0132_0231_0312_1023_1302
Counting sequence:
1, 1, 2, 6, 19, 58, 175, 527, 1585, 4762, 14298, 42915, 128782, 386411, 1159347, 3478241, 10435074, 31305838, 93918595, 281757682, 845276375, 2535834967, 7607515153, 22822563450, 68467721922, 205403221171, 616209760742, 1848629452851, 5545888657979, 16637666499393, 49913000420290, 149739002879062, 449217011476915, 1347651039414122, 4042953126987583, 12128859396309535, 36386578215860337, 109159734694842906, 327479204167467562, 982437612647950211, 2947312838199268734, 8841938515046033723, 26525815545924685635, 79577446639154416417, 238732339919885611330, 716197019763907783102, 2148591059299183244963, 6445773177910640939170, 19337319533754896279527, 58011958601305004453991, 174035875803985762335057, 522107627412081442797946, 1566322882236462206621714, 4698968646709768969501203, 14096905940129977885340630, 42290717820391111138287747, 126872153461175399752193995, 380616460383529825425814657, 1141849381150595839760844418, 3425548143451798686417432230, 10276644430355415656207926867, 30829933291066281358883542426, 92489799873198904427349419727, 277469399619596819190141712431, 832408198858790643426173013169, 2497224596576372256432620130458, 7491673789729117341658408150650, 22475021369187353029397966755427, 67425064107562060850832938204910, 202275192322686185645714695947787, 606825576968058562365359554874323, 1820476730904175696621932755290465, 5461430192712527106582506861508482, 16384290578137581349083299152163086, 49152871734412744098730600086794947, 147458615203238232386534135548995666, 442375845609714697318142153161540599, 1327127536829144092232644319855423863, 3981382610487432277186171269081932945, 11944147831462296832415312312420873082, 35832443494386890498749513498467908450, 107497330483160671498887133422154890867, 322491991449482014501291808063936788678, 967475974348446043512001223421208721875, 2902427923045338130550263453851701926747, 8707283769136014391675814537299331062721, 26121851307408043175071357979027766347842, 78365553922224129525291138279186695841526, 235096661766672388576008653330381333243379, 705289985300017165728263287001812867529418, 2115869955900051497185206340876058489266015, 6347609867700154491556349893851567619073087, 19042829603100463474670332271141536140890865, 57128488809301390424013247601245502725418522, 171385466427904171272043692661014855649350858, 514156399283712513816138009499367200875168835, 1542469197851137541448426192461289417309389662, 4627407593553412624345299923651199594841914331, 13882222780660237873035937231041396289596466787, 41646668341980713619107877430995640350700985761, 124940005025942140857323747654909357853769149634
Generating function in Maple syntax:
(2*x-1)*(x^3+2*x-1)/(3*x-1)/(x^3-x^2+2*x-1)
Generating function in latex syntax:
\frac{\left(2 x -1\right) \left(x^{3}+2 x -1\right)}{\left(3 x -1\right) \left(x^{3}-x^{2}+2 x -1\right)}
Generating function in sympy syntax:
(2*x - 1)*(x**3 + 2*x - 1)/((3*x - 1)*(x**3 - x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(3*x-1)*(x^3-x^2+2*x-1)*F(x)-(2*x-1)*(x^3+2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(3 x -1\right) \left(x^{3}-x^{2}+2 x -1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x^{3}+2 x -1\right) = 0
Explicit closed form in Maple syntax:
-1071/2783*Sum(_alpha^(-n+2),_alpha = RootOf(3*_Z^4-4*_Z^3+7*_Z^2-5*_Z+1))+1412/2783*Sum(_alpha^(-n+1),_alpha = RootOf(3*_Z^4-4*_Z^3+7*_Z^2-5*_Z+1))-1966/2783*Sum(_alpha^(-n),_alpha = RootOf(3*_Z^4-4*_Z^3+7*_Z^2-5*_Z+1))+763/2783*Sum(_alpha^(-n-1),_alpha = RootOf(3*_Z^4-4*_Z^3+7*_Z^2-5*_Z+1))+piecewise(n = 0,2/3,0)
Explicit closed form in latex syntax:
-\frac{1071 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{2783}+\frac{1412 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{2783}-\frac{1966 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{2783}+\frac{763 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(3 \textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{2783}+\left(\left\{\begin{array}{cc}\frac{2}{3} & 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(n+4) = -3*a(n)+4*a(n+1)-7*a(n+2)+5*a(n+3), n >= 5
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(n +4\right) = -3 a \! \left(n \right)+4 a \! \left(n +1\right)-7 a \! \left(n +2\right)+5 a \! \left(n +3\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18584/
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[12,x]*F[4,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,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[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[12,x]+F[20,x]
F[20,x] = F[18,x]
F[21,x] = F[2,x]+F[22,x]
F[22,x] = F[23,x]+F[24,x]+F[56,x]
F[23,x] = 0
F[24,x] = F[12,x]*F[25,x]
F[25,x] = F[26,x]+F[34,x]
F[26,x] = F[14,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[12,x]*F[29,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[12,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[12,x]*F[30,x]
F[33,x] = F[20,x]
F[34,x] = F[35,x]+F[38,x]
F[35,x] = F[23,x]+F[24,x]+F[36,x]
F[36,x] = F[12,x]*F[37,x]
F[37,x] = F[2,x]+F[35,x]
F[38,x] = F[39,x]
F[39,x] = F[12,x]*F[40,x]
F[40,x] = F[41,x]+F[52,x]
F[41,x] = F[42,x]+F[50,x]
F[42,x] = F[23,x]+F[43,x]+F[49,x]
F[43,x] = F[12,x]*F[44,x]
F[44,x] = F[45,x]+F[47,x]
F[45,x] = F[12,x]+F[46,x]
F[46,x] = F[28,x]
F[47,x] = F[42,x]+F[48,x]
F[48,x] = F[39,x]
F[49,x] = F[12,x]*F[2,x]
F[50,x] = F[51,x]
F[51,x] = F[12,x]*F[41,x]
F[52,x] = F[53,x]
F[53,x] = F[54,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[42,x]+F[53,x]
F[56,x] = F[12,x]*F[57,x]
F[57,x] = F[58,x]+F[61,x]
F[58,x] = F[2,x]+F[59,x]
F[59,x] = F[23,x]+F[43,x]+F[60,x]
F[60,x] = F[12,x]*F[58,x]
F[61,x] = F[35,x]+F[62,x]
F[62,x] = F[54,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \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_{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_{17}\! \left(x \right)
F_{14}\! \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_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{2}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{56}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{12}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{34}\! \left(x \right)
F_{26}\! \left(x \right) = F_{14}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \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_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{12}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{20}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{38}\! \left(x \right)
F_{35}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{12}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{52}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)+F_{50}\! \left(x \right)
F_{42}\! \left(x \right) = F_{23}\! \left(x \right)+F_{43}\! \left(x \right)+F_{49}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right) F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)+F_{47}\! \left(x \right)
F_{45}\! \left(x \right) = F_{12}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{28}\! \left(x \right)
F_{47}\! \left(x \right) = F_{42}\! \left(x \right)+F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{39}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right) F_{2}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{12}\! \left(x \right) F_{41}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{42}\! \left(x \right)+F_{53}\! \left(x \right)
F_{56}\! \left(x \right) = F_{12}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)+F_{61}\! \left(x \right)
F_{58}\! \left(x \right) = F_{2}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{23}\! \left(x \right)+F_{43}\! \left(x \right)+F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{12}\! \left(x \right) F_{58}\! \left(x \right)
F_{61}\! \left(x \right) = F_{35}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{54}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_21(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_12(x)*F_8(x))
Eq(F_8(x), F_13(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) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_12(x) + F_20(x))
Eq(F_20(x), F_18(x))
Eq(F_21(x), F_2(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_56(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_12(x)*F_25(x))
Eq(F_25(x), F_26(x) + F_34(x))
Eq(F_26(x), F_14(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_12(x)*F_29(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_12(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_12(x)*F_30(x))
Eq(F_33(x), F_20(x))
Eq(F_34(x), F_35(x) + F_38(x))
Eq(F_35(x), F_23(x) + F_24(x) + F_36(x))
Eq(F_36(x), F_12(x)*F_37(x))
Eq(F_37(x), F_2(x) + F_35(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_12(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_52(x))
Eq(F_41(x), F_42(x) + F_50(x))
Eq(F_42(x), F_23(x) + F_43(x) + F_49(x))
Eq(F_43(x), F_12(x)*F_44(x))
Eq(F_44(x), F_45(x) + F_47(x))
Eq(F_45(x), F_12(x) + F_46(x))
Eq(F_46(x), F_28(x))
Eq(F_47(x), F_42(x) + F_48(x))
Eq(F_48(x), F_39(x))
Eq(F_49(x), F_12(x)*F_2(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_12(x)*F_41(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_54(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_42(x) + F_53(x))
Eq(F_56(x), F_12(x)*F_57(x))
Eq(F_57(x), F_58(x) + F_61(x))
Eq(F_58(x), F_2(x) + F_59(x))
Eq(F_59(x), F_23(x) + F_43(x) + F_60(x))
Eq(F_60(x), F_12(x)*F_58(x))
Eq(F_61(x), F_35(x) + F_62(x))
Eq(F_62(x), F_54(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, 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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 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, 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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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