0213_0231_0312_1320_3021

Counting sequence:
1, 1, 2, 6, 19, 59, 188, 616, 2066, 7060, 24493, 86035, 305369, 1093489, 3945579, 14331321, 52359061, 192282091, 709390516, 2628009346, 9772119616, 36460337056, 136455900286, 512141527486, 1927142761834, 7269020766382, 27478809213176, 104090295706804, 395048648089264, 1501975729294720, 5720009321290580, 21817567776428278, 83339368594453014, 318779628491292564, 1220935956935104957, 4681937103852318843, 17974616110655725417, 69082546248421729465, 265782936790576371055, 1023559636137605104455, 3945530763669321265699, 15222444384086107400639, 58780309297626362801069, 227159512285731578662369, 878548613066830344248149, 3400335231171414865684069, 13169974956319848090598609, 51043743452316741142867789, 197962220185034879406479989, 768232993801506960766634833, 2983071898836889493638162747, 11590028997442283291191906279, 45055295767622208095855634163, 175242203894881366220479062643, 681953273689718140398998349151, 2655128544652879491044770172671, 10342461241419040379319940462135, 40305328194834841411044317691583, 157142765818278607727151868469683, 612933277558337940360485580245227, 2391733099675289542782699223391395, 9336563609004796948504555007214787, 36461174185076445069083337712340971, 142442031212203728685769207893479541, 556678572173425237793955751474815901, 2176330477832587524024481597547392363, 8511273544659939617268306902643695188, 33297313566509379601233738138562292338, 130306043218875426339031130315678865088, 510101483871939582209567175504954725488, 1997475263691517309878046190006638482838, 7824116608321991583595911096809670095038, 30655947408507149588314815605174206994938, 120147926537536799208343119162034541638438, 471017076403498835403859689000760081922668, 1847024741139707330056581440774400986190448, 7244717230132083925514591790202248922014388, 28423655149470498375418827250332614860025668, 111543784980452863628439218982074783618536768, 437839667268835901759997412389036956037503428, 1719042128412323649300719447268791003987083628, 6750829322561681783868928378176016408084057408, 26517008390911890966803793824021620627477527118, 104180490271929150280497451293713226191529925618, 409393087900820481658163692066713214917223785398, 1609106156868964847740178399584244816603112119158, 6325820835504489403642596402365159678552528896698, 24873395067493186849696818790583423615439582518778, 97822170750392631707415942234209533342912897494178, 384788154758229945601989259597004781410445432607578, 1513862559149127578903036417487777170857908785319718, 5957030200309482329439550079912138208591904064878318, 23444990270861952762407541658267352009335109152382518, 92288071119543590789584467360876517749129366905914518, 363341283892903721621240273784831404167614000188316718, 1430724608668624837560842362326241346832905595647918018, 5634663393142315203367454080092759373807527341349587618, 22194690229250581956019855125423194175076152797592358318, 87437549150724332751308091908094208712522040396498852568, 344519080377442706205793996501993803471657895699732707748, 1357668296876120591934248254130182397918920626571473314128

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

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

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

Implicit equation for the generating function in Maple syntax:
(x-1)^2*x*F(x)^2-(x-1)*(2*x^4+x^3-6*x^2+6*x-2)*F(x)+x^7+2*x^6-2*x^5-2*x^4-x^3+8*x^2-7*x+2 = 0

Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} x F \! \left(x \right)^{2}-\left(x -1\right) \left(2 x^{4}+x^{3}-6 x^{2}+6 x -2\right) F \! \left(x \right)+x^{7}+2 x^{6}-2 x^{5}-2 x^{4}-x^{3}+8 x^{2}-7 x +2 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 59
a(6) = 188
a(7) = 616
a(n+5) = -(-3+2*n)/(n+6)*a(n)-1/2*(10+3*n)/(n+6)*a(1+n)+1/2*(25*n+48)/(n+6)*a(n+2)-3*(5*n+16)/(n+6)*a(n+3)+(7*n+32)/(n+6)*a(n+4), n >= 8

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(5\right) = 59
a \! \left(6\right) = 188
a \! \left(7\right) = 616
a \! \left(n +5\right) = -\frac{\left(-3+2 n \right) a \! \left(n \right)}{n +6}-\frac{\left(10+3 n \right) a \! \left(1+n \right)}{2 \left(n +6\right)}+\frac{\left(25 n +48\right) a \! \left(n +2\right)}{12+2 n}-\frac{3 \left(5 n +16\right) a \! \left(n +3\right)}{n +6}+\frac{\left(7 n +32\right) a \! \left(n +4\right)}{n +6}, \quad n \geq 8

Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21813/
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[8,x]
F[4,x] = F[5,x]+F[9,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[5,x]^2*F[8,x]
F[8,x] = x
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[8,x]
F[11,x] = F[12,x]+F[13,x]
F[12,x] = F[5,x]^3
F[13,x] = F[14,x]+F[15,x]
F[14,x] = F[5,x]^2*F[6,x]
F[15,x] = F[16,x]*F[6,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[8,x]
F[18,x] = F[1,x]+F[17,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{5} \left(x \right)^{3}
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{5} \left(x \right)^{2} F_{6}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{6}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{8}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \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_8(x))
Eq(F_4(x), F_5(x) + F_9(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_5(x)**2*F_8(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_8(x))
Eq(F_11(x), F_12(x) + F_13(x))
Eq(F_12(x), F_5(x)**3)
Eq(F_13(x), F_14(x) + F_15(x))
Eq(F_14(x), F_5(x)**2*F_6(x))
Eq(F_15(x), F_16(x)*F_6(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_8(x))
Eq(F_18(x), F_1(x) + F_17(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, 1, 3], "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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 2, 1, 3], "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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 2, 1, 3], "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": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 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, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}], "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, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 2], [0, 2], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [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, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 1]]}], "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, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "rule_class": "EquivalencePathRule", "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, 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