0132_0213_1032_1320_3120
Counting sequence:
1, 1, 2, 6, 19, 54, 140, 341, 797, 1812, 4042, 8895, 19379, 41894, 90008, 192393, 409465, 868200, 1834838, 3866435, 8126255, 17039130, 35651332, 74448621, 155188949, 322961084, 671088290, 1392508551, 2885680747, 5972688462, 12348030512, 25501367825, 52613348849, 108447923664, 223338298798, 459561500043, 944892804455, 1941325217090, 3985729649948, 8177617730805, 16767552322765, 34359738367140, 70368744176762, 144036023237711, 294669116242979, 602532372020214, 1231453023108040, 2515682604350361, 5136918324968297, 10484942882470712, 21392098230008582, 43628621390150355, 88946092640565919, 181269885001661034, 369295169444379188, 752101137770871293, 1531223873305967045, 3116490942140381580, 6341068275337656658, 12898309332789098775, 26228964229805766875, 53322619588066670750, 108374621433043613792, 220208007379907770401, 447333543787456624609, 908502145630195414944, 1844674407370955159390, 3744689046963038975771, 7600058558368335263447, 15421478045621185148562, 31285677949011399538252, 63456799613560857556485, 128684486658197832070589, 260910748178547898054004, 528905046081400263931178, 1071977191611409463506143, 2172288582120036798297235, 4401245562034509339161670, 8915827919657890163454968, 18058329430493523297170345, 36570006043342532534858585, 74046706451396036950749960, 149906801632214017663562422, 303440380723271922851246691, 614134316364231620750733839, 1242775742563838791597945274, 2514565704798428683388842340, 5087159848938359567163584781, 10290376576559723535098966197, 20812866910485455871741522012, 42089961335702929346570219522, 85108377700869893899314786215, 172073665460667858210978262859, 347861151039191857246653902574, 703149942314095996142702554768, 1421155165099616555584194604593, 2872020891142082237765968195025, 5803462904169862728727094357360, 11725768052111121963844504644878, 23689220591765036940469641145515, 47853810158615659906500545997895
Generating function in Maple syntax:
-(x^6-3*x^5+6*x^4-14*x^3+14*x^2-6*x+1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
-\frac{x^{6}-3 x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-x**6 + 3*x**5 - 6*x**4 + 14*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^3*F(x)+x^6-3*x^5+6*x^4-14*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+x^{6}-3 x^{5}+6 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/8*(3*n+2)*2^n-1/2*n^2-1/2*n+1)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(3 n +2\right) 2^{n}}{8}-\frac{n^{2}}{2}-\frac{n}{2}+1 & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 54
a(6) = 140
a(n+2) = -4*a(n)+4*a(n+1)-1/2*(n+1)*(n-4), 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) = 54
a \! \left(6\right) = 140
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{\left(n +1\right) \left(n -4\right)}{2}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21618/
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[39,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[16,x]*F[8,x]
F[8,x] = F[23,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[16,x]
F[12,x] = F[13,x]+F[17,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = x
F[17,x] = F[18,x]+F[21,x]
F[18,x] = F[19,x]
F[19,x] = F[16,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[16,x]*F[17,x]
F[23,x] = F[18,x]+F[24,x]
F[24,x] = F[25,x]+F[26,x]+F[31,x]
F[25,x] = 0
F[26,x] = F[16,x]*F[27,x]
F[27,x] = F[14,x]+F[28,x]
F[28,x] = F[25,x]+F[26,x]+F[29,x]
F[29,x] = F[16,x]*F[30,x]
F[30,x] = F[18,x]+F[28,x]
F[31,x] = F[16,x]*F[32,x]
F[32,x] = F[30,x]+F[33,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[35,x]
F[35,x] = F[16,x]*F[36,x]
F[36,x] = F[18,x]+F[34,x]
F[37,x] = F[38,x]
F[38,x] = F[16,x]*F[33,x]
F[39,x] = F[40,x]
F[40,x] = F[16,x]*F[41,x]
F[41,x] = F[42,x]+F[52,x]
F[42,x] = F[39,x]+F[43,x]
F[43,x] = F[44,x]+F[51,x]
F[44,x] = F[45,x]+F[49,x]
F[45,x] = F[13,x]*F[46,x]
F[46,x] = F[1,x]+F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[16,x]*F[20,x]*F[46,x]
F[49,x] = F[50,x]
F[50,x] = F[16,x]*F[20,x]*F[51,x]
F[51,x] = F[22,x]
F[52,x] = F[53,x]
F[53,x] = F[54,x]*F[58,x]
F[54,x] = F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[13,x]*F[16,x]*F[57,x]
F[57,x] = F[1,x]+F[55,x]
F[58,x] = F[43,x]+F[59,x]
F[59,x] = F[47,x]+F[60,x]
F[60,x] = F[10,x]*F[16,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_{39}\! \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_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{23}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{21}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{16}\! \left(x \right) F_{17}\! \left(x \right)
F_{23}\! \left(x \right) = F_{18}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{25}\! \left(x \right) = 0
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{16}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{18}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{16}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{16}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{18}\! \left(x \right)+F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{16}\! \left(x \right) F_{33}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{16}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{52}\! \left(x \right)
F_{42}\! \left(x \right) = F_{39}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{51}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{45}\! \left(x \right) = F_{13}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{1}\! \left(x \right)+F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right) F_{46}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{22}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right) F_{58}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{1}\! \left(x \right)+F_{55}\! \left(x \right)
F_{58}\! \left(x \right) = F_{43}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{47}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \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_39(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_16(x)*F_8(x))
Eq(F_8(x), F_23(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_16(x))
Eq(F_12(x), F_13(x) + F_17(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), x)
Eq(F_17(x), F_18(x) + F_21(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_16(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_16(x)*F_17(x))
Eq(F_23(x), F_18(x) + F_24(x))
Eq(F_24(x), F_25(x) + F_26(x) + F_31(x))
Eq(F_25(x), 0)
Eq(F_26(x), F_16(x)*F_27(x))
Eq(F_27(x), F_14(x) + F_28(x))
Eq(F_28(x), F_25(x) + F_26(x) + F_29(x))
Eq(F_29(x), F_16(x)*F_30(x))
Eq(F_30(x), F_18(x) + F_28(x))
Eq(F_31(x), F_16(x)*F_32(x))
Eq(F_32(x), F_30(x) + F_33(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_16(x)*F_36(x))
Eq(F_36(x), F_18(x) + F_34(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_16(x)*F_33(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_16(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_52(x))
Eq(F_42(x), F_39(x) + F_43(x))
Eq(F_43(x), F_44(x) + F_51(x))
Eq(F_44(x), F_45(x) + F_49(x))
Eq(F_45(x), F_13(x)*F_46(x))
Eq(F_46(x), F_1(x) + F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_16(x)*F_20(x)*F_46(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_16(x)*F_20(x)*F_51(x))
Eq(F_51(x), F_22(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_54(x)*F_58(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_13(x)*F_16(x)*F_57(x))
Eq(F_57(x), F_1(x) + F_55(x))
Eq(F_58(x), F_43(x) + F_59(x))
Eq(F_59(x), F_47(x) + F_60(x))
Eq(F_60(x), F_10(x)*F_16(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, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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": [1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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