0132_0213_0231_0321_1302_3120
Counting sequence:
1, 1, 2, 6, 18, 47, 115, 269, 610, 1355, 2969, 6445, 13896, 29803, 63639, 135373, 286990, 606555, 1278389, 2687517, 5636756, 11797275, 24642483, 51381341, 106956058, 222299627, 461375185, 956303309, 1979713760, 4093643275, 8455719759, 17448307885, 35970354726, 74088189883, 152471343469, 313532617533, 644245099820, 1322849933115, 2714419337579, 5566277622717, 11407433145906, 23364622098635, 47828755817353, 97856534881901, 200111116265848, 409018325544107, 835628837122055, 1706442046321549, 3483252836808510, 7107243161959195, 14495961300614949, 29554872554636125, 60235645016098756, 122723089845865563, 249949779319083299, 508906757892888093, 1035827914295237450, 2107684625609416875, 4287426845256738369, 8718968878589307917, 17726168133330301456, 36028797018963998795, 73210515542534815423, 148726874094283294061, 302065434206993943638, 613354240450842628987, 1245155224975394773725, 2527203938098208612989, 5128194852491255392860, 10403963657572187157115, 21103075220323727096539, 42796446251006159799165, 86773484122729730853986, 175908151486894284264843, 356538669456658213691129, 722522071879055717755053, 1463933609689590016307880, 2965646151242137194265835, 6006850166210188711888759, 12164816059872206070551117, 24631863574648069434711406, 49868190059103453456705755, 100945305937821536088044693, 204308463514872330525425821, 413452630308203177749597428, 836576667173323388896762267, 1692496147460480844588738195, 3423677921148629822767985629, 6924727094752595912717074810, 14004196694415864359796445035, 28317878398653073788317572529, 57254726816948837714084605005, 115747393673183055703068228416, 233970667424936871955934595723, 472893095007015265011465574959, 955689710328313572222124026413, 1931186461285193228842633919110, 3901987003827518626482039687995, 7883202170169301590557623196749, 15924860665367131856302334160317, 32166633980791321062978843983756
Generating function in Maple syntax:
(6*x^8-11*x^7+9*x^6-8*x^5+19*x^4-28*x^3+20*x^2-7*x+1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
\frac{6 x^{8}-11 x^{7}+9 x^{6}-8 x^{5}+19 x^{4}-28 x^{3}+20 x^{2}-7 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(6*x**8 - 11*x**7 + 9*x**6 - 8*x**5 + 19*x**4 - 28*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**4*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
-(2*x-1)^2*(x-1)^4*F(x)+6*x^8-11*x^7+9*x^6-8*x^5+19*x^4-28*x^3+20*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+6 x^{8}-11 x^{7}+9 x^{6}-8 x^{5}+19 x^{4}-28 x^{3}+20 x^{2}-7 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/24*(6*n+9)*2^n+1/6*n^3-2*n^2+41/6*n-10)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(6 n +9\right) 2^{n}}{24}+\frac{n^{3}}{6}-2 n^{2}+\frac{41 n}{6}-10 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 47
a(6) = 115
a(7) = 269
a(8) = 610
a(n+2) = 1/6*n^3-3*n^2-4*a(n)+4*a(n+1)+89/6*n-23, n >= 9
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) = 18
a \! \left(5\right) = 47
a \! \left(6\right) = 115
a \! \left(7\right) = 269
a \! \left(8\right) = 610
a \! \left(n +2\right) = \frac{n^{3}}{6}-3 n^{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{89 n}{6}-23, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22156/
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[19,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]*F[11,x]*F[16,x]
F[10,x] = x
F[11,x] = F[12,x]+F[14,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[10,x]*F[12,x]
F[14,x] = F[15,x]
F[15,x] = F[10,x]*F[11,x]*F[12,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[10,x]*F[12,x]*F[16,x]
F[19,x] = F[20,x]
F[20,x] = F[10,x]*F[21,x]
F[21,x] = F[22,x]+F[37,x]
F[22,x] = F[23,x]*F[32,x]
F[23,x] = F[1,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[10,x]*F[26,x]
F[26,x] = F[23,x]+F[27,x]
F[27,x] = F[10,x]*F[28,x]
F[28,x] = F[11,x]+F[29,x]
F[29,x] = F[10,x]*F[30,x]
F[30,x] = F[1,x]+F[31,x]
F[31,x] = F[10,x]*F[30,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[12,x]*F[30,x]
F[34,x] = F[35,x]
F[35,x] = F[10,x]*F[12,x]*F[16,x]*F[36,x]
F[36,x] = F[31,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[32,x]*F[39,x]
F[39,x] = F[40,x]
F[40,x] = F[10,x]*F[11,x]
F[41,x] = F[10,x]*F[16,x]*F[36,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_{19}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{16}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{14}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{12}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{37}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{32}\! \left(x \right)
F_{23}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{1}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{10}\! \left(x \right) F_{30}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{16}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{32}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{36}\! \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_19(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_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_11(x)*F_16(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_14(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_10(x)*F_12(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_10(x)*F_11(x)*F_12(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_10(x)*F_12(x)*F_16(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_10(x)*F_21(x))
Eq(F_21(x), F_22(x) + F_37(x))
Eq(F_22(x), F_23(x)*F_32(x))
Eq(F_23(x), F_1(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_10(x)*F_26(x))
Eq(F_26(x), F_23(x) + F_27(x))
Eq(F_27(x), F_10(x)*F_28(x))
Eq(F_28(x), F_11(x) + F_29(x))
Eq(F_29(x), F_10(x)*F_30(x))
Eq(F_30(x), F_1(x) + F_31(x))
Eq(F_31(x), F_10(x)*F_30(x))
Eq(F_32(x), F_33(x) + F_34(x))
Eq(F_33(x), F_12(x)*F_30(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_10(x)*F_12(x)*F_16(x)*F_36(x))
Eq(F_36(x), F_31(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_32(x)*F_39(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_10(x)*F_11(x))
Eq(F_41(x), F_10(x)*F_16(x)*F_36(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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 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": [3, 1, 2, 0], "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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 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": [3, 1, 2, 0], "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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 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": [3, 1, 2, 0], "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, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 1], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", 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