, 2], "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": [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]]}, {"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, 1, 2], "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": [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]]}, {"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"}}, {"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, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"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, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"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, 1, 2], "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": [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]]}, {"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]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 2], [0, 2], [0, 0]]}, {"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, 1, 2], "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": [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]]}, {"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]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "requirements": [[{"patt": [0], "pos": 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