0123_0132_0231_1203_2301
Counting sequence:
1, 1, 2, 6, 19, 55, 143, 340, 756, 1603, 3292, 6626, 13185, 26095, 51565, 101960, 201946, 400779, 796882, 1586998, 3164495, 6315975, 12614491, 25205980, 50382128, 100726099, 201403992, 402747754, 805421005, 1610750687, 3221390361, 6442646800, 12885133174, 25770075419, 51539924974, 103079584254, 206158857595, 412317353143, 824634286631, 1649268088932, 3298535621164, 6597070604899, 13194140482580, 26388280138354, 52776559339737, 105553117620943, 211106234049541, 422212466759768, 844424932019154, 1688849862361771, 3377699722854730, 6755399443631174, 13510798884956263, 27021597767359143, 54043195531896883, 108086391060682348, 216172782117939944, 432345564232117107, 864691128460107280, 1729382256915695866, 3458764513826452133, 6917529027647513023, 13835058055289150769, 27670116110571908128, 55340232221136868846, 110680464442266198587, 221360928884524226790, 442721857769039610382, 885443715538069661203, 1770887431076129000855, 3541774862152246870399, 7083549724304481749748, 14167099448608950596452, 28334198897217887323267, 56668397794435759753292, 113336795588871503530242, 226673591177742989938993, 453347182355485961546671, 906694364710971903484829, 1813388729421943786013800, 3626777458843887549651402, 7253554917687775075430347, 14507109835375550125413058, 29014219670751100223721302, 58028439341502200418595455, 116056878683004400806513031, 232113757366008801580425739, 464227514732017603126233596, 928455029464035206215733152, 1856910058928070412392513939, 3713820117856140824743751368, 7427640235712281649443792522, 14855280471424563298841327741, 29710560942849126597633733791, 59421121885698253195215760201, 118842243771396506390376901936, 237684487542793012780696144742, 475368975085586025561331455835, 950737950171172051122598765278, 1901475900342344102245129928734, 3802951800684688204490188652971
Generating function in Maple syntax:
-(x^2-x+1)*(3*x^4-13*x^3+14*x^2-6*x+1)/(2*x-1)/(-1+x)^6
Generating function in latex syntax:
-\frac{\left(x^{2}-x +1\right) \left(3 x^{4}-13 x^{3}+14 x^{2}-6 x +1\right)}{\left(2 x -1\right) \left(-1+x \right)^{6}}
Generating function in sympy syntax:
(-x**2 + x - 1)*(3*x**4 - 13*x**3 + 14*x**2 - 6*x + 1)/((x - 1)**6*(2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(-1+x)^6*F(x)+(x^2-x+1)*(3*x^4-13*x^3+14*x^2-6*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(-1+x \right)^{6} F \! \left(x \right)+\left(x^{2}-x +1\right) \left(3 x^{4}-13 x^{3}+14 x^{2}-6 x +1\right) = 0
Explicit closed form in Maple syntax:
-143/60*n-2+1/120*n^5-1/24*n^4-1/8*n^3-11/24*n^2+3*2^n
Explicit closed form in latex syntax:
-\frac{143 n}{60}-2+\frac{n^{5}}{120}-\frac{n^{4}}{24}-\frac{n^{3}}{8}-\frac{11 n^{2}}{24}+3 \,2^{n}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 55
a(6) = 143
a(n+1) = 2*a(n)-1/120*(n-1)*(n^4-9*n^3-14*n^2-4*n-120), 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) = 55
a \! \left(6\right) = 143
a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{4}-9 n^{3}-14 n^{2}-4 n -120\right)}{120}, \quad n \geq 7
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/21501/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[10,x]*F[3,x]
F[3,x] = F[0,x]+F[4,x]
F[4,x] = F[5,x]
F[5,x] = F[10,x]*F[6,x]
F[6,x] = F[41,x]+F[43,x]+F[7,x]
F[7,x] = F[11,x]+F[8,x]
F[8,x] = F[1,x]+F[9,x]
F[9,x] = F[10,x]*F[8,x]
F[10,x] = x
F[11,x] = F[12,x]+F[22,x]
F[12,x] = F[13,x]
F[13,x] = F[10,x]*F[14,x]
F[14,x] = F[15,x]+F[8,x]
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[10,x]*F[18,x]
F[18,x] = F[19,x]+F[21,x]
F[19,x] = F[20,x]
F[20,x] = F[9,x]
F[21,x] = F[16,x]
F[22,x] = F[23,x]+F[24,x]+F[36,x]
F[23,x] = 0
F[24,x] = F[10,x]*F[25,x]
F[25,x] = F[26,x]+F[30,x]
F[26,x] = F[20,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[10,x]*F[29,x]
F[29,x] = F[20,x]+F[27,x]
F[30,x] = F[22,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[10,x]*F[33,x]
F[33,x] = F[34,x]+F[35,x]
F[34,x] = F[27,x]
F[35,x] = F[31,x]
F[36,x] = F[10,x]*F[37,x]
F[37,x] = F[20,x]+F[38,x]
F[38,x] = F[23,x]+F[36,x]+F[39,x]
F[39,x] = F[10,x]*F[40,x]
F[40,x] = F[20,x]+F[38,x]
F[41,x] = F[10,x]*F[42,x]*F[8,x]
F[42,x] = F[37,x]+F[8,x]
F[43,x] = F[10,x]*F[44,x]
F[44,x] = F[45,x]+F[6,x]
F[45,x] = F[46,x]
F[46,x] = F[8,x]^2*F[10,x]*F[47,x]
F[47,x] = F[48,x]+F[8,x]
F[48,x] = F[20,x]+F[49,x]
F[49,x] = F[23,x]+F[50,x]+F[52,x]
F[50,x] = F[10,x]*F[51,x]
F[51,x] = F[20,x]+F[49,x]
F[52,x] = F[10,x]*F[48,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_{10}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{0}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{10}\! \left(x \right) F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{41}\! \left(x \right)+F_{43}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{11}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{1}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{22}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{8}\! \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_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)+F_{21}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{9}\! \left(x \right)
F_{21}\! \left(x \right) = F_{16}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{36}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{20}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{22}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{10}\! \left(x \right) F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{35}\! \left(x \right)
F_{34}\! \left(x \right) = F_{27}\! \left(x \right)
F_{35}\! \left(x \right) = F_{31}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{20}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{23}\! \left(x \right)+F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{10}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{20}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right) F_{42}\! \left(x \right) F_{8}\! \left(x \right)
F_{42}\! \left(x \right) = F_{37}\! \left(x \right)+F_{8}\! \left(x \right)
F_{43}\! \left(x \right) = F_{10}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{6}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{8} \left(x \right)^{2} F_{10}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{8}\! \left(x \right)
F_{48}\! \left(x \right) = F_{20}\! \left(x \right)+F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{23}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)
F_{50}\! \left(x \right) = F_{10}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{20}\! \left(x \right)+F_{49}\! \left(x \right)
F_{52}\! \left(x \right) = F_{10}\! \left(x \right) F_{48}\! \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_10(x)*F_3(x))
Eq(F_3(x), F_0(x) + F_4(x))
Eq(F_4(x), F_5(x))
Eq(F_5(x), F_10(x)*F_6(x))
Eq(F_6(x), F_41(x) + F_43(x) + F_7(x))
Eq(F_7(x), F_11(x) + F_8(x))
Eq(F_8(x), F_1(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_8(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_22(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_10(x)*F_14(x))
Eq(F_14(x), F_15(x) + F_8(x))
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_10(x)*F_18(x))
Eq(F_18(x), F_19(x) + F_21(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_9(x))
Eq(F_21(x), F_16(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_36(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_10(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_30(x))
Eq(F_26(x), F_20(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_10(x)*F_29(x))
Eq(F_29(x), F_20(x) + F_27(x))
Eq(F_30(x), F_22(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_10(x)*F_33(x))
Eq(F_33(x), F_34(x) + F_35(x))
Eq(F_34(x), F_27(x))
Eq(F_35(x), F_31(x))
Eq(F_36(x), F_10(x)*F_37(x))
Eq(F_37(x), F_20(x) + F_38(x))
Eq(F_38(x), F_23(x) + F_36(x) + F_39(x))
Eq(F_39(x), F_10(x)*F_40(x))
Eq(F_40(x), F_20(x) + F_38(x))
Eq(F_41(x), F_10(x)*F_42(x)*F_8(x))
Eq(F_42(x), F_37(x) + F_8(x))
Eq(F_43(x), F_10(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_6(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_10(x)*F_47(x)*F_8(x)**2)
Eq(F_47(x), F_48(x) + F_8(x))
Eq(F_48(x), F_20(x) + F_49(x))
Eq(F_49(x), F_23(x) + F_50(x) + F_52(x))
Eq(F_50(x), F_10(x)*F_51(x))
Eq(F_51(x), F_20(x) + F_49(x))
Eq(F_52(x), F_10(x)*F_48(x))
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, 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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 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, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 2], [0, 2], [0, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 2], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}], "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": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], 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