0123_1302

Counting sequence:
1, 1, 2, 6, 22, 86, 338, 1314, 5046, 19190, 72482, 272530, 1021734, 3823622, 14293234, 53394370, 199382550, 744348822, 2778471490, 10370520178, 38705706374, 144456761766, 539130777874, 2012086272674, 7509256255862, 28025026831158, 104591035618146, 390339501517394, 1456767775757798, 5436733279235398, 20290168830844722, 75723949291900802, 282605643369144022, 1054698655323188182, 3936189042348118146, 14690057647213270578, 54824041821382871110, 204606110205253896934, 763600400167863821138, 2849795492871383073378, 10635581576265470797366, 39692530822360982673014, 148134541734069180822434, 552845636156796694099986, 2063248002981078525799590, 7700146375947837316053638, 28737337501179706645347698, 107249203629527453265247042, 400259477018478218787549078, 1493788704447552015372944150, 5574895340778203767168572610, 20805792658678492377206747122, 77648275293962787339422638854, 289787308517227826075919096870, 1081500958775061106954938011026, 4036216526583246285324828842530, 15063365147558392408705623890678, 56217244063651278112618669265334, 209805611107048665596808077224930, 783005200364547347442285725670866, 2922215190351148794622867073387366, 10905855561040064260180623215448006, 40701207053809141680823259386966962, 151898972654196570485481186136391426, 566894683562977278611682037980235862, 2115679761597712825274094089855214166, 7895824362827874594333760606436720898, 29467617689713786714205824979593421234, 109974646396027274623672780746759570886, 410430967894395316576638757171928282470, 1531749225181554001422763118859596812242, 5716565932831820708889323365283098298850, 21334514506145728874274645446664780699574, 79621492091751094869670080250877404435702, 297151453860858650769688502457282419522594, 1108984323351683508544371949861997083827090, 4138785839545875384087820070523934826557990, 15446159034831818029185839345231678624804102, 57645850299781396735451178268261609639166450, 215137242164293768918285713507258334188296898, 802903118357393678949176471047110704273729814, 2996475231265280946901691992708766094519716246, 11182997806703730108704739606752924211618676034, 41735515995549639488012771574052486456756775794, 155759066175494827843539774820595362283361415302, 581300748706429671886538019673884102529293686438, 2169443928650223859703405359212608244572420581906, 8096474965894465766928688870664997103304398442914, 30216455934927639208014599716050504291866783390838, 112769348773816091065136286549995723646873136717878, 420860939160336725052543854339354708133580929066082, 1570674407867530809145066056003277565908429635498322, 5861836692309786511527774839035484112232233174393062, 21876672361371615236966143476802155281864969084129862, 81644852753176674436337021897380208383677125004486706, 304702738651335082508382394722892978131263594215034498, 1137166101852163655597193468118060618181259577611080150, 4243961668757319539880393319804127951239623766126131414, 15838680573177114503924383534822089357201102383459111298, 59110760623951138475817148346159668332676857154385595698, 220604361922627439399344225061623786712259144194521736006

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

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

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

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

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

Explicit closed form in Maple syntax:
1/18*(-2*3^(1/2)+6)*(2-3^(1/2))^(-n)+1/18*(2*3^(1/2)+6)*(2+3^(1/2))^(-n)-1/6*2^n*(n-2)

Explicit closed form in latex syntax:
\frac{\left(-2 \sqrt{3}+6\right) \left(2-\sqrt{3}\right)^{-n}}{18}+\frac{\left(2 \sqrt{3}+6\right) \left(2+\sqrt{3}\right)^{-n}}{18}-\frac{2^{n} \left(n -2\right)}{6}

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = -4*a(n)+20*a(n+1)-21*a(n+2)+8*a(n+3), n >= 4

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(n +4\right) = -4 a \! \left(n \right)+20 a \! \left(n +1\right)-21 a \! \left(n +2\right)+8 a \! \left(n +3\right), \quad n \geq 4

Specification 1
Strategy pack name: regular_insertion_encoding_bottom
Tree: http://permpal.com/tree/14597/
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[21,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[13,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[15,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[14,x]
F[13,x] = x
F[14,x] = F[1,x]+F[11,x]
F[15,x] = F[16,x]+F[17,x]+F[19,x]
F[16,x] = 0
F[17,x] = F[13,x]*F[18,x]
F[18,x] = F[15,x]+F[6,x]
F[19,x] = F[13,x]*F[20,x]
F[20,x] = F[10,x]
F[21,x] = F[16,x]+F[22,x]+F[48,x]
F[22,x] = F[13,x]*F[23,x]
F[23,x] = F[24,x]+F[25,x]
F[24,x] = F[2,x]+F[21,x]
F[25,x] = F[26,x]+F[43,x]
F[26,x] = F[16,x]+F[27,x]+F[29,x]
F[27,x] = F[13,x]*F[28,x]
F[28,x] = F[2,x]+F[26,x]
F[29,x] = F[13,x]*F[30,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[11,x]+F[26,x]
F[32,x] = F[33,x]+F[37,x]
F[33,x] = F[34,x]
F[34,x] = F[13,x]*F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[11,x]+F[33,x]
F[37,x] = 2*F[16,x]+F[38,x]+F[41,x]
F[38,x] = F[13,x]*F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[26,x]+F[37,x]
F[41,x] = F[13,x]*F[42,x]
F[42,x] = F[32,x]
F[43,x] = F[16,x]+F[41,x]+F[44,x]+F[46,x]
F[44,x] = F[13,x]*F[45,x]
F[45,x] = F[21,x]+F[43,x]
F[46,x] = F[13,x]*F[47,x]
F[47,x] = F[25,x]
F[48,x] = F[13,x]*F[49,x]
F[49,x] = F[5,x]+F[50,x]
F[50,x] = F[51,x]+F[52,x]
F[51,x] = F[34,x]
F[52,x] = 2*F[16,x]+F[38,x]+F[53,x]
F[53,x] = F[13,x]*F[54,x]
F[54,x] = F[50,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_{21}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{13}\! \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_{15}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = 0
F_{17}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{6}\! \left(x \right)
F_{19}\! \left(x \right) = F_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right)
F_{21}\! \left(x \right) = F_{16}\! \left(x \right)+F_{22}\! \left(x \right)+F_{48}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{2}\! \left(x \right)+F_{21}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{43}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{2}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{13}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{11}\! \left(x \right)+F_{26}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{13}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{11}\! \left(x \right)+F_{33}\! \left(x \right)
F_{37}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{13}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{26}\! \left(x \right)+F_{37}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{32}\! \left(x \right)
F_{43}\! \left(x \right) = F_{16}\! \left(x \right)+F_{41}\! \left(x \right)+F_{44}\! \left(x \right)+F_{46}\! \left(x \right)
F_{44}\! \left(x \right) = F_{13}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{21}\! \left(x \right)+F_{43}\! \left(x \right)
F_{46}\! \left(x \right) = F_{13}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{25}\! \left(x \right)
F_{48}\! \left(x \right) = F_{13}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{5}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{51}\! \left(x \right) = F_{34}\! \left(x \right)
F_{52}\! \left(x \right) = 2 F_{16}\! \left(x \right)+F_{38}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{13}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{50}\! \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_21(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_13(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_15(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_14(x))
Eq(F_13(x), x)
Eq(F_14(x), F_1(x) + F_11(x))
Eq(F_15(x), F_16(x) + F_17(x) + F_19(x))
Eq(F_16(x), 0)
Eq(F_17(x), F_13(x)*F_18(x))
Eq(F_18(x), F_15(x) + F_6(x))
Eq(F_19(x), F_13(x)*F_20(x))
Eq(F_20(x), F_10(x))
Eq(F_21(x), F_16(x) + F_22(x) + F_48(x))
Eq(F_22(x), F_13(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_25(x))
Eq(F_24(x), F_2(x) + F_21(x))
Eq(F_25(x), F_26(x) + F_43(x))
Eq(F_26(x), F_16(x) + F_27(x) + F_29(x))
Eq(F_27(x), F_13(x)*F_28(x))
Eq(F_28(x), F_2(x) + F_26(x))
Eq(F_29(x), F_13(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_11(x) + F_26(x))
Eq(F_32(x), F_33(x) + F_37(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_13(x)*F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_11(x) + F_33(x))
Eq(F_37(x), 2*F_16(x) + F_38(x) + F_41(x))
Eq(F_38(x), F_13(x)*F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_26(x) + F_37(x))
Eq(F_41(x), F_13(x)*F_42(x))
Eq(F_42(x), F_32(x))
Eq(F_43(x), F_16(x) + F_41(x) + F_44(x) + F_46(x))
Eq(F_44(x), F_13(x)*F_45(x))
Eq(F_45(x), F_21(x) + F_43(x))
Eq(F_46(x), F_13(x)*F_47(x))
Eq(F_47(x), F_25(x))
Eq(F_48(x), F_13(x)*F_49(x))
Eq(F_49(x), F_5(x) + F_50(x))
Eq(F_50(x), F_51(x) + F_52(x))
Eq(F_51(x), F_34(x))
Eq(F_52(x), 2*F_16(x) + F_38(x) + F_53(x))
Eq(F_53(x), F_13(x)*F_54(x))
Eq(F_54(x), F_50(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [1], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "inferral_strats": [], "initial_strats": [{"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "regular_insertion_encoding_bottom", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 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"requirements": [[{"patt": [0], "pos": [[0, 0]]}], [{"patt": [0], "pos": [[1, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 0], "pos": [[0, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [2, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [1, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [3, 0], [3, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [3, 0], [3, 0]]}, {"patt": [1, 0, 2], "pos": [[2, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [1, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [2, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [2, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[2, 0], [2, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[2, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[3, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [3, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [3, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[2, 0], [2, 0], [2, 0], [3, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[3, 0], [3, 0], [3, 0], [3, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}], [{"patt": [0], "pos": [[1, 0]]}], [{"patt": [0], "pos": [[2, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [1, 0], "pos": [[0, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [2, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [1, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [3, 0], [3, 0]]}, {"patt": [1, 0, 2], "pos": [[1, 0], [3, 0], [3, 0]]}, {"patt": [1, 0, 2], "pos": [[2, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [1, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [2, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [2, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[2, 0], [2, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[2, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[3, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [3, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [3, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[2, 0], [2, 0], [2, 0], [3, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[3, 0], [3, 0], [3, 0], [3, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}], [{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[2, 0]]}], "ignore_parent": true, "strategy_class": "RequirementInsertionStrategy"}}, "rule_class": "EquivalenceRule"}]}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [], "pos": []}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "comb_spec_searcher.strategies.strategy", "strategy_class": "EmptyStrategy"}}]}

Specification 2
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14598/
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[12,x]*F[4,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[6,x]
F[14,x] = F[15,x]+F[16,x]+F[20,x]
F[15,x] = 0
F[16,x] = F[12,x]*F[17,x]
F[17,x] = F[18,x]+F[19,x]
F[18,x] = F[10,x]
F[19,x] = F[14,x]
F[20,x] = F[12,x]*F[13,x]
F[21,x] = F[2,x]+F[22,x]
F[22,x] = F[15,x]+F[23,x]+F[53,x]
F[23,x] = F[12,x]*F[24,x]
F[24,x] = F[25,x]+F[31,x]
F[25,x] = F[26,x]+F[6,x]
F[26,x] = F[27,x]
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[18,x]+F[29,x]
F[29,x] = F[30,x]
F[30,x] = F[27,x]
F[31,x] = F[22,x]+F[32,x]
F[32,x] = 2*F[15,x]+F[33,x]+F[37,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[26,x]
F[36,x] = F[32,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[52,x]
F[39,x] = F[40,x]
F[40,x] = F[15,x]+F[41,x]+F[50,x]
F[41,x] = F[12,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[10,x]+F[30,x]
F[44,x] = F[40,x]+F[45,x]
F[45,x] = 2*F[15,x]+F[37,x]+F[46,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[30,x]
F[49,x] = F[45,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[2,x]+F[40,x]
F[52,x] = F[45,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[51,x]+F[55,x]
F[55,x] = F[22,x]+F[56,x]
F[56,x] = F[15,x]+F[46,x]+F[57,x]+F[60,x]
F[57,x] = F[12,x]*F[58,x]
F[58,x] = F[39,x]+F[59,x]
F[59,x] = F[56,x]
F[60,x] = F[12,x]*F[55,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \left(x \right)+F_{5}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \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_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{6}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right)
F_{19}\! \left(x \right) = F_{14}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{53}\! \left(x \right)
F_{23}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{31}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{6}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{18}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{27}\! \left(x \right)
F_{31}\! \left(x \right) = F_{22}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{37}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{26}\! \left(x \right)
F_{36}\! \left(x \right) = F_{32}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{52}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{50}\! \left(x \right)
F_{41}\! \left(x \right) = F_{12}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right)+F_{30}\! \left(x \right)
F_{44}\! \left(x \right) = F_{40}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{30}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{2}\! \left(x \right)+F_{40}\! \left(x \right)
F_{52}\! \left(x \right) = F_{45}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{51}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{22}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{57}\! \left(x \right)+F_{60}\! \left(x \right)
F_{57}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{39}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{56}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_21(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_6(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_20(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_12(x)*F_17(x))
Eq(F_17(x), F_18(x) + F_19(x))
Eq(F_18(x), F_10(x))
Eq(F_19(x), F_14(x))
Eq(F_20(x), F_12(x)*F_13(x))
Eq(F_21(x), F_2(x) + F_22(x))
Eq(F_22(x), F_15(x) + F_23(x) + F_53(x))
Eq(F_23(x), F_12(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_31(x))
Eq(F_25(x), F_26(x) + F_6(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_18(x) + F_29(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_27(x))
Eq(F_31(x), F_22(x) + F_32(x))
Eq(F_32(x), 2*F_15(x) + F_33(x) + F_37(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_26(x))
Eq(F_36(x), F_32(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_52(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_15(x) + F_41(x) + F_50(x))
Eq(F_41(x), F_12(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_10(x) + F_30(x))
Eq(F_44(x), F_40(x) + F_45(x))
Eq(F_45(x), 2*F_15(x) + F_37(x) + F_46(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_30(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_2(x) + F_40(x))
Eq(F_52(x), F_45(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_51(x) + F_55(x))
Eq(F_55(x), F_22(x) + F_56(x))
Eq(F_56(x), F_15(x) + F_46(x) + F_57(x) + F_60(x))
Eq(F_57(x), F_12(x)*F_58(x))
Eq(F_58(x), F_39(x) + F_59(x))
Eq(F_59(x), F_56(x))
Eq(F_60(x), F_12(x)*F_55(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": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
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