0132_0213_0231_0312_1023_1230_3012
Counting sequence:
1, 1, 2, 6, 17, 47, 140, 442, 1451, 4896, 16851, 58875, 208156, 743133, 2674817, 9695455, 35358657, 129646387, 477641284, 1767267371, 6564127185, 24466277966, 91482581351, 343059642307, 1289904193692, 4861946476477, 18367353193545, 69533551112422, 263747952068171, 1002242217165597, 3814986502924344, 14544636040573178, 55534064879226507, 212336130415767688, 812944042155433651, 3116285494916528727, 11959798385875383844, 45950804324645900181, 176733862787045789569, 680425371730039046376, 2622127042276594442975, 10113918591638063714161, 39044429911904711873536, 150853479205085785155137, 583300119592997394496773, 2257117854077249208156890, 8740328711533175226358223, 33868773757191049857644563, 131327898242169370285518876, 509552245179617145833350621, 1978261657756160666210043481, 7684785670514316405595827230, 29869166945772625983093697611, 116157871455782434303870137053, 451959718027953471533877080696, 1759414616608818871132063738417, 6852456927844873497775509898029, 26700952856774851904610656208826, 104088460289122304034089605541959, 405944995127576985731600165393153, 1583850964596120042688320787794816, 6182127958584855650489585577998297, 24139737743045626825715511285811193, 94295850558771979787941942416699967, 368479169875816659479019652923404673, 1440418573150919668872507061924042915, 5632681584560312734993943483739180388, 22033725021956517463358597559627162803, 86218923998960285726185713387161356641, 337485502510215975556783911124089085694, 1321422108420282270489942367582720253735, 5175569924646105559418940502056586886479, 20276890389709399862928999066708652904964, 79463489365077377841208238438864801933893, 311496878311103321137536292823778678955897, 1221395654430378811828760724119447208769070, 4790408930363303911328386211811319083931227, 18793142726809884575211361284615484894134797, 73745243611532458459690151863591723563127064, 289450081175264899454283846043963101289068451, 1136359577947336271931632877028084185016081625, 4462290049988320482463241297544022245872798646, 17526585015616776834735140517976961427117845871, 68854441132780194707888052034767841996079961597, 270557451039395118028642463289329067064487647528, 1063353702922273835973036658043736154220014527105, 4180080073556524734514695828171327654569478803993, 16435314834665426797069144960763566035005229007198, 64633260585762914370496637486147281550459901362931, 254224158304000796523953440778843427065963376740789, 1000134600800354781929399250536544744429655460766920, 3935312233584004685417853572763354169820642055554109, 15487357822491889407128326963778350772127735874182029, 60960876535340415751462563580829661092130144029314738, 239993345518077005168915776623476742746555047356452767, 944973797977428207852605870454939628777665393225003955, 3721443204405954385563870541379246711390215552237017372, 14657929356129575437016877846657032845334098440748178077, 57743358069601357782187700608042856469322583969463357049, 227508830794229349661819540395688854174964678436096216366, 896519947090131496687170070074100632775062369718007824395
Generating function in Maple syntax:
1/2*((-x^2-x+1)*(1-4*x)^(1/2)-2*x^6-4*x^5-2*x^4+x^2+x-1)/(x^2+x-1)/x
Generating function in latex syntax:
\frac{\left(-x^{2}-x +1\right) \sqrt{1-4 x}-2 x^{6}-4 x^{5}-2 x^{4}+x^{2}+x -1}{2 \left(x^{2}+x -1\right) x}
Generating function in sympy syntax:
(-x**6 - 2*x**5 - x**4 + x**2/2 + x/2 + sqrt(1 - 4*x)*(-x**2 - x + 1)/2 - 1/2)/(x*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
x*(x^2+x-1)^2*F(x)^2+(x^2+x-1)*(2*x^6+4*x^5+2*x^4-x^2-x+1)*F(x)+x^11+4*x^10+6*x^9+4*x^8-3*x^6-2*x^5+2*x^4+3*x^3-x^2-2*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{2}+x -1\right)^{2} F \! \left(x \right)^{2}+\left(x^{2}+x -1\right) \left(2 x^{6}+4 x^{5}+2 x^{4}-x^{2}-x +1\right) F \! \left(x \right)+x^{11}+4 x^{10}+6 x^{9}+4 x^{8}-3 x^{6}-2 x^{5}+2 x^{4}+3 x^{3}-x^{2}-2 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 17
a(5) = 47
a(6) = 140
a(7) = 442
a(8) = 1451
a(n+5) = 2*(1+2*n)/(n+6)*a(n)+(10+7*n)/(n+6)*a(1+n)-2*(8+3*n)/(n+6)*a(n+2)-(24+7*n)/(n+6)*a(n+3)+2*(14+3*n)/(n+6)*a(n+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) = 17
a \! \left(5\right) = 47
a \! \left(6\right) = 140
a \! \left(7\right) = 442
a \! \left(8\right) = 1451
a \! \left(n +5\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +6}+\frac{\left(10+7 n \right) a \! \left(1+n \right)}{n +6}-\frac{2 \left(8+3 n \right) a \! \left(n +2\right)}{n +6}-\frac{\left(24+7 n \right) a \! \left(n +3\right)}{n +6}+\frac{2 \left(14+3 n \right) a \! \left(n +4\right)}{n +6}, \quad n \geq 9
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/22775/
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[8,x]
F[4,x] = F[24,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[8,x]
F[13,x] = F[14,x]+F[21,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[8,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[1,x]+F[8,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[14,x]*F[8,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]*F[8,x]
F[23,x] = F[14,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x,1]
F[26,x,k[0]] = F[27,x,k[0]]
F[27,x,k[0]] = F[28,x,k[0]]
F[28,x,k[0]] = F[29,x,k[0]]*F[38,x,k[0]]*F[8,x]
F[29,x,k[0]] = F[30,x,k[0]]+F[41,x,k[0]]
F[30,x,k[0]] = F[31,x]+F[34,x,k[0]]
F[31,x] = F[1,x]+F[32,x]
F[32,x] = F[33,x]
F[33,x] = F[31,x]*F[8,x]
F[34,x,k[0]] = F[35,x,k[0]]+F[39,x,k[0]]
F[35,x,k[0]] = F[36,x,k[0]]
F[36,x,k[0]] = F[37,x,k[0]]*F[38,x,k[0]]
F[37,x,k[0]] = k[0]*x
F[38,x,k[0]] = F[1,x]+F[35,x,k[0]]
F[39,x,k[0]] = F[40,x,k[0]]
F[40,x,k[0]] = F[34,x,k[0]]*F[8,x]
F[41,x,k[0]] = (F[27,x,k[0]]*k[0]-F[27,x,1])/(-1+k[0])
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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{24}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{13}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{21}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{14}\! \left(x \right) F_{8}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{8}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x , 1\right)
F_{26}\! \left(x , y\right) = F_{27}\! \left(x , y\right)
F_{27}\! \left(x , y\right) = F_{28}\! \left(x , y\right)
F_{28}\! \left(x , y\right) = F_{29}\! \left(x , y\right) F_{38}\! \left(x , y\right) F_{8}\! \left(x \right)
F_{29}\! \left(x , y\right) = F_{30}\! \left(x , y\right)+F_{41}\! \left(x , y\right)
F_{30}\! \left(x , y\right) = F_{31}\! \left(x \right)+F_{34}\! \left(x , y\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_{31}\! \left(x \right) F_{8}\! \left(x \right)
F_{34}\! \left(x , y\right) = F_{35}\! \left(x , y\right)+F_{39}\! \left(x , y\right)
F_{35}\! \left(x , y\right) = F_{36}\! \left(x , y\right)
F_{36}\! \left(x , y\right) = F_{37}\! \left(x , y\right) F_{38}\! \left(x , y\right)
F_{37}\! \left(x , y\right) = y x
F_{38}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{35}\! \left(x , y\right)
F_{39}\! \left(x , y\right) = F_{40}\! \left(x , y\right)
F_{40}\! \left(x , y\right) = F_{34}\! \left(x , y\right) F_{8}\! \left(x \right)
F_{41}\! \left(x , y\right) = \frac{y F_{27}\! \left(x , y\right)-F_{27}\! \left(x , 1\right)}{-1+y}
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_8(x))
Eq(F_4(x), F_24(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_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_8(x))
Eq(F_13(x), F_14(x) + F_21(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_8(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_1(x) + F_8(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_14(x)*F_8(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x)*F_8(x))
Eq(F_23(x), F_14(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x, 1))
Eq(F_26(x, k_0), F_27(x, k_0))
Eq(F_27(x, k_0), F_28(x, k_0))
Eq(F_28(x, k_0), F_29(x, k_0)*F_38(x, k_0)*F_8(x))
Eq(F_29(x, k_0), F_30(x, k_0) + F_41(x, k_0))
Eq(F_30(x, k_0), F_31(x) + F_34(x, k_0))
Eq(F_31(x), F_1(x) + F_32(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_31(x)*F_8(x))
Eq(F_34(x, k_0), F_35(x, k_0) + F_39(x, k_0))
Eq(F_35(x, k_0), F_36(x, k_0))
Eq(F_36(x, k_0), F_37(x, k_0)*F_38(x, k_0))
Eq(F_37(x, k_0), k_0*x)
Eq(F_38(x, k_0), F_1(x) + F_35(x, k_0))
Eq(F_39(x, k_0), F_40(x, k_0))
Eq(F_40(x, k_0), F_34(x, k_0)*F_8(x))
Eq(F_41(x, k_0), (-k_0*F_27(x, k_0) + F_27(x, 1))/(1 - k_0))
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.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "point_placements_tracked_fusion", "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"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": [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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 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, 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": [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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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, 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": [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, 2, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, 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