0123_0132_1302_1320_2103
Counting sequence:
1, 1, 2, 6, 19, 50, 120, 288, 681, 1580, 3619, 8229, 18597, 41799, 93547, 208674, 464237, 1030481, 2283214, 5051326, 11161665, 24638173, 54340053, 119764040, 263802566, 580791434, 1278160048, 2811922767, 6184419575, 13598535076, 29895083667, 65710676070, 144415008883, 317350176324, 697305909988, 1532050099255, 3365838833708, 7394165529756, 16242935359622, 35679823663967, 78373007713222, 172146540339931, 378111609739393, 830487868457239, 1824061647850281, 4006266346039107, 8799037620651864, 19325306221482471, 42443786592010133, 93217822250823152, 204729934993621474, 449636651969241437, 987507390727587119, 2168790359049621943, 4763142732489692662, 10460889501724783664, 22974326758858056044, 50456400965202273206, 110812602776013473807, 243366921742736708319, 534482547737342958424, 1173829886934586450973, 2577961974454690565270, 5661709845630135815507, 12434218848922774155528, 27307958549783397030392, 59973558808821063154057, 131713498864145081604324, 289268173067617338006098, 635288414155269389357450, 1395215034234051271258839, 3064159026242717257886646, 6729478427585429689160425, 14779218449184333688805831, 32457980654515310689004703, 71283907312321180372021405, 156553028134449986620270757, 343820230940077204145246984, 755094602721771097711984299, 1658331270501481214225078087, 3642010573817340213248597708, 7998547064304432942510703960, 17566328436795030242158149631, 38578992822110912585566015011, 84726792650771156681953730983, 186076120362221073302534281406, 408658479311375283683772756480, 897491580678059293833393069504, 1971061834816583971607069059522, 4328825828262375634594461616157, 9506922935267048263756796602257, 20879006670779598442334280400910, 45854260216913833190251057217705, 100704655515852939622925400800460, 221166530031897268153047275144865, 485723660559927820370324305891166, 1066741310337325647022173133367710, 2342766297072046117649909830451821, 5145159246510315295250042559503226, 11299745809077035625095716487136002, 24816385486127758713617354390114776
Generating function in Maple syntax:
(1+x)*(x^7+4*x^6+4*x^5+6*x^4-x^3+x^2-2*x+1)/(x^3+x^2+x-1)/(x^5+3*x^4+2*x^3+x^2+x-1)
Generating function in latex syntax:
\frac{\left(1+x \right) \left(x^{7}+4 x^{6}+4 x^{5}+6 x^{4}-x^{3}+x^{2}-2 x +1\right)}{\left(x^{3}+x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(x + 1)*(x**7 + 4*x**6 + 4*x**5 + 6*x**4 - x**3 + x**2 - 2*x + 1)/((x**3 + x**2 + x - 1)*(x**5 + 3*x**4 + 2*x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3+x^2+x-1)*(x^5+3*x^4+2*x^3+x^2+x-1)*F(x)-(1+x)*(x^7+4*x^6+4*x^5+6*x^4-x^3+x^2-2*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}+x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)-\left(1+x \right) \left(x^{7}+4 x^{6}+4 x^{5}+6 x^{4}-x^{3}+x^{2}-2 x +1\right) = 0
Explicit closed form in Maple syntax:
-3395141/2659487*Sum(_alpha^(-n+6),_alpha = RootOf(_Z^8+4*_Z^7+6*_Z^6+5*_Z^5+_Z^4-_Z^3-_Z^2-2*_Z+1))-15918227/2659487*Sum(_alpha^(-n+5),_alpha = RootOf(_Z^8+4*_Z^7+6*_Z^6+5*_Z^5+_Z^4-_Z^3-_Z^2-2*_Z+1))-30169398/2659487*Sum(_alpha^(-n+4),_alpha = RootOf(_Z^8+4*_Z^7+6*_Z^6+5*_Z^5+_Z^4-_Z^3-_Z^2-2*_Z+1))-34007916/2659487*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^8+4*_Z^7+6*_Z^6+5*_Z^5+_Z^4-_Z^3-_Z^2-2*_Z+1))-21207887/2659487*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^8+4*_Z^7+6*_Z^6+5*_Z^5+_Z^4-_Z^3-_Z^2-2*_Z+1))-6348415/2659487*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^8+4*_Z^7+6*_Z^6+5*_Z^5+_Z^4-_Z^3-_Z^2-2*_Z+1))-41851/2659487*Sum(_alpha^(-n),_alpha = RootOf(_Z^8+4*_Z^7+6*_Z^6+5*_Z^5+_Z^4-_Z^3-_Z^2-2*_Z+1))+7591245/2659487*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^8+4*_Z^7+6*_Z^6+5*_Z^5+_Z^4-_Z^3-_Z^2-2*_Z+1))+piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
-\frac{3395141 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+4 \textit{\_Z}^{7}+6 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +6}\right)}{2659487}-\frac{15918227 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+4 \textit{\_Z}^{7}+6 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +5}\right)}{2659487}-\frac{30169398 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+4 \textit{\_Z}^{7}+6 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{2659487}-\frac{34007916 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+4 \textit{\_Z}^{7}+6 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\textit{\_Z}^{4}-\textit{\_Z}^{3}-\textit{\_Z}^{2}-2 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{2659487}-\frac{21207887 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+4 \textit{\_Z}^{7}+6 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\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)}{2659487}-\frac{6348415 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+4 \textit{\_Z}^{7}+6 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\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)}{2659487}-\frac{41851 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+4 \textit{\_Z}^{7}+6 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\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)}{2659487}+\frac{7591245 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+4 \textit{\_Z}^{7}+6 \textit{\_Z}^{6}+5 \textit{\_Z}^{5}+\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)}{2659487}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 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) = 120
a(7) = 288
a(8) = 681
a(n+8) = -a(n)-4*a(n+1)-6*a(n+2)-5*a(n+3)-a(n+4)+a(n+5)+a(n+6)+2*a(n+7), n >= 9
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) = 120
a \! \left(7\right) = 288
a \! \left(8\right) = 681
a \! \left(n +8\right) = -a \! \left(n \right)-4 a \! \left(n +1\right)-6 a \! \left(n +2\right)-5 a \! \left(n +3\right)-a \! \left(n +4\right)+a \! \left(n +5\right)+a \! \left(n +6\right)+2 a \! \left(n +7\right), \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16567/
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[19,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[7,x]
F[12,x] = F[13,x]+F[14,x]+F[18,x]
F[13,x] = 0
F[14,x] = F[15,x]*F[4,x]
F[15,x] = F[16,x]+F[17,x]
F[16,x] = F[4,x]
F[17,x] = F[18,x]
F[18,x] = F[4,x]*F[7,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[13,x]+F[21,x]+F[37,x]
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[27,x]
F[23,x] = F[24,x]+F[7,x]
F[24,x] = F[13,x]+F[14,x]+F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[11,x]
F[27,x] = F[28,x]+F[34,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[31,x]
F[31,x] = F[2,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[2,x]*F[4,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[32,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]
F[39,x] = F[2,x]+F[40,x]
F[40,x] = F[13,x]+F[33,x]+F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[43,x]+F[44,x]
F[43,x] = F[12,x]+F[4,x]
F[44,x] = F[32,x]+F[45,x]
F[45,x] = F[35,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_{19}\! \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_{7}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = 0
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{4}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{37}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{7}\! \left(x \right)
F_{24}\! \left(x \right) = F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{34}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{32}\! \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_{39}\! \left(x \right) = F_{2}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{13}\! \left(x \right)+F_{33}\! \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_{43}\! \left(x \right)+F_{44}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{44}\! \left(x \right) = F_{32}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{35}\! \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_19(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_7(x))
Eq(F_12(x), F_13(x) + F_14(x) + F_18(x))
Eq(F_13(x), 0)
Eq(F_14(x), F_15(x)*F_4(x))
Eq(F_15(x), F_16(x) + F_17(x))
Eq(F_16(x), F_4(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_4(x)*F_7(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_13(x) + F_21(x) + F_37(x))
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_27(x))
Eq(F_23(x), F_24(x) + F_7(x))
Eq(F_24(x), F_13(x) + F_14(x) + F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_11(x))
Eq(F_27(x), F_28(x) + F_34(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_2(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_2(x)*F_4(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_32(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_2(x) + F_40(x))
Eq(F_40(x), F_13(x) + F_33(x) + F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_43(x) + F_44(x))
Eq(F_43(x), F_12(x) + F_4(x))
Eq(F_44(x), F_32(x) + F_45(x))
Eq(F_45(x), F_35(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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 3], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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