02143_02413_02431_04213_04231_20143_20413_20431_21043_21403_24013_24031_24103_40213_40231_42013_42031_42103
Counting sequence:
1, 1, 2, 6, 24, 102, 428, 1762, 7160, 28878, 116004, 465018, 1862096, 7452454, 29817980, 119288274, 477185832, 1908808830, 7635366356, 30541727530, 122167434368, 488670786006, 1954685241132, 7818745158786, 31274989023704, 125099972871982, 500399925042308, 2001599767278042, 8006399203329840, 32025597081754758, 128102388863889884, 512409556529301298, 2049638228264688776, 8198552917353722334, 32794211678004823860, 131176846729199164554, 524707386951156396512, 2098829547873345062710, 8395318191630819204236, 33581272766798154723810, 134325091067742374709048, 537300364272069010463886, 2149201457090475065111012, 8596805828366298306955066, 34387223313473989320842384, 137548893253913549469413862, 550195573015689382249744188, 2200782292062827897743154322, 8803129168251452328460972520, 35212516673006090788820600638, 140850066692024926105235823764, 563400266768100830320850137578, 2253601067072405573083214235456, 9014404268289626795932484312214, 36057617073158516190929191989740, 144230468292634082778115277440834, 576921873170536367141258128727192, 2307687492682145540622626552836590, 9230749970728582306605694287202116, 36922999882914329514653153300520090, 147691999531657318635073365505503728, 590767998126629275693214966628861766, 2363071992506517105078702875729140892, 9452287970026068424926497521343951346, 37809151880104273708929362122230581064, 151236607520417094854164192562631875742, 604946430081668379453550258397946606068, 2419785720326673517887988009886624630602, 9679142881306694071699525992136174935200, 38716571525226776287093251873724052566518, 154866286100907105148963303305254915917644, 619465144403628420597033804841737074973858, 2477860577614513682390496402608383122502136, 9911442310458054729566707976916402135222094, 39645769241832218918276276640631347831315620, 158583076967328875673123996028456869906117114, 634332307869315502692533763045690436786177872, 2537329231477262010770210610046487661468130470, 10149316925909048043080993555913402474519359996, 40597267703636192172324276455108513555371116370, 162389070814544768689297710283343861536071818408, 649556283258179074757192050059195060773461979646, 2598225133032716299028770618088419472352197330772, 10392900532130865196115087308056956347925488147626, 41571602128523460784460358903634382308735350239744, 166286408514093843137841454957350643069008196257622, 665145634056375372551365858515028799944166375627948, 2660582536225501490205463511431367655112932683706882, 10642330144902005960821854200467975531124265097217880, 42569320579608023843287417111356911945842129113652398, 170277282318432095373149669064397667426058653904171524, 681109129273728381492598677495530708989614890515810138, 2724436517094913525970394712458002914529220111861488816, 10897746068379654103881578854783771815258401547042451974, 43590984273518616415526315429038607575316648387362801500, 174363937094074465662105261735961470929832677947837193394, 697455748376297862648421046983459964976462880588120748552, 2789822993505191450593684188013068022420115859946026944350, 11159291974020765802374736752210728414708992114971195677876, 44637167896083063209498947009159826308893025810258958512650, 178548671584332252837995788037273130535686217941784185653088
Generating function in Maple syntax:
(4*x^4-11*x^3+15*x^2-7*x+1)/(4*x-1)/(2*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{4 x^{4}-11 x^{3}+15 x^{2}-7 x +1}{\left(4 x -1\right) \left(2 x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(4*x**4 - 11*x**3 + 15*x**2 - 7*x + 1)/((x - 1)**2*(2*x - 1)*(4*x - 1))
Implicit equation for the generating function in Maple syntax:
(4*x-1)*(2*x-1)*(x-1)^2*F(x)-4*x^4+11*x^3-15*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(4 x -1\right) \left(2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)-4 x^{4}+11 x^{3}-15 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,8/9+2/3*n-2^(-1+n)+1/9*4^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{8}{9}+\frac{2 n}{3}-2^{-1+n}+\frac{4^{n}}{9} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(n+2) = -8*a(n)+6*a(n+1)+2*n, n >= 5
Recurrence in latex format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a{\left(n + 2 \right)} = 2 n - 8 a{\left(n \right)} + 6 a{\left(n + 1 \right)}, \quad n \geq 5
Specification 1
Strategy pack name: regular_insertion_encoding_left
Tree: http://permpal.com/tree/24899/
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[16,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[2,x]+F[6,x]
F[6,x] = F[28,x]+F[7,x]+F[8,x]
F[7,x] = 0
F[8,x] = F[16,x]*F[9,x]
F[9,x] = F[10,x]+F[27,x]
F[10,x] = F[11,x]+F[2,x]
F[11,x] = F[12,x]+F[17,x]+F[7,x]
F[12,x] = F[13,x]*F[16,x]
F[13,x] = F[14,x]+F[15,x]
F[14,x] = F[2,x]
F[15,x] = F[11,x]
F[16,x] = x
F[17,x] = F[16,x]*F[18,x]
F[18,x] = F[19,x]+F[26,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[21,x]
F[21,x] = F[16,x]*F[22,x]
F[22,x] = F[1,x]+F[20,x]
F[23,x] = F[17,x]+F[24,x]+F[7,x]
F[24,x] = F[16,x]*F[25,x]
F[25,x] = F[2,x]+F[23,x]
F[26,x] = F[23,x]
F[27,x] = F[6,x]
F[28,x] = F[16,x]*F[29,x]
F[29,x] = F[30,x]+F[39,x]
F[30,x] = F[31,x]+F[35,x]
F[31,x] = F[32,x]
F[32,x] = F[16,x]*F[33,x]
F[33,x] = F[22,x]+F[34,x]
F[34,x] = F[31,x]
F[35,x] = F[28,x]+F[36,x]+F[7,x]
F[36,x] = F[16,x]*F[37,x]
F[37,x] = F[25,x]+F[38,x]
F[38,x] = F[35,x]
F[39,x] = F[35,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_{16}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{28}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)
F_{7}\! \left(x \right) = 0
F_{8}\! \left(x \right) = F_{16}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{27}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{2}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{17}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{15}\! \left(x \right)
F_{14}\! \left(x \right) = F_{2}\! \left(x \right)
F_{15}\! \left(x \right) = F_{11}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{16}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{26}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{16}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{17}\! \left(x \right)+F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{16}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)
F_{27}\! \left(x \right) = F_{6}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{39}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{35}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{22}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{31}\! \left(x \right)
F_{35}\! \left(x \right) = F_{28}\! \left(x \right)+F_{36}\! \left(x \right)+F_{7}\! \left(x \right)
F_{36}\! \left(x \right) = F_{16}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{25}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{35}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \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_16(x)*F_4(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_2(x) + F_6(x))
Eq(F_6(x), F_28(x) + F_7(x) + F_8(x))
Eq(F_7(x), 0)
Eq(F_8(x), F_16(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_27(x))
Eq(F_10(x), F_11(x) + F_2(x))
Eq(F_11(x), F_12(x) + F_17(x) + F_7(x))
Eq(F_12(x), F_13(x)*F_16(x))
Eq(F_13(x), F_14(x) + F_15(x))
Eq(F_14(x), F_2(x))
Eq(F_15(x), F_11(x))
Eq(F_16(x), x)
Eq(F_17(x), F_16(x)*F_18(x))
Eq(F_18(x), F_19(x) + F_26(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_16(x)*F_22(x))
Eq(F_22(x), F_1(x) + F_20(x))
Eq(F_23(x), F_17(x) + F_24(x) + F_7(x))
Eq(F_24(x), F_16(x)*F_25(x))
Eq(F_25(x), F_2(x) + F_23(x))
Eq(F_26(x), F_23(x))
Eq(F_27(x), F_6(x))
Eq(F_28(x), F_16(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_39(x))
Eq(F_30(x), F_31(x) + F_35(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_16(x)*F_33(x))
Eq(F_33(x), F_22(x) + F_34(x))
Eq(F_34(x), F_31(x))
Eq(F_35(x), F_28(x) + F_36(x) + F_7(x))
Eq(F_36(x), F_16(x)*F_37(x))
Eq(F_37(x), F_25(x) + F_38(x))
Eq(F_38(x), F_35(x))
Eq(F_39(x), F_35(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, 2, 1, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 4, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 4, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 4, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 4, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 2, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [], "assumptions": []}, "rules": [{"class_module": "comb_spec_searcher.strategies.rule", "rule_class": "Rule", "comb_class": {"class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 4, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 4, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 4, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 4, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 4, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 4, 1, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 2, 1, 0, 3], "pos": 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