0213_0231_1203_1230_2301
Counting sequence:
1, 1, 2, 6, 19, 58, 166, 446, 1137, 2782, 6596, 15266, 34679, 77634, 171778, 376502, 818781, 1768950, 3800448, 8125690, 17300579, 36699066, 77593342, 163576366, 343931209, 721418318, 1509947196, 3154114066, 6576665807, 13690205042, 28454154746, 59055796326, 122406563509, 253403065574, 523986004728, 1082331752682, 2233382987451, 4604204934250, 9483287781878, 19516331384670, 40132174404769, 82463372073406, 169324790666932, 347445674365826, 712483534786599, 1460151441673378, 2990671627536626, 6122080743441686, 12525636463608077, 25614222880652502, 52354345668163696, 106960491150029786, 218424581927448339, 445856363109657114, 909727124728816878, 1855483046476619662, 3783023686991190521, 7710162562058261550, 15708555500268260908, 31993571752839972850, 65140065010286821759, 132585973029787368146, 269783632078002156522, 548790636192859122886, 1116028016459427833189, 2268949521066274807238, 4611686018427387860456, 9370945989444452175306, 19037039884068257219947, 38664375578495220137162, 78509342777707851625446, 159379868796850525907646, 323482104076570697081169, 656408941118880684644254, 1331707348169239950200228, 2701193628201437062169442, 5477945120128788447819991, 11107005967709405542542850, 22516243390322468378829538, 45636949690452251345082230, 92482825200519131864943549, 187383502040267522079375286, 379602707358993560857654112, 768876821274904155113039546, 1557096455663642377021462979, 3152878537554952887633611898, 6383128327565242042448510686, 12920999160041156619259506926, 26151483329903658307243893417, 52921936679450006751937451022, 107081813398185393778774131996, 216639506874941548107346621906, 438230773907024617314289853999, 886365068128332276827772818994, 1792537176885230638053931746778, 3624688435027593444904635594022, 7328605032569451227402815267861, 14815666390167431129992718570150, 29948245430391919610359613079768, 60530316160897953921467577904810, 122328282922024137244431859162139
Generating function in Maple syntax:
-(2*x^6+5*x^5-20*x^4+28*x^3-20*x^2+7*x-1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
-\frac{2 x^{6}+5 x^{5}-20 x^{4}+28 x^{3}-20 x^{2}+7 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-2*x**6 - 5*x**5 + 20*x**4 - 28*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**4*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^4*F(x)+2*x^6+5*x^5-20*x^4+28*x^3-20*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+2 x^{6}+5 x^{5}-20 x^{4}+28 x^{3}-20 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,5+2^n*n-7*2^(-1+n)+1/6*n+n^2-1/6*n^3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ 5+2^{n} n -7 \,2^{-1+n}+\frac{n}{6}+n^{2}-\frac{n^{3}}{6} & \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) = 58
a(6) = 166
a(n+2) = -1/6*n^3+2*n^2-4*a(n)+4*a(n+1)-23/6*n+4, 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) = 58
a \! \left(6\right) = 166
a \! \left(n +2\right) = -\frac{n^{3}}{6}+2 n^{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{23 n}{6}+4, \quad n \geq 7
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://www.permpal.com/tree/21869/
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[12,x]*F[29,x]
F[29,x] = F[30,x]+F[39,x]+F[49,x]
F[30,x] = F[31,x]+F[34,x]
F[31,x] = F[1,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[12,x]*F[31,x]
F[34,x] = F[10,x]+F[35,x]
F[35,x] = F[19,x]+F[36,x]+F[38,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[32,x]+F[35,x]
F[38,x] = F[12,x]*F[34,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[29,x]+F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[16,x]*F[43,x]*F[9,x]
F[43,x] = F[44,x]+F[9,x]
F[44,x] = F[32,x]+F[45,x]
F[45,x] = F[19,x]+F[46,x]+F[48,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[10,x]+F[45,x]
F[48,x] = F[12,x]*F[44,x]
F[49,x] = F[50,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = 2*F[53,x]+F[52,x]+F[54,x]
F[52,x] = F[27,x]+F[9,x]
F[53,x] = F[12,x]*F[51,x]
F[54,x] = F[55,x]
F[55,x] = F[9,x]^2*F[12,x]*F[43,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_{12}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{39}\! \left(x \right)+F_{49}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{34}\! \left(x \right)
F_{31}\! \left(x \right) = F_{1}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{19}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{32}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{29}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right) F_{43}\! \left(x \right) F_{9}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{9}\! \left(x \right)
F_{44}\! \left(x \right) = F_{32}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{19}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{10}\! \left(x \right)+F_{45}\! \left(x \right)
F_{48}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{53}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{27}\! \left(x \right)+F_{9}\! \left(x \right)
F_{53}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \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_{12}\! \left(x \right) F_{43}\! \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_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_39(x) + F_49(x))
Eq(F_30(x), F_31(x) + F_34(x))
Eq(F_31(x), F_1(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_12(x)*F_31(x))
Eq(F_34(x), F_10(x) + F_35(x))
Eq(F_35(x), F_19(x) + F_36(x) + F_38(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_32(x) + F_35(x))
Eq(F_38(x), F_12(x)*F_34(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_29(x) + F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_16(x)*F_43(x)*F_9(x))
Eq(F_43(x), F_44(x) + F_9(x))
Eq(F_44(x), F_32(x) + F_45(x))
Eq(F_45(x), F_19(x) + F_46(x) + F_48(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_10(x) + F_45(x))
Eq(F_48(x), F_12(x)*F_44(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x) + 2*F_53(x) + F_54(x))
Eq(F_52(x), F_27(x) + F_9(x))
Eq(F_53(x), F_12(x)*F_51(x))
Eq(F_54(x), F_55(x))
Eq(F_55(x), F_12(x)*F_43(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, 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, 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, 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, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], 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