02341_02413_02431_03241_03421_04213_04231_04321_20341_20413_20431_24013_24031_40213_40231_40321_42013_42031
Counting sequence:
1, 1, 2, 6, 24, 102, 428, 1768, 7248, 29644, 121208, 495688, 2027488, 8293576, 33926224, 138781152, 567707584, 2322299536, 9499734560, 38860162912, 158963623808, 650265780576, 2660014798528, 10881210401408, 44511308766464, 182080534731456, 744829169049472, 3046841288145024, 12463585235822080, 50984262794530944, 208559174869529856, 853144226045545984, 3489921126180068352, 14276073253661937920, 58398531134435869184, 238888409863328589312, 977210749285052897280, 3997434823500131419648, 16352138144020985949184, 66891002277059877234688, 273628203616030266183680, 1119319359336510597008384, 4578752524880801679308800, 18730109963013189472143360, 76618471367525751822016512, 313419951419867144851507200, 1282093784889408700437385216, 5244607006686728815172378624, 21453892826537758642090524672, 87760535122219338787257552896, 358998321992700705190772088832, 1468539304302223226772631937024, 6007291834428983165038013677568, 24573775504867503333626219552768, 100522907694066837515136340508672, 411204821549230030227561429762048, 1682098231578654145518493902897152, 6880888336911898902894650228916224, 28147359896226305142910147233808384, 115141218740261760930930842098761728, 471003330396547385547214789415403520, 1926713471264190303258198808492277760, 7881525587569640978092186618291355648, 32240624521484603295346101473749368832, 131885363815190389938309265773553254400, 539497898903113130149826335759205990400, 2206901315666310381132207067664379674624, 9027678192986540657899291175639505567744, 36929142675107999063344581596927713869824, 151064487409173204882576476105326335492096, 617953131405304452752360267680805819252736, 2527834828441837519165257875384290083602432, 10340507386623339942595454302688742044860416, 42299477722886393927837977992643548761817088, 173032690634074860883698006045224905978937344, 707817534396366013781507642743545891157180416, 2895439354049373441593832032509796174423654400, 11844251725308621819311300597089647474676072448, 48450781307603959172936541973687800228784111616, 198195568935875325530329912565394807795061096448, 810750260071665685196438145365399840176237182976, 3316501916442658471276112053388516291679043452928, 13566674601862676382315383193268759264613862735872, 55496623970067309318524249665918383915775731695616, 227017700538950711740236570535197225909246769496064, 928651738992082609021324232295459392452589606928384, 3798796526815552405451652932529288289554414335688704, 15539576836208279443411374829396864633116392635236352, 63567091983957085715588400636310676520677079917789184, 260031223880020848796004701702438030850043348269400064, 1063698767431524995364288153788371537018948221050814464, 4351227713935623777230917763058478455165960847284502528, 17799383808856624886102896759883569219422560547852255232, 72811189118031683976798985774022298346359289056210190336, 297845662395563840147493820687340059169959609026186575872, 1218384697220700437620021139064151149511280756450748530688, 4983994927044093420660236104425982278977278898933837332480, 20387817976920681686096064954405742422479893555500512444416, 83399587669037116725539003742047800600275439321714025037824, 341159178056186746362004915916485459225851734932537892405248, 1395565470105839182255627758268229096524509329351535214198784
Generating function in Maple syntax:
(x-1)*(6*x^3-10*x^2+6*x-1)/(14*x^4-28*x^3+22*x^2-8*x+1)
Generating function in latex syntax:
\frac{\left(x -1\right) \left(6 x^{3}-10 x^{2}+6 x -1\right)}{14 x^{4}-28 x^{3}+22 x^{2}-8 x +1}
Generating function in sympy syntax:
(x - 1)*(6*x**3 - 10*x**2 + 6*x - 1)/(14*x**4 - 28*x**3 + 22*x**2 - 8*x + 1)
Implicit equation for the generating function in Maple syntax:
(14*x^4-28*x^3+22*x^2-8*x+1)*F(x)-(x-1)*(6*x^3-10*x^2+6*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(14 x^{4}-28 x^{3}+22 x^{2}-8 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(6 x^{3}-10 x^{2}+6 x -1\right) = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/28*(2^(1/2)+4)*(1/2-1/14*I*(7+14*2^(1/2))^(1/2))^(-n)+1/28*(2^(1/2)+4)*(1/2+1/14*I*(7+14*2^(1/2))^(1/2))^(-n)-1/28*((1/2-1/14*(-7+14*2^(1/2))^(1/2))^(-n)+(1/2+1/14*(-7+14*2^(1/2))^(1/2))^(-n))*(2^(1/2)-4))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\sqrt{2}+4\right) \left(\frac{1}{2}-\frac{i \sqrt{7+14 \sqrt{2}}}{14}\right)^{-n}}{28}+\\\frac{\left(\sqrt{2}+4\right) \left(\frac{1}{2}+\frac{i \sqrt{7+14 \sqrt{2}}}{14}\right)^{-n}}{28}-\\\frac{\left(\left(\frac{1}{2}-\frac{\sqrt{-7+14 \sqrt{2}}}{14}\right)^{-n}+\left(\frac{1}{2}+\frac{\sqrt{-7+14 \sqrt{2}}}{14}\right)^{-n}\right) \left(\sqrt{2}-4\right)}{28} & \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+4) = -14*a(n)+28*a(n+1)-22*a(n+2)+8*a(n+3), n >= 5
Recurrence in latex format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a{\left(n + 4 \right)} = - 14 a{\left(n \right)} + 28 a{\left(n + 1 \right)} - 22 a{\left(n + 2 \right)} + 8 a{\left(n + 3 \right)}, \quad n \geq 5
Specification 1
Strategy pack name: regular_insertion_encoding_left
Tree: http://permpal.com/tree/24990/
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[0,x]+F[5,x]
F[5,x] = F[2,x]+F[6,x]
F[6,x] = F[13,x]+F[7,x]+F[8,x]
F[7,x] = 0
F[8,x] = F[12,x]*F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = F[2,x]+F[6,x]
F[11,x] = F[6,x]
F[12,x] = x
F[13,x] = F[12,x]*F[14,x]
F[14,x] = F[15,x]+F[30,x]
F[15,x] = F[16,x]+F[2,x]
F[16,x] = F[17,x]+F[21,x]+F[7,x]
F[17,x] = F[12,x]*F[18,x]
F[18,x] = F[19,x]+F[20,x]
F[19,x] = F[16,x]+F[2,x]
F[20,x] = F[16,x]
F[21,x] = F[12,x]*F[22,x]
F[22,x] = F[23,x]+F[24,x]
F[23,x] = F[2,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[2,x]+F[25,x]
F[29,x] = F[25,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]+F[36,x]+F[7,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[2,x]+F[31,x]
F[35,x] = F[31,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[2,x]+F[25,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_{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_{13}\! \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_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{2}\! \left(x \right)+F_{6}\! \left(x \right)
F_{11}\! \left(x \right) = F_{6}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{12}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{2}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{21}\! \left(x \right)+F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right)+F_{2}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right)
F_{21}\! \left(x \right) = F_{12}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)
F_{23}\! \left(x \right) = F_{2}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{2}\! \left(x \right)+F_{25}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{36}\! \left(x \right)+F_{7}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{31}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{2}\! \left(x \right)+F_{25}\! \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_0(x) + F_5(x))
Eq(F_5(x), F_2(x) + F_6(x))
Eq(F_6(x), F_13(x) + F_7(x) + F_8(x))
Eq(F_7(x), 0)
Eq(F_8(x), F_12(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), F_2(x) + F_6(x))
Eq(F_11(x), F_6(x))
Eq(F_12(x), x)
Eq(F_13(x), F_12(x)*F_14(x))
Eq(F_14(x), F_15(x) + F_30(x))
Eq(F_15(x), F_16(x) + F_2(x))
Eq(F_16(x), F_17(x) + F_21(x) + F_7(x))
Eq(F_17(x), F_12(x)*F_18(x))
Eq(F_18(x), F_19(x) + F_20(x))
Eq(F_19(x), F_16(x) + F_2(x))
Eq(F_20(x), F_16(x))
Eq(F_21(x), F_12(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_24(x))
Eq(F_23(x), F_2(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_2(x) + F_25(x))
Eq(F_29(x), F_25(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x) + F_36(x) + F_7(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_2(x) + F_31(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_2(x) + F_25(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, 3, 4, 1], "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, 3, 2, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 2, 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": [0, 4, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 4, 1], "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, 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": [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, 0, 3, 2, 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]]}], "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, 3, 4, 1], "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, 3, 2, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 4, 2, 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