0123_0132_1203_1320_3102
Counting sequence:
1, 1, 2, 6, 19, 50, 108, 214, 413, 784, 1474, 2754, 5125, 9512, 17622, 32604, 60265, 111310, 205468, 379090, 699139, 1288946, 2375622, 4377326, 8063879, 14852346, 27350966, 50360032, 92713503, 170667400, 314133888, 578150534, 1063980405, 1957929036, 3602752646, 6629018842, 12196749821, 22439927228, 41284150970, 75950688876, 139723082865, 257036099210, 472836363088, 869800213642, 1600003836395, 2943176241894, 5413847280986, 9958430176926, 18317723506335, 33693673588246, 61975769701850, 113996781850584, 209681782624983, 385679506715676, 709398801213604, 1304825993114678, 2400010933626525, 4414408263097576, 8119524357563898, 14934395257155666, 27469058748409701, 50524160936589264, 92929528386206958, 170925844088718012, 314384542873078409, 578248020827079398, 1063571522729515764, 1956225306849389282, 3598079185766339843, 6617931571125315738, 12172325954881470846, 22388482158693622990, 41178975022761331847, 75740163921317844242, 139308237225815139630, 256228373077918074544, 471278387256117157503, 866817607498940229296, 1594328590803131417752, 2932431418467434618278, 5393588672648908035157, 9920366570708121654436, 18246415606492513660638, 33560417683306240285930, 61727275638631621889149, 113534231540011819059236, 208822123251655870745650, 384083951431586944428556, 706440825614248456478961, 1299347740689772726632866, 2389873877518883404765256, 4395664643998461320081306, 8084889822165949460908171, 14870434103817682927387342, 27350997890075314459437810, 50306336896286556340426558, 92527793290500383971005567, 170185167557410694507316654, 313019361625066904798948978, 575732425834395692993917736, 1058937122259160271997029719
Generating function in Maple syntax:
-(2*x^12+4*x^11-2*x^10-10*x^9-9*x^8+8*x^7+14*x^6+3*x^5-3*x^4-3*x^3-2*x^2+3*x-1)/(x^2+x-1)/(x^3+x^2+x-1)/(x-1)^2
Generating function in latex syntax:
-\frac{2 x^{12}+4 x^{11}-2 x^{10}-10 x^{9}-9 x^{8}+8 x^{7}+14 x^{6}+3 x^{5}-3 x^{4}-3 x^{3}-2 x^{2}+3 x -1}{\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**12 - 4*x**11 + 2*x**10 + 10*x**9 + 9*x**8 - 8*x**7 - 14*x**6 - 3*x**5 + 3*x**4 + 3*x**3 + 2*x**2 - 3*x + 1)/((x - 1)**2*(x**2 + x - 1)*(x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^2+x-1)*(x^3+x^2+x-1)*(x-1)^2*F(x)+2*x^12+4*x^11-2*x^10-10*x^9-9*x^8+8*x^7+14*x^6+3*x^5-3*x^4-3*x^3-2*x^2+3*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+2 x^{12}+4 x^{11}-2 x^{10}-10 x^{9}-9 x^{8}+8 x^{7}+14 x^{6}+3 x^{5}-3 x^{4}-3 x^{3}-2 x^{2}+3 x -1 = 0
Explicit closed form in Maple syntax:
-9/2-291/55*Sum(_alpha^(3-n),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3-_Z^2-2*_Z+1))-1453/110*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3-_Z^2-2*_Z+1))-1393/110*Sum(_alpha^(1-n),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3-_Z^2-2*_Z+1))-189/110*Sum(_alpha^(-n),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3-_Z^2-2*_Z+1))+508/55*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^5+2*_Z^4+_Z^3-_Z^2-2*_Z+1))-2*n-piecewise(n = 0,-2,n = 1,-3,n = 3,2,n = 4,4,n = 5,2,0)
Explicit closed form in latex syntax:
-\frac{9}{2}-\frac{291 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{3-n}\right)}{55}-\frac{1453 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{110}-\frac{1393 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{1-n}\right)}{110}-\frac{189 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{110}+\frac{508 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{5}+2 \textit{\_Z}^{4}+\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{55}-2 n -\left(\left\{\begin{array}{cc}-2 & n =0 \\ -3 & n =1 \\ 2 & n =3 \\ 4 & n =4 \\ 2 & n =5 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 50
a(6) = 108
a(7) = 214
a(8) = 413
a(9) = 784
a(10) = 1474
a(11) = 2754
a(12) = 5125
a(n+5) = -a(n)-2*a(n+1)-a(n+2)+a(n+3)+2*a(n+4)-4*n-5, n >= 13
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) = 50
a \! \left(6\right) = 108
a \! \left(7\right) = 214
a \! \left(8\right) = 413
a \! \left(9\right) = 784
a \! \left(10\right) = 1474
a \! \left(11\right) = 2754
a \! \left(12\right) = 5125
a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)-4 n -5, \quad n \geq 13
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16431/
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[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[24,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[11,x]
F[10,x] = F[1,x]+F[4,x]
F[11,x] = F[12,x]+F[18,x]
F[12,x] = F[13,x]
F[13,x] = F[14,x]*F[4,x]
F[14,x] = F[10,x]+F[15,x]
F[15,x] = F[12,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[12,x]*F[4,x]
F[18,x] = F[17,x]+F[19,x]+F[20,x]
F[19,x] = 0
F[20,x] = F[21,x]*F[4,x]
F[21,x] = F[22,x]+F[23,x]
F[22,x] = F[4,x]
F[23,x] = F[18,x]
F[24,x] = F[25,x]+F[36,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[29,x]
F[28,x] = F[1,x]+F[12,x]
F[29,x] = F[25,x]+F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]
F[33,x] = F[25,x]+F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[25,x]*F[4,x]
F[36,x] = F[19,x]+F[37,x]+F[53,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[49,x]
F[39,x] = F[40,x]+F[7,x]
F[40,x] = F[19,x]+F[20,x]+F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[15,x]+F[43,x]
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[45,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[22,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[44,x]
F[49,x] = F[36,x]+F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[4,x]*F[52,x]
F[52,x] = F[34,x]
F[53,x] = F[4,x]*F[54,x]
F[54,x] = F[33,x]+F[55,x]
F[55,x] = F[56,x]+F[60,x]
F[56,x] = F[57,x]
F[57,x] = F[4,x]*F[58,x]
F[58,x] = F[59,x]
F[59,x] = F[12,x]+F[44,x]
F[60,x] = F[61,x]
F[61,x] = F[4,x]*F[56,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_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{24}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \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_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{17}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)
F_{19}\! \left(x \right) = 0
F_{20}\! \left(x \right) = F_{21}\! \left(x \right) F_{4}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{23}\! \left(x \right)
F_{22}\! \left(x \right) = F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{18}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{36}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)
F_{28}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{25}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{19}\! \left(x \right)+F_{37}\! \left(x \right)+F_{53}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{49}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{7}\! \left(x \right)
F_{40}\! \left(x \right) = F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{15}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{22}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{44}\! \left(x \right)
F_{49}\! \left(x \right) = F_{36}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{4}\! \left(x \right) F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{34}\! \left(x \right)
F_{53}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{33}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{60}\! \left(x \right)
F_{56}\! \left(x \right) = F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{4}\! \left(x \right) F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{12}\! \left(x \right)+F_{44}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{56}\! \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_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_24(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_11(x))
Eq(F_10(x), F_1(x) + F_4(x))
Eq(F_11(x), F_12(x) + F_18(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_14(x)*F_4(x))
Eq(F_14(x), F_10(x) + F_15(x))
Eq(F_15(x), F_12(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_12(x)*F_4(x))
Eq(F_18(x), F_17(x) + F_19(x) + F_20(x))
Eq(F_19(x), 0)
Eq(F_20(x), F_21(x)*F_4(x))
Eq(F_21(x), F_22(x) + F_23(x))
Eq(F_22(x), F_4(x))
Eq(F_23(x), F_18(x))
Eq(F_24(x), F_25(x) + F_36(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_29(x))
Eq(F_28(x), F_1(x) + F_12(x))
Eq(F_29(x), F_25(x) + F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_25(x) + F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_25(x)*F_4(x))
Eq(F_36(x), F_19(x) + F_37(x) + F_53(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_49(x))
Eq(F_39(x), F_40(x) + F_7(x))
Eq(F_40(x), F_19(x) + F_20(x) + F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_15(x) + F_43(x))
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_22(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_44(x))
Eq(F_49(x), F_36(x) + F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_4(x)*F_52(x))
Eq(F_52(x), F_34(x))
Eq(F_53(x), F_4(x)*F_54(x))
Eq(F_54(x), F_33(x) + F_55(x))
Eq(F_55(x), F_56(x) + F_60(x))
Eq(F_56(x), F_57(x))
Eq(F_57(x), F_4(x)*F_58(x))
Eq(F_58(x), F_59(x))
Eq(F_59(x), F_12(x) + F_44(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_4(x)*F_56(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, 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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 2], "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, 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": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 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