0231_1302_2301
Counting sequence:
1, 1, 2, 6, 21, 77, 286, 1068, 4006, 15093, 57104, 216875, 826448, 3158726, 12104591, 46494761, 178964400, 690154864, 2666030048, 10314637719, 39962629319, 155029147429, 602124421192, 2341166890859, 9112047297626, 35498101445794, 138411229433499, 540118368436995, 2109283246232919, 8243057358492040, 32235192794044489, 126137300331057228, 493869708807281073, 1934739395084131865, 7583341387567257950, 29738199643136561340, 116673844325460555369, 457959218682889398631, 1798311947154666490039, 7064465969417921684820, 27762688272121680427363, 109145020197913883292736, 429239263383589402360252, 1688656875039767634166005, 6645438173172383162349545, 26160150935495283617941659, 103011526225420191106207398, 405746910858373985005962253, 1598614041387740810385153490, 6300086096106761166617554464, 24834734704514185845763499139, 97921645937598321268875641511, 386189273134113178916217012201, 1523422690033600613527448647114, 6010850487991561150093274903587, 23721572985659929298552949393732, 93635389285142647610657928736432, 369677157284855572467306173750235, 1459783652225106609488305723989807, 5765474032213080084605241918784853, 22775067702826142867908611080349366, 89982917434714705275270590771231875, 355577208459927188815260732873405764, 1405332325457885820566757629661401456, 5555118760667979230387674785496443818, 21962150456989392046856079088714400213, 86840372281091236088985944247721349984, 343425049252958233037826886440580061419, 1358325945141827503302698053664882044636, 5373238105496913758003398301550916032038, 21258207209460054407792497368915708486315, 84115127610823005579402705248404965423413, 332871816664241593264071865494801638264635, 1317449484608058068543427152899196426216486, 5214871148720391308643150225040486120350935, 20644508423590536006636722909217801211092656, 81736418241081330002385215301850313186418860, 323649945029546791023547045647546663068899583, 1281690548341677648854983231428485216725992615, 5076184611263411929348520410709707281890255201, 20106530216889967378979746955336139531556216408, 79649178736057624297803680913252228113247610759, 315550527910985876219385392731709662787255171786, 1250256099105360572539803086351737291168900244122, 4954166199796666289396241310594350392120106891009, 19632823892802503629877627373780093924110983927691, 77809871607254980079391146745177753019519596547191, 308407907112770307776944822088005937515017055945818, 1222515472054304492281364883170901722777914034449443, 4846413684269453086540735026673630619238494517561034, 19214234517894123901055932622652468667873278422506430, 76183589659484062369950639723223916472286055546254621, 302088918086074221299029926039372471405612134427210583, 1197960424533845898659137289655984902265406977033276459, 4750986368681191426198819649862885002858772902407407174, 18843348254413193130957247093149079784266668763687935295, 74741997892339629740985100091707687580472337232290993012, 296485219663463556203280465627635098251530686010567300662, 1176176528011780707914449874817298768775269987665230222191, 4666298310386872055767787745449441477196640509057877426921, 18514093504839881691826172608263227007813853918463739325913
Generating function in Maple syntax:
(4*(x-1)^3*(x-1/2)^2*(1-4*x)^(1/2)-12*x^6+64*x^5-114*x^4+101*x^3-47*x^2+11*x-1)/(8*x^7-64*x^6+192*x^5-276*x^4+218*x^3-96*x^2+22*x-2)
Generating function in latex syntax:
\frac{4 \left(x -1\right)^{3} \left(x -\frac{1}{2}\right)^{2} \sqrt{1-4 x}-12 x^{6}+64 x^{5}-114 x^{4}+101 x^{3}-47 x^{2}+11 x -1}{8 x^{7}-64 x^{6}+192 x^{5}-276 x^{4}+218 x^{3}-96 x^{2}+22 x -2}
Generating function in sympy syntax:
(-12*x**6 + 64*x**5 - 114*x**4 + 101*x**3 - 47*x**2 + 11*x + 4*sqrt(1 - 4*x)*(x - 1)**3*(x - 1/2)**2 - 1)/(8*x**7 - 64*x**6 + 192*x**5 - 276*x**4 + 218*x**3 - 96*x**2 + 22*x - 2)
Implicit equation for the generating function in Maple syntax:
(4*x^7-32*x^6+96*x^5-138*x^4+109*x^3-48*x^2+11*x-1)*F(x)^2+(12*x^6-64*x^5+114*x^4-101*x^3+47*x^2-11*x+1)*F(x)+x*(9*x^4-20*x^3+18*x^2-7*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(4 x^{7}-32 x^{6}+96 x^{5}-138 x^{4}+109 x^{3}-48 x^{2}+11 x -1\right) F \! \left(x \right)^{2}+\left(12 x^{6}-64 x^{5}+114 x^{4}-101 x^{3}+47 x^{2}-11 x +1\right) F \! \left(x \right)+x \left(9 x^{4}-20 x^{3}+18 x^{2}-7 x +1\right) = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 77
a(6) = 286
a(7) = 1068
a(8) = 4006
a(9) = 15093
a(n+10) = -16*(2*n+3)/(n+10)*a(n)+8*(39*n+82)/(n+10)*a(n+1)-4*(311*n+886)/(n+10)*a(n+2)+4*(669*n+2488)/(n+10)*a(n+3)-4*(877*n+4049)/(n+10)*a(n+4)+4*(743*n+4100)/(n+10)*a(n+5)-(10662+1661*n)/(n+10)*a(n+6)+(607*n+4442)/(n+10)*a(n+7)-(1142+139*n)/(n+10)*a(n+8)+2*(9*n+82)/(n+10)*a(n+9), n >= 10
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) = 21
a \! \left(5\right) = 77
a \! \left(6\right) = 286
a \! \left(7\right) = 1068
a \! \left(8\right) = 4006
a \! \left(9\right) = 15093
a \! \left(n +10\right) = -\frac{16 \left(2 n +3\right) a \! \left(n \right)}{n +10}+\frac{8 \left(39 n +82\right) a \! \left(n +1\right)}{n +10}-\frac{4 \left(311 n +886\right) a \! \left(n +2\right)}{n +10}+\frac{4 \left(669 n +2488\right) a \! \left(n +3\right)}{n +10}-\frac{4 \left(877 n +4049\right) a \! \left(n +4\right)}{n +10}+\frac{4 \left(743 n +4100\right) a \! \left(n +5\right)}{n +10}-\frac{\left(10662+1661 n \right) a \! \left(n +6\right)}{n +10}+\frac{\left(607 n +4442\right) a \! \left(n +7\right)}{n +10}-\frac{\left(1142+139 n \right) a \! \left(n +8\right)}{n +10}+\frac{2 \left(9 n +82\right) a \! \left(n +9\right)}{n +10}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/373/
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[10,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[10,x]*F[11,x]*F[7,x]
F[7,x] = F[1,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[7,x]
F[10,x] = x
F[11,x] = F[12,x]+F[4,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[14,x]*F[4,x]
F[14,x] = F[15,x]+F[20,x]
F[15,x] = F[16,x]+F[7,x]
F[16,x] = F[17,x]
F[17,x] = F[10,x]*F[15,x]*F[18,x]*F[7,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[10,x]*F[15,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]+F[8,x]
F[22,x] = F[23,x]+F[24,x]+F[26,x]
F[23,x] = 0
F[24,x] = F[10,x]*F[25,x]
F[25,x] = F[22,x]+F[8,x]
F[26,x] = F[10,x]*F[21,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_{10}\! \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_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{7}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{7}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{7}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{8}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{22}\! \left(x \right)+F_{8}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right) F_{21}\! \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_10(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_10(x)*F_11(x)*F_7(x))
Eq(F_7(x), F_1(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_7(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_4(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_14(x)*F_4(x))
Eq(F_14(x), F_15(x) + F_20(x))
Eq(F_15(x), F_16(x) + F_7(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_10(x)*F_15(x)*F_18(x)*F_7(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_10(x)*F_15(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x) + F_8(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_26(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_10(x)*F_25(x))
Eq(F_25(x), F_22(x) + F_8(x))
Eq(F_26(x), F_10(x)*F_21(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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [2, 1], [2, 1], [2, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1], [2, 1]], [[1, 0]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 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