0132_0213_0231_1023_2301
Counting sequence:
1, 1, 2, 6, 19, 54, 135, 305, 642, 1291, 2528, 4888, 9421, 18204, 35365, 69131, 135924, 268553, 532598, 1059178, 2110487, 4210866, 8408947, 16801941, 33584214, 67144439, 134259900, 268485100, 536928977, 1073809336, 2147561713, 4295057103, 8590037416, 17179986389, 34359871410, 68719627166, 137439122939, 274878097198, 549756026783, 1099511865273, 2199023519722, 4398046804131, 8796093346392, 17592186402176, 35184372482709, 70368744610324, 140737488829565, 281474977229395, 562949953987612, 1125899907459681, 2251799814356398, 4503599628099218, 9007199255530911, 18014398510336874, 36028797019887755, 72057594038924701, 144115188076929854, 288230376152867343, 576460752304665268, 1152921504608179668, 2305843009215122457, 4611686018428917296, 9223372036856411337, 18446744073711298711, 36893488147420967504, 73786976294840193709, 147573952589678529130, 295147905179355077190, 590295810358708044547, 1180591620717413844326, 2361183241434825302583, 4722366482869648071233, 9444732965739293453906, 18889465931478584057659, 37778931862957165096400, 75557863725914326997736, 151115727451828650616669, 302231454903657297662988, 604462909807314591556053, 1208925819614629179134363, 2417851639229258354074692, 4835703278458516703730361, 9671406556917033402807782, 19342813113834066800719546, 38685626227668133596290599, 77371252455336267187170594, 154742504910672534368658595, 309485009821345068731352485, 618970019642690137456447782, 1237940039285380274906335271, 2475880078570760549805796268, 4951760157141521099604393148, 9903520314283042199201250401, 19807040628566084398394616744, 39614081257132168796780989345, 79228162514264337593553362271, 158456325028528675187097723384, 316912650057057350374186048133, 633825300114114700748362287138, 1267650600228229401496714341358, 2535301200456458802993418012427
Generating function in Maple syntax:
-(2*x^6+5*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}+5 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 - 5*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+5*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}+5 x^{5}-12 x^{4}+18 x^{3}-15 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,15/8*n^2-49/12*n-11/12*n^3+1/8*n^4+2^(n+1))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{15 n^{2}}{8}-\frac{49 n}{12}-\frac{11 n^{3}}{12}+\frac{n^{4}}{8}+2^{n +1} & \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) = 54
a(6) = 135
a(n+1) = 2*a(n)-1/24*(n-1)*(3*n^3-31*n^2+62*n-72), 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) = 54
a \! \left(6\right) = 135
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(3 n^{3}-31 n^{2}+62 n -72\right)}{24}, \quad n \geq 7
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/21559/
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[4,x]
F[3,x] = x
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[3,x]*F[7,x]
F[7,x] = F[44,x]+F[54,x]+F[8,x]
F[8,x] = F[11,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[3,x]*F[9,x]
F[11,x] = F[12,x]+F[23,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[3,x]
F[14,x] = F[15,x]+F[17,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[15,x]*F[3,x]
F[17,x] = F[12,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[3,x]
F[20,x] = F[21,x]+F[22,x]
F[21,x] = F[10,x]
F[22,x] = F[19,x]
F[23,x] = F[24,x]+F[25,x]+F[39,x]
F[24,x] = 0
F[25,x] = F[26,x]*F[3,x]
F[26,x] = F[27,x]+F[32,x]
F[27,x] = F[21,x]+F[28,x]
F[28,x] = F[29,x]
F[29,x] = F[3,x]*F[30,x]
F[30,x] = F[3,x]+F[31,x]
F[31,x] = F[29,x]
F[32,x] = F[23,x]+F[33,x]
F[33,x] = F[34,x]
F[34,x] = F[3,x]*F[35,x]
F[35,x] = F[36,x]+F[38,x]
F[36,x] = F[37,x]
F[37,x] = F[21,x]*F[3,x]
F[38,x] = F[34,x]
F[39,x] = F[3,x]*F[40,x]
F[40,x] = F[21,x]+F[41,x]
F[41,x] = F[24,x]+F[39,x]+F[42,x]
F[42,x] = F[3,x]*F[43,x]
F[43,x] = F[21,x]+F[41,x]
F[44,x] = F[3,x]*F[45,x]
F[45,x] = F[46,x]+F[7,x]
F[46,x] = F[47,x]
F[47,x] = F[9,x]^2*F[3,x]*F[48,x]
F[48,x] = F[49,x]+F[9,x]
F[49,x] = F[21,x]+F[50,x]
F[50,x] = F[24,x]+F[51,x]+F[53,x]
F[51,x] = F[3,x]*F[52,x]
F[52,x] = F[21,x]+F[50,x]
F[53,x] = F[3,x]*F[49,x]
F[54,x] = F[55,x]
F[55,x] = F[9,x]^2*F[15,x]*F[3,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_{4}\! \left(x \right)
F_{3}\! \left(x \right) = x
F_{4}\! \left(x \right) = F_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{3}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{44}\! \left(x \right)+F_{54}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{11}\! \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_{3}\! \left(x \right) F_{9}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{23}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{3}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{17}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{15}\! \left(x \right) F_{3}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{3}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)
F_{22}\! \left(x \right) = F_{19}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{39}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{3}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{32}\! \left(x \right)
F_{27}\! \left(x \right) = F_{21}\! \left(x \right)+F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{3}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{3}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{23}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{3}\! \left(x \right) F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{21}\! \left(x \right) F_{3}\! \left(x \right)
F_{38}\! \left(x \right) = F_{34}\! \left(x \right)
F_{39}\! \left(x \right) = F_{3}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{21}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{24}\! \left(x \right)+F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{3}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{21}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{3}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{7}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{3}\! \left(x \right) F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{9}\! \left(x \right)
F_{49}\! \left(x \right) = F_{21}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{24}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)
F_{51}\! \left(x \right) = F_{3}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{21}\! \left(x \right)+F_{50}\! \left(x \right)
F_{53}\! \left(x \right) = F_{3}\! \left(x \right) F_{49}\! \left(x \right)
F_{54}\! \left(x \right) = F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{15}\! \left(x \right) F_{3}\! \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)*F_4(x))
Eq(F_3(x), x)
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_3(x)*F_7(x))
Eq(F_7(x), F_44(x) + F_54(x) + F_8(x))
Eq(F_8(x), F_11(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_3(x)*F_9(x))
Eq(F_11(x), F_12(x) + F_23(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_3(x))
Eq(F_14(x), F_15(x) + F_17(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_15(x)*F_3(x))
Eq(F_17(x), F_12(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_3(x))
Eq(F_20(x), F_21(x) + F_22(x))
Eq(F_21(x), F_10(x))
Eq(F_22(x), F_19(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_39(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_26(x)*F_3(x))
Eq(F_26(x), F_27(x) + F_32(x))
Eq(F_27(x), F_21(x) + F_28(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_3(x)*F_30(x))
Eq(F_30(x), F_3(x) + F_31(x))
Eq(F_31(x), F_29(x))
Eq(F_32(x), F_23(x) + F_33(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_3(x)*F_35(x))
Eq(F_35(x), F_36(x) + F_38(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_21(x)*F_3(x))
Eq(F_38(x), F_34(x))
Eq(F_39(x), F_3(x)*F_40(x))
Eq(F_40(x), F_21(x) + F_41(x))
Eq(F_41(x), F_24(x) + F_39(x) + F_42(x))
Eq(F_42(x), F_3(x)*F_43(x))
Eq(F_43(x), F_21(x) + F_41(x))
Eq(F_44(x), F_3(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_7(x))
Eq(F_46(x), F_47(x))
Eq(F_47(x), F_3(x)*F_48(x)*F_9(x)**2)
Eq(F_48(x), F_49(x) + F_9(x))
Eq(F_49(x), F_21(x) + F_50(x))
Eq(F_50(x), F_24(x) + F_51(x) + F_53(x))
Eq(F_51(x), F_3(x)*F_52(x))
Eq(F_52(x), F_21(x) + F_50(x))
Eq(F_53(x), F_3(x)*F_49(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_15(x)*F_3(x)*F_9(x)**2)
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "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": "row_and_col_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, 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": [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, 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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 1], "pos": [[1, 2], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 2], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [2, 3, 0, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", 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