0123_0132_2301_2310
Counting sequence:
1, 1, 2, 6, 20, 60, 152, 332, 646, 1150, 1910, 3002, 4512, 6536, 9180, 12560, 16802, 22042, 28426, 36110, 45260, 56052, 68672, 83316, 100190, 119510, 141502, 166402, 194456, 225920, 261060, 300152, 343482, 391346, 444050, 501910, 565252, 634412, 709736, 791580, 880310, 976302, 1079942, 1191626, 1311760, 1440760, 1579052, 1727072, 1885266, 2054090, 2234010, 2425502, 2629052, 2845156, 3074320, 3317060, 3573902, 3845382, 4132046, 4434450, 4753160, 5088752, 5441812, 5812936, 6202730, 6611810, 7040802, 7490342, 7961076, 8453660, 8968760, 9507052, 10069222, 10655966, 11267990, 11906010, 12570752, 13262952, 13983356, 14732720, 15511810, 16321402, 17162282, 18035246, 18941100, 19880660, 20854752, 21864212, 22909886, 23992630, 25113310, 26272802, 27471992, 28711776, 29993060, 31316760, 32683802, 34095122, 35551666, 37054390, 38604260
Generating function in Maple syntax:
-(x^6+4*x^5+5*x^4-4*x^3+7*x^2-4*x+1)/(x-1)^5
Generating function in latex syntax:
-\frac{x^{6}+4 x^{5}+5 x^{4}-4 x^{3}+7 x^{2}-4 x +1}{\left(x -1\right)^{5}}
Generating function in sympy syntax:
(-x**6 - 4*x**5 - 5*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+4*x^5+5*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}+4 x^{5}+5 x^{4}-4 x^{3}+7 x^{2}-4 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,10+5/12*n^4-19/6*n^3+127/12*n^2-95/6*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 10+\frac{5}{12} n^{4}-\frac{19}{6} n^{3}+\frac{127}{12} n^{2}-\frac{95}{6} n & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 60
a(6) = 152
a(n) = 1/12*(5*n^2-13*n+12)*(n^2-5*n+10), 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) = 20
a \! \left(5\right) = 60
a \! \left(6\right) = 152
a \! \left(n \right) = \frac{\left(5 n^{2}-13 n +12\right) \left(n^{2}-5 n +10\right)}{12}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/14887/
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[19,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[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[12,x]
F[18,x] = x^2
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[56,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[37,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[13,x]+F[14,x]+F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[30,x]
F[27,x] = F[28,x]+F[4,x]
F[28,x] = F[13,x]+F[18,x]+F[29,x]
F[29,x] = x^2
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[13,x]+F[25,x]+F[29,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[35,x]+F[36,x]
F[35,x] = F[28,x]
F[36,x] = F[32,x]
F[37,x] = F[20,x]+F[38,x]
F[38,x] = F[13,x]+F[39,x]+F[49,x]+F[55,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[43,x]
F[41,x] = F[24,x]+F[42,x]
F[42,x] = F[33,x]
F[43,x] = F[38,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[42,x]
F[48,x] = F[44,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[35,x]+F[51,x]
F[51,x] = F[52,x]
F[52,x] = F[13,x]+F[49,x]+F[53,x]+F[54,x]
F[53,x] = F[12,x]*F[4,x]
F[54,x] = 0
F[55,x] = 0
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[27,x]+F[58,x]
F[58,x] = F[52,x]+F[59,x]
F[59,x] = F[13,x]+F[56,x]+F[60,x]
F[60,x] = F[4,x]*F[7,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_{19}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)
F_{18}\! \left(x \right) = x^{2}
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{56}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{37}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \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_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{13}\! \left(x \right)+F_{18}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = x^{2}
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \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_{35}\! \left(x \right)+F_{36}\! \left(x \right)
F_{35}\! \left(x \right) = F_{28}\! \left(x \right)
F_{36}\! \left(x \right) = F_{32}\! \left(x \right)
F_{37}\! \left(x \right) = F_{20}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{13}\! \left(x \right)+F_{39}\! \left(x \right)+F_{49}\! \left(x \right)+F_{55}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{24}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{33}\! \left(x \right)
F_{43}\! \left(x \right) = F_{38}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{42}\! \left(x \right)
F_{48}\! \left(x \right) = F_{44}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{35}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{13}\! \left(x \right)+F_{49}\! \left(x \right)+F_{53}\! \left(x \right)+F_{54}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{54}\! \left(x \right) = 0
F_{55}\! \left(x \right) = 0
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{27}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{52}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{13}\! \left(x \right)+F_{56}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \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_19(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_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_12(x))
Eq(F_18(x), x**2)
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_56(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_37(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_13(x) + F_14(x) + F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_30(x))
Eq(F_27(x), F_28(x) + F_4(x))
Eq(F_28(x), F_13(x) + F_18(x) + F_29(x))
Eq(F_29(x), x**2)
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_13(x) + F_25(x) + F_29(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_35(x) + F_36(x))
Eq(F_35(x), F_28(x))
Eq(F_36(x), F_32(x))
Eq(F_37(x), F_20(x) + F_38(x))
Eq(F_38(x), F_13(x) + F_39(x) + F_49(x) + F_55(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_43(x))
Eq(F_41(x), F_24(x) + F_42(x))
Eq(F_42(x), F_33(x))
Eq(F_43(x), F_38(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_42(x))
Eq(F_48(x), F_44(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_35(x) + F_51(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_13(x) + F_49(x) + F_53(x) + F_54(x))
Eq(F_53(x), F_12(x)*F_4(x))
Eq(F_54(x), 0)
Eq(F_55(x), 0)
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_27(x) + F_58(x))
Eq(F_58(x), F_52(x) + F_59(x))
Eq(F_59(x), F_13(x) + F_56(x) + F_60(x))
Eq(F_60(x), F_4(x)*F_7(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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "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, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "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, 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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "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, 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": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 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