0132_0213_1032_2130_3120
Counting sequence:
1, 1, 2, 6, 19, 52, 125, 275, 572, 1151, 2276, 4468, 8761, 17214, 33935, 67129, 133194, 264913, 527838, 1053058, 2102735, 4201176, 8396977, 16787311, 33566504, 67123187, 134234600, 268455200, 536893877, 1073768386, 2147514211, 4295002293, 8589974486, 17179914469, 34359789570, 68719534414, 137439018219, 274877979388, 549755894693, 1099511717643, 2199023355220, 4398046621351, 8796093143852, 17592186178316, 35184372235889, 70368744338822, 140737488531575, 281474976903025, 562949953630882, 1125899907070521, 2251799813932646, 4503599627638618, 9007199255031111, 18014398509795424, 36028797019302105, 72057594038292199, 144115188076247744, 288230376152132763, 576460752303875248, 1152921504607331128, 2305843009214212205, 4611686018427942026, 9223372036855367627, 18446744073710183021, 36893488147419776174, 73786976294838922957, 147573952589677175050, 295147905179353635750, 590295810358706511587, 1180591620717412215556, 2361183241434823573581, 4722366482869646237443, 9444732965739291510636, 18889465931478582000079, 37778931862957162919540, 75557863725914324696484, 151115727451828648185769, 302231454903657295097038, 604462909807314588849503, 1208925819614629176281513, 2417851639229258351069690, 4835703278458516700567201, 9671406556917033399480302, 19342813113834066797221426, 38685626227668133592615359, 77371252455336267183311592, 154742504910672534364609025, 309485009821345068727105375, 618970019642690137451995992, 1237940039285380274901671491, 2475880078570760549800913016, 4951760157141521099599282768, 9903520314283042199195905061, 19807040628566084398389028434, 39614081257132168796775149875, 79228162514264337593547263269, 158456325028528675187091356294, 316912650057057350374179404213, 633825300114114700748355357458, 1267650600228229401496707116798, 2535301200456458802993410483675
Generating function in Maple syntax:
(2*x^6-7*x^5+12*x^4-18*x^3+15*x^2-6*x+1)/(2*x-1)/(x-1)^5
Generating function in latex syntax:
\frac{2 x^{6}-7 x^{5}+12 x^{4}-18 x^{3}+15 x^{2}-6 x +1}{\left(2 x -1\right) \left(x -1\right)^{5}}
Generating function in sympy syntax:
(2*x**6 - 7*x**5 + 12*x**4 - 18*x**3 + 15*x**2 - 6*x + 1)/((x - 1)**5*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)^5*F(x)-2*x^6+7*x^5-12*x^4+18*x^3-15*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)-2 x^{6}+7 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,-2+2^(n+1)-1/12*n^3-25/24*n^2+1/12*n+1/24*n^4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ -2+2^{n +1}-\frac{n^{3}}{12}-\frac{25 n^{2}}{24}+\frac{n}{12}+\frac{n^{4}}{24} & \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) = 125
a(n+1) = 2*a(n)-1/24*(n-1)*(n+4)*(n^2-9*n+6), 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) = 125
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n +4\right) \left(n^{2}-9 n +6\right)}{24}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18365/
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[12,x]*F[4,x]
F[4,x] = F[20,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,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[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[18,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[14,x]
F[18,x] = F[19,x]
F[19,x] = F[12,x]*F[13,x]
F[20,x] = F[21,x]+F[26,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[25,x]
F[24,x] = F[1,x]+F[14,x]
F[25,x] = F[21,x]
F[26,x] = F[27,x]+F[28,x]+F[42,x]
F[27,x] = 0
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[38,x]
F[30,x] = F[10,x]+F[31,x]
F[31,x] = F[19,x]+F[27,x]+F[32,x]
F[32,x] = F[12,x]*F[33,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[10,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[12,x]*F[34,x]
F[37,x] = F[18,x]
F[38,x] = F[39,x]
F[39,x] = F[27,x]+F[28,x]+F[40,x]
F[40,x] = F[12,x]*F[41,x]
F[41,x] = F[21,x]+F[39,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[41,x]+F[44,x]
F[44,x] = F[45,x]+F[57,x]
F[45,x] = F[27,x]+F[46,x]+F[53,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]
F[48,x] = F[10,x]+F[49,x]
F[49,x] = F[27,x]+F[50,x]+F[52,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[34,x]
F[52,x] = F[12,x]*F[17,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[55,x]+F[56,x]
F[55,x] = F[21,x]
F[56,x] = F[45,x]
F[57,x] = F[58,x]
F[58,x] = F[12,x]*F[44,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{20}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \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_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{26}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{25}\! \left(x \right) = F_{21}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{42}\! \left(x \right)
F_{27}\! \left(x \right) = 0
F_{28}\! \left(x \right) = F_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{38}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{19}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \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_{10}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{37}\! \left(x \right) = F_{18}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{21}\! \left(x \right)+F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{57}\! \left(x \right)
F_{45}\! \left(x \right) = F_{27}\! \left(x \right)+F_{46}\! \left(x \right)+F_{53}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{10}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{27}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{34}\! \left(x \right)
F_{52}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)+F_{56}\! \left(x \right)
F_{55}\! \left(x \right) = F_{21}\! \left(x \right)
F_{56}\! \left(x \right) = F_{45}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_20(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_12(x)*F_8(x))
Eq(F_8(x), F_13(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_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_12(x)*F_13(x))
Eq(F_20(x), F_21(x) + F_26(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_25(x))
Eq(F_24(x), F_1(x) + F_14(x))
Eq(F_25(x), F_21(x))
Eq(F_26(x), F_27(x) + F_28(x) + F_42(x))
Eq(F_27(x), 0)
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_38(x))
Eq(F_30(x), F_10(x) + F_31(x))
Eq(F_31(x), F_19(x) + F_27(x) + F_32(x))
Eq(F_32(x), F_12(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_10(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_12(x)*F_34(x))
Eq(F_37(x), F_18(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_27(x) + F_28(x) + F_40(x))
Eq(F_40(x), F_12(x)*F_41(x))
Eq(F_41(x), F_21(x) + F_39(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_41(x) + F_44(x))
Eq(F_44(x), F_45(x) + F_57(x))
Eq(F_45(x), F_27(x) + F_46(x) + F_53(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_10(x) + F_49(x))
Eq(F_49(x), F_27(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_34(x))
Eq(F_52(x), F_12(x)*F_17(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_55(x) + F_56(x))
Eq(F_55(x), F_21(x))
Eq(F_56(x), F_45(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_12(x)*F_44(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, 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": [2, 1, 3, 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": [2, 1, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 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