01324_01342_03124_03142_10324_10342_13024_13042_30124_30142_31024_31042

Counting sequence:
1, 1, 2, 6, 24, 108, 504, 2376, 11232, 53136, 251424, 1189728, 5629824, 26640576, 126064512, 596543616, 2822874624, 13357986048, 63210668544, 299116094976, 1415432558592, 6697898781696, 31694797338624, 149981391341568, 709719564017664, 3358429036056576, 15892256832233472, 75202966777061376, 355864259668967424, 1683967757351436288, 7968620986094813184, 37707919372460261376, 178435790318192689152, 844367225674394566656, 3995588612137211265024, 18907328318776900190208, 89470438239838133551104, 423378659526367400165376, 2003449327719175599685632, 9480424009156849197121536, 44861848088626041584615424, 212288544476815154324963328, 1004560178329134676442087424, 4753629803113917132702744576, 22494417748708694737563942912, 106444727673568665629167190016, 503701859549159825349619482624, 2383542791253546958322713755648, 11279045590226322797838565638144, 53373016793836655037095111294976, 252563827221661993435539273940992, 1195144862566952030390664975876096, 5655486212071740221730754211610624, 26762048097028729148040535414407168, 126639371309741933557858687216779264, 599263939276279226458908910814232576, 2835747407799223757406301341584719872, 13418900811137667185684354584622923776, 63498920420030660569668319458229223424, 300480117653357960303903789241637797888, 1421887183399963798405412818700451446784, 6728442394479635028609054176752881893376, 31839331266478027381221848148314582679552, 150665333231990354115676763829370204717056, 712956011793073960406729494086333732225024, 3373744071366501637746316381541781165047808, 15964728357440566064037521324732684596936704, 75545905716444386557747229659145420591333376, 357487064154022922962258250006476415966380032, 1691646950625471218427066122083985972250279936, 8004959318828689772788847232465057337703399424, 37879874209219311326170686662286428192718716928, 179249489342343729320291036578928225130091905024, 848217690798746507964722099499850781624239128576, 4013809208738416671866586377525535338964883341312, 18993549107638020983411185668154107344043865276416, 89878439393397625869267595743771432030473891610624, 425309341714557629315138460453703948118580158005248, 2012585413926960020675225188259595096528637598367744, 9523656433274414348160520366835346890460344642174976, 45066426116084725964911771071454510763590242262843392, 213256618096861869700507504227714983238779385724010496, 1009141151884662862413574398937562834851134860767002624, 4775307202726805956278401368259087109674132850257952768, 22596996305052858563188961815929145648937987936945700864, 106930134613956315641463362686020351235583130520126488576, 505998829853420742469646405220547233519870855499084726272, 2394412171436786560969098255207161293705726349873749426176, 11330480049500194910996711099919684361115132966247988199424, 53616407268380450100165677068275138404456439698245432639488, 253715563313281531135013795810132724260047840391984666640384, 1200594936269406486209088712451145515133548404162435404005376, 5681276237736749730444449499846076745241003382622704424189952, 26884087808804059465412164724369587380644729870761614121107456, 127216869426403858409806291347141063812422358928833458181505024, 601996689705598793666364759736628858590665774348431064362385408, 2848678921675169611539350810336926768669460492517585637085282304, 13480093391817424907237916303601787460472768309014927436337381376, 63788486820853531774191392959589164150819846898984050795512594432, 301850360574216641201720859935924260142082471539814740155051278336, 1428371242520178656565176801858010575947575747844984136157232103424

Generating function in Maple syntax:
(2*x^2-5*x+1)/(6*x^2-6*x+1)

Generating function in latex syntax:
\frac{2 x^{2}-5 x +1}{6 x^{2}-6 x +1}

Generating function in sympy syntax:
(2*x**2 - 5*x + 1)/(6*x**2 - 6*x + 1)

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

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

Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/6*(2-3^(1/2))*(1/2-1/6*3^(1/2))^(-n)+1/6*(1/2+1/6*3^(1/2))^(-n)*(2+3^(1/2)))

Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(2-\sqrt{3}\right) \left(\frac{1}{2}-\frac{\sqrt{3}}{6}\right)^{-n}}{6}+\frac{\left(\frac{1}{2}+\frac{\sqrt{3}}{6}\right)^{-n} \left(2+\sqrt{3}\right)}{6} & \mathit{\text{otherwise}} \end{array}\right.

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(n+2) = -6*a(n)+6*a(n+1), n >= 3

Recurrence in latex format:
a(0) = 1
a(1) = 1
a(2) = 2
a{\left(n + 2 \right)} = - 6 a{\left(n \right)} + 6 a{\left(n + 1 \right)}, \quad n \geq 3

Specification 1
Strategy pack name: regular_insertion_encoding_left
Tree: http://permpal.com/tree/25957/
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[15,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[22,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[15,x]*F[8,x]
F[8,x] = F[10,x]+F[9,x]
F[9,x] = F[1,x]+F[6,x]
F[10,x] = F[11,x]+F[6,x]
F[11,x] = F[12,x]+F[13,x]+F[16,x]
F[12,x] = 0
F[13,x] = F[14,x]*F[15,x]
F[14,x] = F[6,x]
F[15,x] = x
F[16,x] = F[15,x]*F[17,x]
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[12,x]+F[16,x]+F[20,x]
F[20,x] = F[15,x]*F[6,x]
F[21,x] = F[19,x]
F[22,x] = F[12,x]+F[23,x]+F[47,x]
F[23,x] = F[15,x]*F[24,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[2,x]+F[22,x]
F[26,x] = F[22,x]+F[27,x]
F[27,x] = F[12,x]+F[28,x]+F[30,x]+F[44,x]
F[28,x] = F[15,x]*F[29,x]
F[29,x] = F[22,x]
F[30,x] = F[15,x]*F[31,x]
F[31,x] = F[32,x]+F[43,x]
F[32,x] = F[33,x]+F[39,x]
F[33,x] = F[12,x]+F[34,x]+F[35,x]
F[34,x] = F[15,x]*F[2,x]
F[35,x] = F[15,x]*F[36,x]
F[36,x] = F[37,x]+F[38,x]
F[37,x] = F[15,x]+F[33,x]
F[38,x] = F[19,x]+F[39,x]
F[39,x] = F[12,x]+F[30,x]+F[40,x]+F[41,x]
F[40,x] = F[15,x]*F[22,x]
F[41,x] = F[15,x]*F[42,x]
F[42,x] = F[38,x]
F[43,x] = F[39,x]
F[44,x] = F[15,x]*F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[11,x]+F[27,x]
F[47,x] = F[15,x]*F[48,x]
F[48,x] = F[46,x]+F[5,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_{15}\! \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_{22}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{15}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{10}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{6}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = 0
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{6}\! \left(x \right)
F_{15}\! \left(x \right) = x
F_{16}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right) F_{6}\! \left(x \right)
F_{21}\! \left(x \right) = F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right)+F_{23}\! \left(x \right)+F_{47}\! \left(x \right)
F_{23}\! \left(x \right) = F_{15}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{22}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)+F_{44}\! \left(x \right)
F_{28}\! \left(x \right) = F_{15}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{22}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{43}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{39}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right)+F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{15}\! \left(x \right) F_{2}\! \left(x \right)
F_{35}\! \left(x \right) = F_{15}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{38}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right)+F_{33}\! \left(x \right)
F_{38}\! \left(x \right) = F_{19}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right)+F_{30}\! \left(x \right)+F_{40}\! \left(x \right)+F_{41}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right) F_{22}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \left(x \right)
F_{43}\! \left(x \right) = F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{11}\! \left(x \right)+F_{27}\! \left(x \right)
F_{47}\! \left(x \right) = F_{15}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{46}\! \left(x \right)+F_{5}\! \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_15(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_22(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_15(x)*F_8(x))
Eq(F_8(x), F_10(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_6(x))
Eq(F_10(x), F_11(x) + F_6(x))
Eq(F_11(x), F_12(x) + F_13(x) + F_16(x))
Eq(F_12(x), 0)
Eq(F_13(x), F_14(x)*F_15(x))
Eq(F_14(x), F_6(x))
Eq(F_15(x), x)
Eq(F_16(x), F_15(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_15(x) + F_19(x))
Eq(F_19(x), F_12(x) + F_16(x) + F_20(x))
Eq(F_20(x), F_15(x)*F_6(x))
Eq(F_21(x), F_19(x))
Eq(F_22(x), F_12(x) + F_23(x) + F_47(x))
Eq(F_23(x), F_15(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_2(x) + F_22(x))
Eq(F_26(x), F_22(x) + F_27(x))
Eq(F_27(x), F_12(x) + F_28(x) + F_30(x) + F_44(x))
Eq(F_28(x), F_15(x)*F_29(x))
Eq(F_29(x), F_22(x))
Eq(F_30(x), F_15(x)*F_31(x))
Eq(F_31(x), F_32(x) + F_43(x))
Eq(F_32(x), F_33(x) + F_39(x))
Eq(F_33(x), F_12(x) + F_34(x) + F_35(x))
Eq(F_34(x), F_15(x)*F_2(x))
Eq(F_35(x), F_15(x)*F_36(x))
Eq(F_36(x), F_37(x) + F_38(x))
Eq(F_37(x), F_15(x) + F_33(x))
Eq(F_38(x), F_19(x) + F_39(x))
Eq(F_39(x), F_12(x) + F_30(x) + F_40(x) + F_41(x))
Eq(F_40(x), F_15(x)*F_22(x))
Eq(F_41(x), F_15(x)*F_42(x))
Eq(F_42(x), F_38(x))
Eq(F_43(x), F_39(x))
Eq(F_44(x), F_15(x)*F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_11(x) + F_27(x))
Eq(F_47(x), F_15(x)*F_48(x))
Eq(F_48(x), F_46(x) + F_5(x))
Pack JSON:
{"name": "regular_insertion_encoding_left", "initial_strats": [{"class_module": "tilings.strategies.factor", "strategy_class": "FactorFactory", "interleaving": null, "unions": false, "ignore_parent": true, "workable": true, "tracked": false}, {"class_module": "tilings.strategies.requirement_insertion", "strategy_class": "CellInsertionFactory", "ignore_parent": true, "maxreqlen": 1, "extra_basis": [], "one_cell_only": true}], "inferral_strats": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}], "expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "strategy_class": "RowAndColumnPlacementFactory", "partial": false, "ignore_parent": false, "dirs": [0], "place_row": false, "place_col": true}]], "symmetries": [], "iterative": false}
Specification JSON:
{"root": {"class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2, 4], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}], 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