0213_1302_3120
Counting sequence:
1, 1, 2, 6, 21, 73, 242, 762, 2290, 6610, 18434, 49922, 131842, 340738, 864258, 2156546, 5304322, 12881922, 30932994, 73531394, 173211650, 404684802, 938475522, 2161639426, 4948230146, 11262754818, 25501368322, 57461964802, 128899350530, 287947358210, 640755433474, 1420694650882, 3139352657922, 6915165782018, 15187004358658, 33260226740226, 72649371811842, 158291019694082, 344078420017154, 746259157614594, 1615113861726210, 3488544236503042, 7520659533987842, 16183711649234946, 34765458158649346, 74558982991118338, 159649088353075202, 341332389726781442, 728721122517843970, 1553618726140510210, 3307893926303629314, 7034059667999293442, 14939284388918067202, 31691549102829469698, 67153174043721465858, 142140359639253909506, 300547721132570050562, 634849921472281968642, 1339694788353156186114, 2824441513504661307394, 5949291136553444179970, 12520367252060168519682, 26326962557697350696962, 55313138565770393944066, 116121101022497020575746, 243590408414839235936258, 510605875960280388730882, 1069542221393679802695682, 2238752201017612312772610, 4682945790784127901892610, 9789170571083595175165954, 20450060480114408921497602, 42694620223719283024199682, 89081835889137449252683778, 185759007970160364125945858, 387134881899170644973584386, 806367688782406268464660482, 1678683225498087132338585602, 3492813366457841428696399874, 7263717450358910524736929794, 15098199003303377848582012930, 31367469336494701795837214722, 65136923160836219933168762882, 135199010110782439124090486786, 280493759741166646486152249346, 581675867033398442039051288578, 1205734255450846553684951695362, 2498269384750023482113806827522, 5174260536389954369942819176450, 10712301359011786034104823185410, 22169030223522589962893516603394, 45861036180376019008981680783362, 94837038984603876134678282895362, 196043588956340855128781284704258, 405108450215904982679089007558658, 836830135396367070507959387684866, 1728040500838462325731687504281602, 3567173740802394677332822416424962, 7361247035525328134491355513618434, 15185820365434074115229756048801794, 31317545763125950156988833320665090
Generating function in Maple syntax:
-(x^6-4*x^5+23*x^4-36*x^3+25*x^2-8*x+1)/(x-1)/(2*x-1)^4
Generating function in latex syntax:
-\frac{x^{6}-4 x^{5}+23 x^{4}-36 x^{3}+25 x^{2}-8 x +1}{\left(x -1\right) \left(2 x -1\right)^{4}}
Generating function in sympy syntax:
(-x**6 + 4*x**5 - 23*x**4 + 36*x**3 - 25*x**2 + 8*x - 1)/((x - 1)*(2*x - 1)**4)
Implicit equation for the generating function in Maple syntax:
(x-1)*(2*x-1)^4*F(x)+x^6-4*x^5+23*x^4-36*x^3+25*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(2 x -1\right)^{4} F \! \left(x \right)+x^{6}-4 x^{5}+23 x^{4}-36 x^{3}+25 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,2+1/192*(5*n^3-27*n^2+136*n-204)*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2+\frac{\left(5 n^{3}-27 n^{2}+136 n -204\right) 2^{n}}{192} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 73
a(6) = 242
a(n+4) = -16*a(n)+32*a(n+1)-24*a(n+2)+8*a(n+3)+2, 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) = 21
a \! \left(5\right) = 73
a \! \left(6\right) = 242
a \! \left(n +4\right) = -16 a \! \left(n \right)+32 a \! \left(n +1\right)-24 a \! \left(n +2\right)+8 a \! \left(n +3\right)+2, \quad n \geq 7
Specification 1
Strategy pack name: point_and_row_placements
Tree: http://permpal.com/tree/335/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[13,x]*F[3,x]
F[3,x] = F[34,x]+F[4,x]
F[4,x] = F[1,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[1,x]+F[33,x]+F[8,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[13,x]*F[27,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[14,x]
F[13,x] = x
F[14,x] = F[15,x]+F[18,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[13,x]*F[15,x]
F[18,x] = F[19,x]+F[22,x]
F[19,x] = F[20,x]
F[20,x] = F[13,x]*F[21,x]
F[21,x] = F[1,x]+F[19,x]
F[22,x] = F[23,x]+F[24,x]+F[26,x]
F[23,x] = 0
F[24,x] = F[13,x]*F[25,x]
F[25,x] = F[16,x]+F[22,x]
F[26,x] = F[13,x]*F[18,x]
F[27,x] = F[21,x]+F[28,x]
F[28,x] = F[16,x]+F[29,x]
F[29,x] = F[23,x]+F[30,x]+F[32,x]
F[30,x] = F[13,x]*F[31,x]
F[31,x] = F[19,x]+F[29,x]
F[32,x] = F[13,x]*F[28,x]
F[33,x] = F[13,x]*F[7,x]
F[34,x] = F[23,x]+F[35,x]+F[49,x]
F[35,x] = F[13,x]*F[36,x]
F[36,x] = F[37,x]
F[37,x] = F[13,x]*F[38,x]*F[44,x]*F[7,x]
F[38,x] = F[15,x]+F[39,x]
F[39,x] = F[19,x]+F[40,x]
F[40,x] = F[23,x]+F[41,x]+F[43,x]
F[41,x] = F[13,x]*F[42,x]
F[42,x] = F[16,x]+F[40,x]
F[43,x] = F[13,x]*F[39,x]
F[44,x] = F[1,x]+F[45,x]
F[45,x] = F[46,x]
F[46,x] = F[13,x]*F[47,x]
F[47,x] = F[44,x]+F[48,x]
F[48,x] = F[45,x]
F[49,x] = F[13,x]*F[50,x]
F[50,x] = F[1,x]+F[51,x]+F[59,x]+F[60,x]
F[51,x] = F[13,x]*F[52,x]
F[52,x] = F[50,x]+F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[10,x]*F[16,x]*F[55,x]
F[55,x] = F[1,x]+F[56,x]+F[57,x]
F[56,x] = F[13,x]*F[55,x]
F[57,x] = F[58,x]
F[58,x] = F[13,x]*F[27,x]*F[44,x]
F[59,x] = F[13,x]*F[50,x]
F[60,x] = F[61,x]
F[61,x] = F[13,x]*F[44,x]*F[62,x]
F[62,x] = 2*F[66,x]+F[1,x]+F[63,x]
F[63,x] = F[64,x]
F[64,x] = F[21,x]^2*F[10,x]*F[13,x]*F[65,x]
F[65,x] = F[1,x]+F[13,x]
F[66,x] = F[13,x]*F[62,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_{13}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{34}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{1}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{1}\! \left(x \right)+F_{33}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{27}\! \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_{13}\! \left(x \right) F_{14}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{18}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{13}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right)+F_{22}\! \left(x \right)
F_{26}\! \left(x \right) = F_{13}\! \left(x \right) F_{18}\! \left(x \right)
F_{27}\! \left(x \right) = F_{21}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{23}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{13}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{19}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{13}\! \left(x \right) F_{28}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{34}\! \left(x \right) = F_{23}\! \left(x \right)+F_{35}\! \left(x \right)+F_{49}\! \left(x \right)
F_{35}\! \left(x \right) = F_{13}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{13}\! \left(x \right) F_{38}\! \left(x \right) F_{44}\! \left(x \right) F_{7}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{19}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{23}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)
F_{41}\! \left(x \right) = F_{13}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{16}\! \left(x \right)+F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{13}\! \left(x \right) F_{39}\! \left(x \right)
F_{44}\! \left(x \right) = F_{1}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{13}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{44}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{45}\! \left(x \right)
F_{49}\! \left(x \right) = F_{13}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{1}\! \left(x \right)+F_{51}\! \left(x \right)+F_{59}\! \left(x \right)+F_{60}\! \left(x \right)
F_{51}\! \left(x \right) = F_{13}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{50}\! \left(x \right)+F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{1}\! \left(x \right)+F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{13}\! \left(x \right) F_{55}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{13}\! \left(x \right) F_{27}\! \left(x \right) F_{44}\! \left(x \right)
F_{59}\! \left(x \right) = F_{13}\! \left(x \right) F_{50}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{13}\! \left(x \right) F_{44}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = 2 F_{66}\! \left(x \right)+F_{1}\! \left(x \right)+F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{21} \left(x \right)^{2} F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{66}\! \left(x \right) = F_{13}\! \left(x \right) F_{62}\! \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_13(x)*F_3(x))
Eq(F_3(x), F_34(x) + F_4(x))
Eq(F_4(x), F_1(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_13(x)*F_7(x))
Eq(F_7(x), F_1(x) + F_33(x) + F_8(x))
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_13(x)*F_27(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_14(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x) + F_18(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_13(x)*F_15(x))
Eq(F_18(x), F_19(x) + F_22(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_13(x)*F_21(x))
Eq(F_21(x), F_1(x) + F_19(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_26(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_13(x)*F_25(x))
Eq(F_25(x), F_16(x) + F_22(x))
Eq(F_26(x), F_13(x)*F_18(x))
Eq(F_27(x), F_21(x) + F_28(x))
Eq(F_28(x), F_16(x) + F_29(x))
Eq(F_29(x), F_23(x) + F_30(x) + F_32(x))
Eq(F_30(x), F_13(x)*F_31(x))
Eq(F_31(x), F_19(x) + F_29(x))
Eq(F_32(x), F_13(x)*F_28(x))
Eq(F_33(x), F_13(x)*F_7(x))
Eq(F_34(x), F_23(x) + F_35(x) + F_49(x))
Eq(F_35(x), F_13(x)*F_36(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_13(x)*F_38(x)*F_44(x)*F_7(x))
Eq(F_38(x), F_15(x) + F_39(x))
Eq(F_39(x), F_19(x) + F_40(x))
Eq(F_40(x), F_23(x) + F_41(x) + F_43(x))
Eq(F_41(x), F_13(x)*F_42(x))
Eq(F_42(x), F_16(x) + F_40(x))
Eq(F_43(x), F_13(x)*F_39(x))
Eq(F_44(x), F_1(x) + F_45(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_13(x)*F_47(x))
Eq(F_47(x), F_44(x) + F_48(x))
Eq(F_48(x), F_45(x))
Eq(F_49(x), F_13(x)*F_50(x))
Eq(F_50(x), F_1(x) + F_51(x) + F_59(x) + F_60(x))
Eq(F_51(x), F_13(x)*F_52(x))
Eq(F_52(x), F_50(x) + F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_10(x)*F_16(x)*F_55(x))
Eq(F_55(x), F_1(x) + F_56(x) + F_57(x))
Eq(F_56(x), F_13(x)*F_55(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_13(x)*F_27(x)*F_44(x))
Eq(F_59(x), F_13(x)*F_50(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_13(x)*F_44(x)*F_62(x))
Eq(F_62(x), F_1(x) + F_63(x) + 2*F_66(x))
Eq(F_63(x), F_64(x))
Eq(F_64(x), F_10(x)*F_13(x)*F_21(x)**2*F_65(x))
Eq(F_65(x), F_1(x) + F_13(x))
Eq(F_66(x), F_13(x)*F_62(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"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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_and_row_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, 2, 1, 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": [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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, 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