0123_0132_0213_0231_0312_1023_1230_3012
Counting sequence:
1, 1, 2, 6, 16, 46, 139, 441, 1450, 4895, 16850, 58874, 208155, 743132, 2674816, 9695454, 35358656, 129646386, 477641283, 1767267370, 6564127184, 24466277965, 91482581350, 343059642306, 1289904193691, 4861946476476, 18367353193544, 69533551112421, 263747952068170, 1002242217165596, 3814986502924343, 14544636040573177, 55534064879226506, 212336130415767687, 812944042155433650, 3116285494916528726, 11959798385875383843, 45950804324645900180, 176733862787045789568, 680425371730039046375, 2622127042276594442974, 10113918591638063714160, 39044429911904711873535, 150853479205085785155136, 583300119592997394496772, 2257117854077249208156889, 8740328711533175226358222, 33868773757191049857644562, 131327898242169370285518875, 509552245179617145833350620, 1978261657756160666210043480, 7684785670514316405595827229, 29869166945772625983093697610, 116157871455782434303870137052, 451959718027953471533877080695, 1759414616608818871132063738416, 6852456927844873497775509898028, 26700952856774851904610656208825, 104088460289122304034089605541958, 405944995127576985731600165393152, 1583850964596120042688320787794815, 6182127958584855650489585577998296, 24139737743045626825715511285811192, 94295850558771979787941942416699966, 368479169875816659479019652923404672, 1440418573150919668872507061924042914, 5632681584560312734993943483739180387, 22033725021956517463358597559627162802, 86218923998960285726185713387161356640, 337485502510215975556783911124089085693, 1321422108420282270489942367582720253734, 5175569924646105559418940502056586886478, 20276890389709399862928999066708652904963, 79463489365077377841208238438864801933892, 311496878311103321137536292823778678955896, 1221395654430378811828760724119447208769069, 4790408930363303911328386211811319083931226, 18793142726809884575211361284615484894134796, 73745243611532458459690151863591723563127063, 289450081175264899454283846043963101289068450, 1136359577947336271931632877028084185016081624, 4462290049988320482463241297544022245872798645, 17526585015616776834735140517976961427117845870, 68854441132780194707888052034767841996079961596, 270557451039395118028642463289329067064487647527, 1063353702922273835973036658043736154220014527104, 4180080073556524734514695828171327654569478803992, 16435314834665426797069144960763566035005229007197, 64633260585762914370496637486147281550459901362930, 254224158304000796523953440778843427065963376740788, 1000134600800354781929399250536544744429655460766919, 3935312233584004685417853572763354169820642055554108, 15487357822491889407128326963778350772127735874182028, 60960876535340415751462563580829661092130144029314737, 239993345518077005168915776623476742746555047356452766, 944973797977428207852605870454939628777665393225003954, 3721443204405954385563870541379246711390215552237017371, 14657929356129575437016877846657032845334098440748178076, 57743358069601357782187700608042856469322583969463357048, 227508830794229349661819540395688854174964678436096216365, 896519947090131496687170070074100632775062369718007824394
Generating function in Maple syntax:
1/2*((-x^3+2*x-1)*(1-4*x)^(1/2)+2*x^4+x^3-2*x+1)/x/(x^2+x-1)/(x-1)
Generating function in latex syntax:
\frac{\left(-x^{3}+2 x -1\right) \sqrt{1-4 x}+2 x^{4}+x^{3}-2 x +1}{2 x \left(x^{2}+x -1\right) \left(x -1\right)}
Generating function in sympy syntax:
(x**4 + x**3/2 - x + sqrt(1 - 4*x)*(-x**3 + 2*x - 1)/2 + 1/2)/(x*(x - 1)*(x**2 + x - 1))
Implicit equation for the generating function in Maple syntax:
x*(x-1)^2*(x^2+x-1)^2*F(x)^2-(x-1)*(x^2+x-1)*(2*x^4+x^3-2*x+1)*F(x)+x^7+2*x^6-6*x^4+3*x^3+4*x^2-4*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{2} F \! \left(x \right)^{2}-\left(x -1\right) \left(x^{2}+x -1\right) \left(2 x^{4}+x^{3}-2 x +1\right) F \! \left(x \right)+x^{7}+2 x^{6}-6 x^{4}+3 x^{3}+4 x^{2}-4 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 16
a(5) = 46
a(6) = 139
a(7) = 441
a(n+5) = 2*(1+2*n)/(n+6)*a(n)+(10+7*n)/(n+6)*a(n+1)-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)+3*(n-2)/(n+6), n >= 8
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) = 16
a \! \left(5\right) = 46
a \! \left(6\right) = 139
a \! \left(7\right) = 441
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(n +1\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}+\frac{3 n -6}{n +6}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/23243/
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[14,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[11,x]
F[10,x] = F[1,x]+F[8,x]
F[11,x] = F[12,x]+F[6,x]
F[12,x] = F[13,x]
F[13,x] = F[6,x]*F[8,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x,1]
F[16,x,k[0]] = F[17,x,k[0]]
F[17,x,k[0]] = F[18,x,k[0]]
F[18,x,k[0]] = F[19,x,k[0]]*F[21,x,k[0]]*F[8,x]
F[19,x,k[0]] = (F[20,x,k[0]]*k[0]-F[20,x,1])/(-1+k[0])
F[20,x,k[0]] = F[17,x,k[0]]+F[21,x,k[0]]
F[21,x,k[0]] = F[1,x]+F[22,x,k[0]]
F[22,x,k[0]] = F[23,x,k[0]]
F[23,x,k[0]] = F[21,x,k[0]]*F[24,x,k[0]]
F[24,x,k[0]] = k[0]*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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{14}\! \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_{11}\! \left(x \right)
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{6}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{6}\! \left(x \right) F_{8}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x , 1\right)
F_{16}\! \left(x , y\right) = F_{17}\! \left(x , y\right)
F_{17}\! \left(x , y\right) = F_{18}\! \left(x , y\right)
F_{18}\! \left(x , y\right) = F_{19}\! \left(x , y\right) F_{21}\! \left(x , y\right) F_{8}\! \left(x \right)
F_{19}\! \left(x , y\right) = \frac{y F_{20}\! \left(x , y\right)-F_{20}\! \left(x , 1\right)}{-1+y}
F_{20}\! \left(x , y\right) = F_{17}\! \left(x , y\right)+F_{21}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{22}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{23}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = F_{21}\! \left(x , y\right) F_{24}\! \left(x , y\right)
F_{24}\! \left(x , y\right) = y x
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_14(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_11(x))
Eq(F_10(x), F_1(x) + F_8(x))
Eq(F_11(x), F_12(x) + F_6(x))
Eq(F_12(x), F_13(x))
Eq(F_13(x), F_6(x)*F_8(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x, 1))
Eq(F_16(x, k_0), F_17(x, k_0))
Eq(F_17(x, k_0), F_18(x, k_0))
Eq(F_18(x, k_0), F_19(x, k_0)*F_21(x, k_0)*F_8(x))
Eq(F_19(x, k_0), (-k_0*F_20(x, k_0) + F_20(x, 1))/(1 - k_0))
Eq(F_20(x, k_0), F_17(x, k_0) + F_21(x, k_0))
Eq(F_21(x, k_0), F_1(x) + F_22(x, k_0))
Eq(F_22(x, k_0), F_23(x, k_0))
Eq(F_23(x, k_0), F_21(x, k_0)*F_24(x, k_0))
Eq(F_24(x, k_0), k_0*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.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, 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": [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, 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": [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, 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": [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": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[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], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 2]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 0]]}], 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