0132_0213_1203_2013_2301
Counting sequence:
1, 1, 2, 6, 19, 56, 152, 385, 923, 2119, 4702, 10160, 21507, 44818, 92300, 188435, 382267, 771977, 1554034, 3121498, 6260627, 12544092, 25117408, 50271797, 100589915, 201237307, 402545318, 805176916, 1610458339, 3221042390, 6442235028, 12884648551, 25769507963, 51539263709, 103078817146, 206157971486, 412316333683, 824633118208, 1649266754600, 3298534102569, 6597068882203, 13194138534383, 26388277941614, 52776556869688, 105553114851011, 211106230951258, 422212463302684, 844424928170747, 1688849858087355, 3377699718117361, 6755399438391554, 13510798879172642, 27021597760987219, 54043195524889700, 108086391052990192, 216172782109510237, 432345564222894299, 864691128450032739, 1729382256904707766, 3458764513814485340, 6917529027634498979, 13835058055275017374, 27670116110556579620, 55340232221120265679, 110680464442248237307, 221360928884504819909, 442721857769018666250, 885443715538047083878, 1770887431076104689971, 3541774862152220721032, 7083549724304453652280, 14167099448608920436433, 28334198897217854981275, 56668397794435725104791, 113336795588871466445438, 226673591177742950282688, 453347182355485919178115, 906694364710971858257570, 1813388729421943737775532, 3626777458843887498243811, 7253554917687775020688955, 14507109835375550067167065, 29014219670751100161793426, 58028439341502200352801770, 116056878683004400736662803, 232113757366008801506321260, 464227514732017603047670016, 928455029464035206132498309, 1856910058928070412304388187, 3713820117856140824650507403, 7427640235712281649345195206, 14855280471424563298737133924, 29710560942849126597523692131, 59421121885698253195099610982, 118842243771396506390254376884, 237684487542793012780566966839, 475368975085586025561195339131, 950737950171172051122455414701, 1901475900342344102244979039898, 3802951800684688204490029911982, 7605903601369376408980135430131
Generating function in Maple syntax:
-(4*x^6-15*x^5+30*x^4-33*x^3+21*x^2-7*x+1)/(2*x-1)/(-1+x)^6
Generating function in latex syntax:
-\frac{4 x^{6}-15 x^{5}+30 x^{4}-33 x^{3}+21 x^{2}-7 x +1}{\left(2 x -1\right) \left(-1+x \right)^{6}}
Generating function in sympy syntax:
(-4*x**6 + 15*x**5 - 30*x**4 + 33*x**3 - 21*x**2 + 7*x - 1)/((x - 1)**6*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(-1+x)^6*F(x)+4*x^6-15*x^5+30*x^4-33*x^3+21*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(-1+x \right)^{6} F \! \left(x \right)+4 x^{6}-15 x^{5}+30 x^{4}-33 x^{3}+21 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
-5+6*2^n-11/24*n^3-n^2-68/15*n-1/120*n^5
Explicit closed form in latex syntax:
-5+6 \,2^{n}-\frac{11 n^{3}}{24}-n^{2}-\frac{68 n}{15}-\frac{n^{5}}{120}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 56
a(6) = 152
a(n+1) = 2*a(n)+1/120*(n-1)*(n^4-4*n^3+41*n^2-14*n+120), n >= 7
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) = 19
a \! \left(5\right) = 56
a \! \left(6\right) = 152
a \! \left(n +1\right) = 2 a \! \left(n \right)+\frac{\left(n -1\right) \left(n^{4}-4 n^{3}+41 n^{2}-14 n +120\right)}{120}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21624/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[0,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[4,x]*F[8,x]
F[8,x] = F[20,x]+F[9,x]
F[9,x] = F[10,x]*F[15,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[1,x]+F[12,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[4,x]
F[14,x] = F[15,x]*F[17,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[15,x]*F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[19,x]*F[4,x]
F[19,x] = F[1,x]+F[18,x]
F[20,x] = F[21,x]
F[21,x] = F[10,x]*F[15,x]*F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[32,x]
F[23,x] = F[15,x]+F[24,x]
F[24,x] = F[25,x]+F[27,x]
F[25,x] = F[26,x]
F[26,x] = F[15,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]+F[31,x]
F[28,x] = 0
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[25,x]+F[27,x]
F[31,x] = F[24,x]*F[4,x]
F[32,x] = F[25,x]+F[33,x]
F[33,x] = F[28,x]+F[34,x]+F[35,x]
F[34,x] = F[32,x]*F[4,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[25,x]+F[33,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{0}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{20}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right) F_{4}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right) F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{32}\! \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_{27}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = 0
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right)+F_{27}\! \left(x \right)
F_{31}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{25}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{28}\! \left(x \right)+F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{25}\! \left(x \right)+F_{33}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_0(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_4(x)*F_8(x))
Eq(F_8(x), F_20(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_15(x))
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_1(x) + F_12(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_4(x))
Eq(F_14(x), F_15(x)*F_17(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_15(x)*F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_19(x)*F_4(x))
Eq(F_19(x), F_1(x) + F_18(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_10(x)*F_15(x)*F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_32(x))
Eq(F_23(x), F_15(x) + F_24(x))
Eq(F_24(x), F_25(x) + F_27(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_15(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x) + F_31(x))
Eq(F_28(x), 0)
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_25(x) + F_27(x))
Eq(F_31(x), F_24(x)*F_4(x))
Eq(F_32(x), F_25(x) + F_33(x))
Eq(F_33(x), F_28(x) + F_34(x) + F_35(x))
Eq(F_34(x), F_32(x)*F_4(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_25(x) + F_33(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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"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]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": []}], "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": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 0], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 0], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 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