0132_0231_1203_1230_2301
Counting sequence:
1, 1, 2, 6, 19, 57, 158, 407, 989, 2302, 5198, 11496, 25071, 54159, 116222, 248189, 527961, 1119468, 2366878, 4991206, 10499915, 22038885, 46161886, 96500595, 201366773, 419480906, 872478062, 1812016756, 3758190983, 7784742891, 16106265342, 33286161465, 68719672625, 141734152088, 292058047806, 601295738898, 1236950950435, 2542621066657, 5222680724702, 10720238936655, 21990233202829, 45079977476694, 92358977571470, 189116000926784, 387028094048927, 791648373205255, 1618481117440766, 3307330977866805, 6755399442749577, 13792273860709380, 28147497673163486, 57420895251300734, 117093590314208379, 238690780253481117, 486388759759149790, 990791918024960107, 2017612633065772709, 4107282860166048546, 8358680908404190126, 17005592192955965004, 34587645138210834743, 70328211781023576803, 142962266571255456254, 290536219160932425073, 590295810358713232481, 1199038364791129069104, 2434970217729669702270, 4943727411754169440554, 10035028776098006450771, 20365205457375356167769, 41320706725109407665630, 83822005070936215503303, 170005193383307241622333, 344732753249484115555246, 698910239464707507668174, 1416709944860893581297816, 2871198821584744308328783, 5817955506895402922955135, 11787026741242634474417150, 23876284937388926222902253, 48357032784585167012201657, 97922991388784963176733212, 198263834416799184679006238, 401363372112056886031389462, 812398150781030805433323435, 1644139114675895677633098453, 3326963855579459488826116574, 6731298963614255244800828003, 13617340432139183023929428565, 27544165874099711116546904058, 55707301767842112370504417774, 112652543574969605015866683172, 227780967228509970581487904487, 460513694614161462262526048475, 930930909542605966724196168958, 1881668859713778017846726618025, 3802951800684688204490170592657, 7685131763883640746573827476424, 15528719852795810168334682019774, 31374352355648677687043475694402, 63382530011411470074835235389571
Generating function in Maple syntax:
-(3*x^7-30*x^6+74*x^5-96*x^4+75*x^3-35*x^2+9*x-1)/(2*x-1)^2/(x-1)^6
Generating function in latex syntax:
-\frac{3 x^{7}-30 x^{6}+74 x^{5}-96 x^{4}+75 x^{3}-35 x^{2}+9 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{6}}
Generating function in sympy syntax:
(-3*x**7 + 30*x**6 - 74*x**5 + 96*x**4 - 75*x**3 + 35*x**2 - 9*x + 1)/((x - 1)**6*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^6*F(x)+3*x^7-30*x^6+74*x^5-96*x^4+75*x^3-35*x^2+9*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)+3 x^{7}-30 x^{6}+74 x^{5}-96 x^{4}+75 x^{3}-35 x^{2}+9 x -1 = 0
Explicit closed form in Maple syntax:
1+2^(-1+n)*n+1/120*n^5-1/12*n^4+1/8*n^3-5/12*n^2-19/30*n
Explicit closed form in latex syntax:
1+2^{-1+n} n +\frac{n^{5}}{120}-\frac{n^{4}}{12}+\frac{n^{3}}{8}-\frac{5 n^{2}}{12}-\frac{19 n}{30}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 57
a(6) = 158
a(7) = 407
a(n+2) = -4*a(n)+4*a(n+1)+1/120*(n+1)*(n^4-21*n^3+116*n^2-216*n+240), n >= 8
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) = 57
a \! \left(6\right) = 158
a \! \left(7\right) = 407
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{\left(n +1\right) \left(n^{4}-21 n^{3}+116 n^{2}-216 n +240\right)}{120}, \quad n \geq 8
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/21713/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[12,x]*F[3,x]
F[3,x] = F[28,x]+F[4,x]
F[4,x] = F[1,x]+F[5,x]
F[5,x] = F[12,x]*F[6,x]
F[6,x] = F[4,x]+F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[7,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[14,x] = F[12,x]*F[15,x]*F[9,x]
F[15,x] = F[16,x]+F[23,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[19,x]+F[20,x]+F[22,x]
F[19,x] = 0
F[20,x] = F[12,x]*F[21,x]
F[21,x] = F[10,x]+F[18,x]
F[22,x] = F[12,x]*F[17,x]
F[23,x] = F[10,x]+F[24,x]
F[24,x] = F[19,x]+F[25,x]+F[26,x]
F[25,x] = F[12,x]*F[23,x]
F[26,x] = F[12,x]*F[27,x]
F[27,x] = F[10,x]+F[24,x]
F[28,x] = F[29,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]+F[43,x]+F[52,x]
F[31,x] = F[32,x]*F[9,x]
F[32,x] = F[1,x]+F[33,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[37,x]
F[35,x] = F[1,x]+F[36,x]
F[36,x] = F[12,x]*F[35,x]
F[37,x] = F[10,x]+F[38,x]
F[38,x] = F[19,x]+F[39,x]+F[42,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[38,x]+F[41,x]
F[41,x] = F[36,x]
F[42,x] = F[12,x]*F[37,x]
F[43,x] = F[44,x]
F[44,x] = F[12,x]*F[45,x]
F[45,x] = F[46,x]+F[47,x]
F[46,x] = F[16,x]*F[32,x]*F[9,x]
F[47,x] = F[48,x]
F[48,x] = F[12,x]*F[35,x]*F[49,x]*F[9,x]
F[49,x] = F[50,x]+F[51,x]
F[50,x] = F[32,x]*F[34,x]
F[51,x] = F[9,x]^2*F[10,x]
F[52,x] = F[53,x]
F[53,x] = F[12,x]*F[54,x]
F[54,x] = F[30,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[9,x]^2*F[12,x]*F[57,x]
F[57,x] = F[27,x]+F[9,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_{12}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{28}\! \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_{12}\! \left(x \right) F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{4}\! \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_{7}\! \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_{14}\! \left(x \right) = F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{9}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{23}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \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_{20}\! \left(x \right)+F_{22}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{19}\! \left(x \right)+F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{12}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{10}\! \left(x \right)+F_{24}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{43}\! \left(x \right)+F_{52}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{9}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{37}\! \left(x \right)
F_{35}\! \left(x \right) = F_{1}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right)
F_{37}\! \left(x \right) = F_{10}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{19}\! \left(x \right)+F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{36}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{12}\! \left(x \right) F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)+F_{47}\! \left(x \right)
F_{46}\! \left(x \right) = F_{16}\! \left(x \right) F_{32}\! \left(x \right) F_{9}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{35}\! \left(x \right) F_{49}\! \left(x \right) F_{9}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)+F_{51}\! \left(x \right)
F_{50}\! \left(x \right) = F_{32}\! \left(x \right) F_{34}\! \left(x \right)
F_{51}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{10}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{30}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{27}\! \left(x \right)+F_{9}\! \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_12(x)*F_3(x))
Eq(F_3(x), F_28(x) + F_4(x))
Eq(F_4(x), F_1(x) + F_5(x))
Eq(F_5(x), F_12(x)*F_6(x))
Eq(F_6(x), F_4(x) + F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_7(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))
Eq(F_14(x), F_12(x)*F_15(x)*F_9(x))
Eq(F_15(x), F_16(x) + F_23(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_19(x) + F_20(x) + F_22(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_12(x)*F_21(x))
Eq(F_21(x), F_10(x) + F_18(x))
Eq(F_22(x), F_12(x)*F_17(x))
Eq(F_23(x), F_10(x) + F_24(x))
Eq(F_24(x), F_19(x) + F_25(x) + F_26(x))
Eq(F_25(x), F_12(x)*F_23(x))
Eq(F_26(x), F_12(x)*F_27(x))
Eq(F_27(x), F_10(x) + F_24(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_43(x) + F_52(x))
Eq(F_31(x), F_32(x)*F_9(x))
Eq(F_32(x), F_1(x) + F_33(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_37(x))
Eq(F_35(x), F_1(x) + F_36(x))
Eq(F_36(x), F_12(x)*F_35(x))
Eq(F_37(x), F_10(x) + F_38(x))
Eq(F_38(x), F_19(x) + F_39(x) + F_42(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_38(x) + F_41(x))
Eq(F_41(x), F_36(x))
Eq(F_42(x), F_12(x)*F_37(x))
Eq(F_43(x), F_44(x))
Eq(F_44(x), F_12(x)*F_45(x))
Eq(F_45(x), F_46(x) + F_47(x))
Eq(F_46(x), F_16(x)*F_32(x)*F_9(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_12(x)*F_35(x)*F_49(x)*F_9(x))
Eq(F_49(x), F_50(x) + F_51(x))
Eq(F_50(x), F_32(x)*F_34(x))
Eq(F_51(x), F_10(x)*F_9(x)**2)
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_12(x)*F_54(x))
Eq(F_54(x), F_30(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_12(x)*F_57(x)*F_9(x)**2)
Eq(F_57(x), F_27(x) + F_9(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, 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": [1, 2, 3, 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], "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, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], 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