0321_1230_1302_2031_2103
Counting sequence:
1, 1, 2, 6, 19, 52, 136, 347, 862, 2100, 5045, 11982, 28192, 65827, 152724, 352416, 809451, 1851772, 4221542, 9594627, 21747788, 49177286, 110965899, 249912074, 561877724, 1261323697, 2827519788, 6330465042, 14156842837, 31625865254, 70583500206, 157392533555, 350684835970, 780783035166, 1737201055785, 3862775759980, 8584199449114, 19066468427261, 42328066058520, 93926972083840, 208339008397133, 461936894533952, 1023855803114496, 2268560300392117, 5024902442211458, 11127056215803350, 24632987620652767, 54518787679989420, 120635492464333720, 266876768739918007, 590282117120573574, 1305351946630402300, 2886159127156139313, 6380323436625284422, 14102614582039411080, 31167036268821588479, 68870684309553809476, 152166874103528317232, 336167875430876552191, 742584641676843853508, 1640183525057799215646, 3622417572464973060231, 7999573305091943450668, 17664439242352040992878, 39003171344202142743479, 86113090071528476541162, 190112040483180192604724, 419684707167542504356869, 926427603804462243092932, 2044919818620508057695146, 4513558345146541907827161, 9961878519016610217811822, 21985895322765431456110510, 48520922253712541896969031, 107077217206164645657671594, 236292168334995281578944270, 521418956695508592624903277, 1150563228591368917653778700, 2538757592663249760463750786, 5601700441305223377823536505, 12359679271390727242211879296, 27269923627202051451970467408, 60165941913367532229877969369, 132741961427202383259257460944, 292857969172045221458965473904, 646096997204046729296888598825, 1425380386348134388553287208722, 3144536933037694697949121935254, 6937067841545271283065606511275, 15303435216846158659488141099028, 33759504305254928825066935680952, 74472804685688578461269180416371, 164283605033462480485715491800062, 362398116075509819549343315655604, 799416552391069128080771134604429, 1763421476197403356256366162540078, 3889870714258379086601379147761120, 8580458825889744226226417380811419, 18927026667064837437930424507812196, 41749476490415844245314055113301424, 92090883123589065033352657549374323
Generating function in Maple syntax:
(x^8+x^7-6*x^4-2*x^2+3*x-1)/(x-1)/(x^3+2*x-1)/(x^4+2*x^3+x^2+x-1)
Generating function in latex syntax:
\frac{x^{8}+x^{7}-6 x^{4}-2 x^{2}+3 x -1}{\left(x -1\right) \left(x^{3}+2 x -1\right) \left(x^{4}+2 x^{3}+x^{2}+x -1\right)}
Generating function in sympy syntax:
(x**8 + x**7 - 6*x**4 - 2*x**2 + 3*x - 1)/((x - 1)*(x**3 + 2*x - 1)*(x**4 + 2*x**3 + x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
(x-1)*(x^3+2*x-1)*(x^4+2*x^3+x^2+x-1)*F(x)-x^8-x^7+6*x^4+2*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right) \left(x^{3}+2 x -1\right) \left(x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)-x^{8}-x^{7}+6 x^{4}+2 x^{2}-3 x +1 = 0
Explicit closed form in Maple syntax:
2185047/88618*Sum(_alpha^(-n+6),_alpha = RootOf(_Z^8+_Z^7+_Z^6+_Z^5-5*_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))+6418075/177236*Sum(_alpha^(-n+5),_alpha = RootOf(_Z^8+_Z^7+_Z^6+_Z^5-5*_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))+7390891/177236*Sum(_alpha^(-n+4),_alpha = RootOf(_Z^8+_Z^7+_Z^6+_Z^5-5*_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))+3931897/88618*Sum(_alpha^(-n+3),_alpha = RootOf(_Z^8+_Z^7+_Z^6+_Z^5-5*_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))-4533943/44309*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^8+_Z^7+_Z^6+_Z^5-5*_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))+267937/177236*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^8+_Z^7+_Z^6+_Z^5-5*_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))-4346970/44309*Sum(_alpha^(-n),_alpha = RootOf(_Z^8+_Z^7+_Z^6+_Z^5-5*_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))+9301479/177236*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^8+_Z^7+_Z^6+_Z^5-5*_Z^4+2*_Z^3-4*_Z^2+4*_Z-1))+piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
\frac{2185047 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +6}\right)}{88618}+\frac{6418075 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +5}\right)}{177236}+\frac{7390891 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +4}\right)}{177236}+\frac{3931897 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +3}\right)}{88618}-\frac{4533943 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{44309}+\frac{267937 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{177236}-\frac{4346970 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{44309}+\frac{9301479 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{8}+\textit{\_Z}^{7}+\textit{\_Z}^{6}+\textit{\_Z}^{5}-5 \textit{\_Z}^{4}+2 \textit{\_Z}^{3}-4 \textit{\_Z}^{2}+4 \textit{\_Z} -1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{177236}+\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) = 52
a(6) = 136
a(7) = 347
a(8) = 862
a(n+7) = -a(n)-2*a(n+1)-3*a(n+2)-4*a(n+3)+a(n+4)-a(n+5)+3*a(n+6)+4, 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) = 52
a \! \left(6\right) = 136
a \! \left(7\right) = 347
a \! \left(8\right) = 862
a \! \left(n +7\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)-4 a \! \left(n +3\right)+a \! \left(n +4\right)-a \! \left(n +5\right)+3 a \! \left(n +6\right)+4, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21097/
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[22,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[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]+F[16,x]+F[19,x]
F[15,x] = 0
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[18,x]+F[7,x]
F[18,x] = F[16,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[11,x]+F[18,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[15,x]+F[24,x]+F[37,x]
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[30,x]
F[26,x] = F[27,x]+F[7,x]
F[27,x] = F[28,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[26,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[33,x]+F[34,x]
F[33,x] = F[2,x]
F[34,x] = F[35,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[2,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[43,x]
F[39,x] = F[2,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[26,x]
F[43,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_{22}\! \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_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = 0
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{7}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{11}\! \left(x \right)+F_{18}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{37}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{7}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{26}\! \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_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{2}\! \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_{2}\! \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_{43}\! \left(x \right)
F_{39}\! \left(x \right) = F_{2}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \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_{26}\! \left(x \right)
F_{43}\! \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_22(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_6(x))
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_11(x))
Eq(F_14(x), F_15(x) + F_16(x) + F_19(x))
Eq(F_15(x), 0)
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_18(x) + F_7(x))
Eq(F_18(x), F_16(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_11(x) + F_18(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_15(x) + F_24(x) + F_37(x))
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_30(x))
Eq(F_26(x), F_27(x) + F_7(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_26(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) + F_34(x))
Eq(F_33(x), F_2(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_2(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_43(x))
Eq(F_39(x), F_2(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_26(x))
Eq(F_43(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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 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