0132_0231_0312_0321_1032

Counting sequence:
1, 1, 2, 6, 19, 64, 225, 816, 3031, 11473, 44096, 171631, 675130, 2679728, 10719237, 43168826, 174885089, 712222799, 2914150406, 11973792218, 49385167369, 204386777160, 848530495383, 3532844222611, 14747626307436, 61712139464939, 258815543183250, 1087703471299997, 4580023434516276, 19319905221037506, 81634001102022263, 345477340743566698, 1464226744876759237, 6214397852850104917, 26409290555485092888, 112369733987427758846, 478682467910036789613, 2041381669140133316794, 8714741340518018014553, 37240488729347203384012, 159288898409831167907685, 681938395710706386723561, 2921971028563390175436038, 12530277641932646566240065, 53775389340443054495819314, 230956197980441998692125566, 992624382880300627150824433, 4269101376595052490569036873, 18372657073593993607362728326, 79118846344110790267120468407, 340917729377162978642412662226, 1469842198424692865511624492536, 6340649123314001391471058035387, 27367148988479799727864125801768, 118181543760144840773724259664671, 510606020930216262008206019990401, 2207144209257554835968179083603938, 9545017491372836746626130133893415, 41296866173209253440212209988093634, 178749678541637317302882766749211263, 774025073664341025125660687954581762, 3353053333673777560144781924579934982, 14531014461580403295844175406546938355, 62996451396471699521298540575643150598, 273209502821629836163218328727564404777, 1185305227207831891813283306685033011741, 5144161836555234739011287758265897599336, 22332874207291919938365451469523652712116, 96987537519370633125582931274108221907675, 421332043912903262812986266206070737419470, 1830907452901936891505543590602406377390641, 7958622490994518788336693966108998239873824, 34604728549043529910076861917847511089051197, 150506613641590743119027276379068096908321293, 654779421405499007714041925573773785407289606, 2849381464223350962192732676325262548467183011, 12402786363348587642710550559970434324915251624, 54000540193620835958934646396875216566884556986, 235171293501227498631502012549916432088850014381, 1024413208554011668846965111451412058242788922638, 4463423397901475986201095604417721026328224565111, 19451833745818830564937298555745089769238708479399, 84791075346846165082235503709474604880637083912828, 369687306399604186939051688740486954180017520915244, 1612172428925119264078565547179860447410066689665569, 7031999447100109344705123152296395548844256251429942, 30678524018401912026876028961175042599655429427288115, 133867874966652398532153063770542960478766757700288961, 584255218547992593885124221499535686764578573107725364, 2550417426896425830002091237057917292747691921772090851, 11135264738302603209563617976584466623050100848628938724, 48626014439723922019421540555881205763985125487032021753, 212380160396381017191224615976587271184276428209137866546, 927757991805326921939757318167764143529554489419837980800, 4053492317463929546115240450800103821659360094967178174171, 17713172245945177061515417175791370589417192959273156489813, 77416614964929187790454500284821493030010306084007414839478, 338408603611016505657608384928522029062335482350585425377945, 1479505518715697207926165701870746978111223792896415431971616, 6469316382599284908671973379263263357065338621268892141423310, 28292118814572652770610620018214307105929154907217176452392649

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

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

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

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

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

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+5) = -n/(6+n)*a(n)-n/(6+n)*a(1+n)+2*(n+3)/(6+n)*a(n+2)+6*(n+4)/(6+n)*a(n+3)+3*(n+4)/(6+n)*a(n+4), 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 +5\right) = -\frac{n a \! \left(n \right)}{6+n}-\frac{n a \! \left(1+n \right)}{6+n}+\frac{2 \left(n +3\right) a \! \left(n +2\right)}{6+n}+\frac{6 \left(n +4\right) a \! \left(n +3\right)}{6+n}+\frac{3 \left(n +4\right) a \! \left(n +4\right)}{6+n}, \quad n \geq 5

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21641/
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[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[0,x]*F[4,x]*F[7,x]*F[8,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = 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_{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_{0}\! \left(x \right) F_{4}\! \left(x \right) F_{7}\! \left(x \right) F_{8}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = x
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_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_0(x)*F_4(x)*F_7(x)*F_8(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), 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, 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]]}], "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]]}], "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]]}], "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, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 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3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "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}}, {"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"}}, {"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": [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]]}], "requirements": []}, {"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": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": 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