0123_1302_1320_2301
Counting sequence:
1, 1, 2, 6, 20, 61, 162, 379, 799, 1549, 2806, 4808, 7866, 12377, 18838, 27861, 40189, 56713, 78490, 106762, 142976, 188805, 246170, 317263, 404571, 510901, 639406, 793612, 977446, 1195265, 1451886, 1752617, 2103289, 2510289, 2980594, 3521806, 4142188, 4850701, 5657042, 6571683, 7605911, 8771869, 10082598, 11552080, 13195282, 15028201, 17067910, 19332605, 21841653, 24615641, 27676426, 31047186, 34752472, 38818261, 43272010, 48142711, 53460947, 59258949, 65570654, 72431764, 79879806, 87954193, 96696286, 106149457, 116359153, 127372961, 139240674, 152014358, 165748420, 180499677, 196327426, 213293515, 231462415, 250901293, 271680086, 293871576, 317551466, 342798457, 369694326, 398324005, 428775661, 461140777, 495514234, 531994394, 570683184, 611686181, 655112698, 701075871, 749692747, 801084373, 855375886, 912696604, 973180118, 1036964385, 1104191822, 1175009401, 1249568745, 1328026225, 1410543058, 1497285406, 1588424476
Generating function in Maple syntax:
-(x^6-9*x^5+20*x^4-22*x^3+16*x^2-6*x+1)/(x-1)^7
Generating function in latex syntax:
-\frac{x^{6}-9 x^{5}+20 x^{4}-22 x^{3}+16 x^{2}-6 x +1}{\left(x -1\right)^{7}}
Generating function in sympy syntax:
(-x**6 + 9*x**5 - 20*x**4 + 22*x**3 - 16*x**2 + 6*x - 1)/(x - 1)**7
Implicit equation for the generating function in Maple syntax:
(x-1)^7*F(x)+x^6-9*x^5+20*x^4-22*x^3+16*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{7} F \! \left(x \right)+x^{6}-9 x^{5}+20 x^{4}-22 x^{3}+16 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
1-1/4*n+11/48*n^3+4/45*n^2+1/720*n^6+1/48*n^5-13/144*n^4
Explicit closed form in latex syntax:
1-\frac{1}{4} n +\frac{11}{48} n^{3}+\frac{4}{45} n^{2}+\frac{1}{720} n^{6}+\frac{1}{48} n^{5}-\frac{13}{144} n^{4}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 61
a(6) = 162
a(n) = 1-1/4*n+11/48*n^3+4/45*n^2+1/720*n^6+1/48*n^5-13/144*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) = 20
a \! \left(5\right) = 61
a \! \left(6\right) = 162
a \! \left(n \right) = 1-\frac{1}{4} n +\frac{11}{48} n^{3}+\frac{4}{45} n^{2}+\frac{1}{720} n^{6}+\frac{1}{48} n^{5}-\frac{13}{144} n^{4}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15207/
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[28,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[16,x]+F[20,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[22,x]
F[18,x] = F[11,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[11,x]
F[22,x] = F[14,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[19,x]
F[27,x] = F[23,x]
F[28,x] = F[2,x]+F[29,x]
F[29,x] = F[15,x]+F[30,x]+F[34,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[32,x]+F[40,x]
F[32,x] = F[33,x]+F[7,x]
F[33,x] = F[15,x]+F[16,x]+F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[36,x]
F[36,x] = F[11,x]+F[37,x]
F[37,x] = F[15,x]+F[20,x]+F[38,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[11,x]+F[37,x]
F[40,x] = F[29,x]+F[41,x]
F[41,x] = F[15,x]+F[42,x]+F[46,x]+F[52,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[53,x]
F[44,x] = F[33,x]+F[45,x]
F[45,x] = F[15,x]+F[24,x]+F[46,x]+F[52,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[37,x]+F[49,x]
F[49,x] = F[50,x]
F[50,x] = F[4,x]*F[51,x]
F[51,x] = F[19,x]+F[49,x]
F[52,x] = 0
F[53,x] = F[41,x]+F[54,x]
F[54,x] = F[15,x]+F[55,x]+F[59,x]+F[67,x]+F[68,x]
F[55,x] = F[4,x]*F[56,x]
F[56,x] = F[57,x]+F[61,x]
F[57,x] = F[45,x]+F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[4,x]*F[60,x]
F[60,x] = F[49,x]
F[61,x] = F[54,x]+F[62,x]
F[62,x] = F[63,x]
F[63,x] = F[4,x]*F[64,x]
F[64,x] = F[65,x]+F[66,x]
F[65,x] = F[58,x]
F[66,x] = F[62,x]
F[67,x] = 0
F[68,x] = 0
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_{28}\! \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_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right)
F_{22}\! \left(x \right) = F_{14}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
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_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{19}\! \left(x \right)
F_{27}\! \left(x \right) = F_{23}\! \left(x \right)
F_{28}\! \left(x \right) = F_{2}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)+F_{40}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)+F_{7}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{11}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{11}\! \left(x \right)+F_{37}\! \left(x \right)
F_{40}\! \left(x \right) = F_{29}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{46}\! \left(x \right)+F_{52}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{53}\! \left(x \right)
F_{44}\! \left(x \right) = F_{33}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{46}\! \left(x \right)+F_{52}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{37}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{4}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{19}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = 0
F_{53}\! \left(x \right) = F_{41}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{59}\! \left(x \right)+F_{67}\! \left(x \right)+F_{68}\! \left(x \right)
F_{55}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)+F_{61}\! \left(x \right)
F_{57}\! \left(x \right) = F_{45}\! \left(x \right)+F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{4}\! \left(x \right) F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{49}\! \left(x \right)
F_{61}\! \left(x \right) = F_{54}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{4}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{66}\! \left(x \right)
F_{65}\! \left(x \right) = F_{58}\! \left(x \right)
F_{66}\! \left(x \right) = F_{62}\! \left(x \right)
F_{67}\! \left(x \right) = 0
F_{68}\! \left(x \right) = 0
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_28(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) + F_16(x) + F_20(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_22(x))
Eq(F_18(x), F_11(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_11(x))
Eq(F_22(x), F_14(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_19(x))
Eq(F_27(x), F_23(x))
Eq(F_28(x), F_2(x) + F_29(x))
Eq(F_29(x), F_15(x) + F_30(x) + F_34(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_32(x) + F_40(x))
Eq(F_32(x), F_33(x) + F_7(x))
Eq(F_33(x), F_15(x) + F_16(x) + F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_11(x) + F_37(x))
Eq(F_37(x), F_15(x) + F_20(x) + F_38(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_11(x) + F_37(x))
Eq(F_40(x), F_29(x) + F_41(x))
Eq(F_41(x), F_15(x) + F_42(x) + F_46(x) + F_52(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_53(x))
Eq(F_44(x), F_33(x) + F_45(x))
Eq(F_45(x), F_15(x) + F_24(x) + F_46(x) + F_52(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_37(x) + F_49(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_4(x)*F_51(x))
Eq(F_51(x), F_19(x) + F_49(x))
Eq(F_52(x), 0)
Eq(F_53(x), F_41(x) + F_54(x))
Eq(F_54(x), F_15(x) + F_55(x) + F_59(x) + F_67(x) + F_68(x))
Eq(F_55(x), F_4(x)*F_56(x))
Eq(F_56(x), F_57(x) + F_61(x))
Eq(F_57(x), F_45(x) + F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_4(x)*F_60(x))
Eq(F_60(x), F_49(x))
Eq(F_61(x), F_54(x) + F_62(x))
Eq(F_62(x), F_63(x))
Eq(F_63(x), F_4(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_66(x))
Eq(F_65(x), F_58(x))
Eq(F_66(x), F_62(x))
Eq(F_67(x), 0)
Eq(F_68(x), 0)
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": [1, 3, 0, 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": [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, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": 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