0132_0213_0231_1320

Counting sequence:
1, 1, 2, 6, 20, 67, 221, 725, 2394, 8011, 27222, 93880, 328053, 1159448, 4137936, 14891569, 53979192, 196895631, 722164597, 2661647906, 9852549489, 36613041432, 136535889101, 510787122692, 1916428908405, 7209391350940, 27187187744275, 102756082775586, 389184470566231, 1476873468512742, 5614522746411259, 21380212258800031, 81544324090613854, 311469426111544371, 1191347270738538202, 4562756922138591180, 17496455433573549901, 67170266642586683284, 258155389317300158432, 993201716385544691418, 3824923973466123616451, 14744019660138780622564, 56884896325685628044408, 219658318046561740995116, 848888922895662004265695, 3283149716317457281090530, 12707269898954970765272862, 49217740283632578820556414, 190759389146088176872317343, 739831487231052724669321816, 2871116904780926556184161426, 11148831355181618417488136912, 43316970823047989966014497731, 168394349493650623016104342714, 654980896726023091603696936740, 2548900682785501162133756997324, 9924126604039942110763917008597, 38657979486751789390061879814394, 150655936583118184703130920617522, 587390250183722052834870839592336, 2291153003280643358196469668530579, 8940506828172394178822129083503206, 34901572132394007618910561201683172, 136300372964408366238321774372635781, 532491851262282423751636323746908056, 2081074698851753547814050142891296191, 8136100421910125927396500710648725173, 31819564097046220990410587655805118650, 124484998530503629656725971950023961705, 487169897717633041243273044445006433216, 1907130696577957722928161582092961361301, 7468152448135843122107248979249185665808, 29253301392820100051548270910936456051145, 114620428881556623676691562253604886772440, 449232523114406821598558384996069283572351, 1761161132250527047844296487771790943515466, 6906254657049606237328654509066144859729663, 27089354409932906166587921487272890863287070, 106283142548017923792094571825098496993045375, 417096926107166771356371976350842744946099536, 1637245376115375390975777093146914385144714301, 6428240967312498189306037136344830260478463528, 25244665631602451849049065767518792174756782985, 99161661174653783914163513836946984323523269990, 389594075681840090664732030617070943804153928991, 1530992421142882962349177260437198198294561749886, 6017605705517177915191708954584834048702726126093, 23657144608288281801375972528269908382321900749690, 93022246097085746414125478195918320423833261545355, 365843445038442139252721473988212493519947800029086, 1439082915912643806073663122808511269072918619081781, 5661827405830659041758204132909191016509788325871958, 22279526160093026086766195280732606594863993627371675, 87686369933525280974391856588387589557420847268188530, 345170289987810038101679946032073498098366396080608601, 1358965111634336594821735405834842584783851421251549376, 5351250118421187919516493088101562306012905722304855413, 21075250430303977092933526192014685479029769861098666460, 83015529214002604790807313539716330555849522052781066195, 327049612125211120611257197622684362830451181846894881546, 1288647917197517307691546822816428649207369461216204221711

Generating function in Maple syntax:
1/2*((-x^6+3*x^5-3*x^4-5*x^3+9*x^2-5*x+1)*(1-4*x)^(1/2)+3*x^6-15*x^5+17*x^4-3*x^3-7*x^2+5*x-1)/x/(2*x-1)/(x-1)^4

Generating function in latex syntax:
\frac{\left(-x^{6}+3 x^{5}-3 x^{4}-5 x^{3}+9 x^{2}-5 x +1\right) \sqrt{1-4 x}+3 x^{6}-15 x^{5}+17 x^{4}-3 x^{3}-7 x^{2}+5 x -1}{2 x \left(2 x -1\right) \left(x -1\right)^{4}}

Generating function in sympy syntax:
(3*x**6/2 - 15*x**5/2 + 17*x**4/2 - 3*x**3/2 - 7*x**2/2 + 5*x/2 + sqrt(1 - 4*x)*(-x**6 + 3*x**5 - 3*x**4 - 5*x**3 + 9*x**2 - 5*x + 1)/2 - 1/2)/(x*(x - 1)**4*(2*x - 1))

Implicit equation for the generating function in Maple syntax:
x*(2*x-1)^2*(x-1)^8*F(x)^2-(2*x-1)*(3*x^6-15*x^5+17*x^4-3*x^3-7*x^2+5*x-1)*(x-1)^4*F(x)+x^12-4*x^11-6*x^10+70*x^9-169*x^8+188*x^7-50*x^6-135*x^5+199*x^4-135*x^3+52*x^2-11*x+1 = 0

Implicit equation for the generating function in latex syntax:
x \left(2 x -1\right)^{2} \left(x -1\right)^{8} F \! \left(x \right)^{2}-\left(2 x -1\right) \left(3 x^{6}-15 x^{5}+17 x^{4}-3 x^{3}-7 x^{2}+5 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{12}-4 x^{11}-6 x^{10}+70 x^{9}-169 x^{8}+188 x^{7}-50 x^{6}-135 x^{5}+199 x^{4}-135 x^{3}+52 x^{2}-11 x +1 = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 67
a(6) = 221
a(7) = 725
a(8) = 2394
a(9) = 8011
a(10) = 27222
a(11) = 93880
a(12) = 328053
a(n+9) = -4*(2*n-1)/(n+10)*a(n)+2*(19*n+4)/(n+10)*a(n+1)-(73*n+67)/(n+10)*a(n+2)+3*(8*n-49)/(n+10)*a(n+3)+(787+118*n)/(n+10)*a(n+4)-(198*n+1255)/(n+10)*a(n+5)+(1023+146*n)/(n+10)*a(n+6)-(58*n+459)/(n+10)*a(n+7)+(107+12*n)/(n+10)*a(n+8)+(n^2-n-4)/(n+10), n >= 13

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) = 20
a \! \left(5\right) = 67
a \! \left(6\right) = 221
a \! \left(7\right) = 725
a \! \left(8\right) = 2394
a \! \left(9\right) = 8011
a \! \left(10\right) = 27222
a \! \left(11\right) = 93880
a \! \left(12\right) = 328053
a \! \left(n +9\right) = -\frac{4 \left(2 n -1\right) a \! \left(n \right)}{n +10}+\frac{2 \left(19 n +4\right) a \! \left(n +1\right)}{n +10}-\frac{\left(73 n +67\right) a \! \left(n +2\right)}{n +10}+\frac{3 \left(8 n -49\right) a \! \left(n +3\right)}{n +10}+\frac{\left(787+118 n \right) a \! \left(n +4\right)}{n +10}-\frac{\left(198 n +1255\right) a \! \left(n +5\right)}{n +10}+\frac{\left(1023+146 n \right) a \! \left(n +6\right)}{n +10}-\frac{\left(58 n +459\right) a \! \left(n +7\right)}{n +10}+\frac{\left(107+12 n \right) a \! \left(n +8\right)}{n +10}+\frac{n^{2}-n -4}{n +10}, \quad n \geq 13

Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21168/
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[13,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[14,x]+F[8,x]
F[8,x] = F[9,x]^2*F[0,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[9,x]^2*F[13,x]
F[13,x] = x
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]*F[9,x]
F[16,x] = F[17,x]+F[37,x]
F[17,x] = F[18,x]*F[25,x]
F[18,x] = F[19,x]+F[34,x]
F[19,x] = F[20,x]+F[22,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[13,x]*F[20,x]
F[22,x] = F[23,x]+F[27,x]
F[23,x] = F[24,x]
F[24,x] = F[13,x]*F[25,x]
F[25,x] = F[1,x]+F[26,x]
F[26,x] = F[13,x]*F[25,x]
F[27,x] = F[28,x]+F[29,x]+F[33,x]
F[28,x] = 0
F[29,x] = F[13,x]*F[30,x]
F[30,x] = F[27,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[13,x]*F[20,x]
F[33,x] = F[13,x]*F[22,x]
F[34,x] = F[25,x]*F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[13,x]*F[20,x]*F[25,x]
F[37,x] = F[19,x]*F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[11,x]*F[13,x]*F[25,x]*F[9,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_{13}\! \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_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{14}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{0}\! \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_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{9}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{37}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{25}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{34}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{25}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)
F_{28}\! \left(x \right) = 0
F_{29}\! \left(x \right) = F_{13}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right) F_{22}\! \left(x \right)
F_{34}\! \left(x \right) = F_{25}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right) F_{25}\! \left(x \right)
F_{37}\! \left(x \right) = F_{19}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{25}\! \left(x \right) F_{9}\! \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_13(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_13(x)*F_7(x))
Eq(F_7(x), F_14(x) + F_8(x))
Eq(F_8(x), F_0(x)*F_9(x)**2)
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_9(x)**2)
Eq(F_13(x), x)
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x)*F_9(x))
Eq(F_16(x), F_17(x) + F_37(x))
Eq(F_17(x), F_18(x)*F_25(x))
Eq(F_18(x), F_19(x) + F_34(x))
Eq(F_19(x), F_20(x) + F_22(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_13(x)*F_20(x))
Eq(F_22(x), F_23(x) + F_27(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_13(x)*F_25(x))
Eq(F_25(x), F_1(x) + F_26(x))
Eq(F_26(x), F_13(x)*F_25(x))
Eq(F_27(x), F_28(x) + F_29(x) + F_33(x))
Eq(F_28(x), 0)
Eq(F_29(x), F_13(x)*F_30(x))
Eq(F_30(x), F_27(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_13(x)*F_20(x))
Eq(F_33(x), F_13(x)*F_22(x))
Eq(F_34(x), F_25(x)*F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_13(x)*F_20(x)*F_25(x))
Eq(F_37(x), F_19(x)*F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_11(x)*F_13(x)*F_25(x)*F_9(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, 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": [1, 3, 2, 0], "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, 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": [1, 3, 2, 0], "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, 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": [1, 3, 2, 0], "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": 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