0132_0231_1023_2031_2310
Counting sequence:
1, 1, 2, 6, 19, 52, 120, 242, 441, 744, 1182, 1790, 2607, 3676, 5044, 6762, 8885, 11472, 14586, 18294, 22667, 27780, 33712, 40546, 48369, 57272, 67350, 78702, 91431, 105644, 121452, 138970, 158317, 179616, 202994, 228582, 256515, 286932, 319976, 355794, 394537, 436360, 481422, 529886, 581919, 637692, 697380, 761162, 829221, 901744, 978922, 1060950, 1148027, 1240356, 1338144, 1441602, 1550945, 1666392, 1788166, 1916494, 2051607, 2193740, 2343132, 2500026, 2664669, 2837312, 3018210, 3207622, 3405811, 3613044, 3829592, 4055730, 4291737, 4537896, 4794494, 5061822, 5340175, 5629852, 5931156, 6244394, 6569877, 6907920, 7258842, 7622966, 8000619, 8392132, 8797840, 9218082, 9653201, 10103544, 10569462, 11051310, 11549447, 12064236, 12596044, 13145242, 13712205, 14297312, 14900946, 15523494, 16165347
Generating function in Maple syntax:
(x^6-x^5-4*x^4+4*x^3-7*x^2+4*x-1)/(x-1)^5
Generating function in latex syntax:
\frac{x^{6}-x^{5}-4 x^{4}+4 x^{3}-7 x^{2}+4 x -1}{\left(x -1\right)^{5}}
Generating function in sympy syntax:
(x**6 - x**5 - 4*x**4 + 4*x**3 - 7*x**2 + 4*x - 1)/(x - 1)**5
Implicit equation for the generating function in Maple syntax:
-(x-1)^5*F(x)+x^6-x^5-4*x^4+4*x^3-7*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x -1\right)^{5} F \! \left(x \right)+x^{6}-x^{5}-4 x^{4}+4 x^{3}-7 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/6*(-1+n)*(n^3-2*n^2-3*n+18))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-1+n \right) \left(n^{3}-2 n^{2}-3 n +18\right)}{6} & \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) = 52
a(6) = 120
a(n) = 1/6*(-1+n)*(n^3-2*n^2-3*n+18), 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) = 52
a \! \left(6\right) = 120
a \! \left(n \right) = \frac{\left(-1+n \right) \left(n^{3}-2 n^{2}-3 n +18\right)}{6}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18808/
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[21,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[14,x]+F[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[20,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[15,x]
F[21,x] = F[2,x]+F[22,x]
F[22,x] = F[23,x]+F[24,x]+F[48,x]
F[23,x] = 0
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[31,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[1,x]+F[27,x]
F[30,x] = F[15,x]
F[31,x] = F[32,x]+F[39,x]
F[32,x] = F[23,x]+F[24,x]+F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[11,x]+F[35,x]
F[35,x] = F[23,x]+F[33,x]+F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[27,x]+F[38,x]
F[38,x] = F[19,x]
F[39,x] = F[40,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[42,x]+F[47,x]
F[42,x] = F[43,x]+F[45,x]
F[43,x] = F[19,x]+F[23,x]+F[44,x]
F[44,x] = F[11,x]*F[4,x]
F[45,x] = F[46,x]
F[46,x] = F[4,x]*F[42,x]
F[47,x] = F[40,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[50,x]+F[53,x]
F[50,x] = F[11,x]+F[51,x]
F[51,x] = F[19,x]+F[23,x]+F[52,x]
F[52,x] = F[4,x]*F[50,x]
F[53,x] = F[54,x]+F[55,x]
F[54,x] = F[23,x]+F[36,x]+F[48,x]
F[55,x] = F[40,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_{21}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{48}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{30}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{39}\! \left(x \right)
F_{32}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{11}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{23}\! \left(x \right)+F_{33}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{27}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{19}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{47}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)+F_{45}\! \left(x \right)
F_{43}\! \left(x \right) = F_{19}\! \left(x \right)+F_{23}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{47}\! \left(x \right) = F_{40}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{50}\! \left(x \right) = F_{11}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{19}\! \left(x \right)+F_{23}\! \left(x \right)+F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{55}\! \left(x \right)
F_{54}\! \left(x \right) = F_{23}\! \left(x \right)+F_{36}\! \left(x \right)+F_{48}\! \left(x \right)
F_{55}\! \left(x \right) = F_{40}\! \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_21(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_14(x) + F_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_20(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_15(x))
Eq(F_21(x), F_2(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_48(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_31(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_1(x) + F_27(x))
Eq(F_30(x), F_15(x))
Eq(F_31(x), F_32(x) + F_39(x))
Eq(F_32(x), F_23(x) + F_24(x) + F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_11(x) + F_35(x))
Eq(F_35(x), F_23(x) + F_33(x) + F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_27(x) + F_38(x))
Eq(F_38(x), F_19(x))
Eq(F_39(x), F_40(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_42(x) + F_47(x))
Eq(F_42(x), F_43(x) + F_45(x))
Eq(F_43(x), F_19(x) + F_23(x) + F_44(x))
Eq(F_44(x), F_11(x)*F_4(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_4(x)*F_42(x))
Eq(F_47(x), F_40(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_50(x) + F_53(x))
Eq(F_50(x), F_11(x) + F_51(x))
Eq(F_51(x), F_19(x) + F_23(x) + F_52(x))
Eq(F_52(x), F_4(x)*F_50(x))
Eq(F_53(x), F_54(x) + F_55(x))
Eq(F_54(x), F_23(x) + F_36(x) + F_48(x))
Eq(F_55(x), F_40(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": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 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, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 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": true, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": 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