0132_0213_0231_0312_1302_3021
Counting sequence:
1, 1, 2, 6, 18, 54, 165, 520, 1692, 5657, 19322, 67101, 236106, 839681, 3012848, 10892268, 39636160, 145058164, 533557635, 1971377503, 7313184388, 27228310290, 101710487299, 381078871345, 1431718463142, 5392586519890, 20358599785011, 77025442349189, 292003794593672, 1109048055169212, 4219543936611741, 16079960085057656, 61371168521790260, 234565174111841529, 897729952162909570, 3440150470229385017, 13198604536566107698, 50695483432217958477, 194928462113345783372, 750277717079625490077, 2890596810363504271734, 11146824713981530517991, 43022335344414508222938, 166187462154216524979955, 642462105078385284547522, 2485572223683082651948359, 9623221867850277402756166, 37283476676111669170333633, 144544475142150425499984390, 560743112321394751534427749, 2176670239551324554766384220, 8454289387584116809112664159, 32855456182619578240310492518, 127754003442270283104647621041, 497014868914202276050196792004, 1934567071450214505648859116403, 7533727280704735691411914765714, 29352172625180977183063807927041, 114411008121285631621483607922652, 446155117906011197503136669693083, 1740556377634601167739222371614846, 6793106357460041799387072482645125, 26522904718253386788319060431086632, 103595437720455951890685487462982436, 404782592589188637658447378654261144, 1582193877771041884502648825949280492, 6186563761686611373003388785573183579, 24198400996705249018166275972763065855, 94681883474457813408417504435141264220, 370583434188726393908181388763831173074, 1450908538128237550321703462008218337267, 5682313482766146110573392743649924912549, 22260648658021936986077215829681539296870, 87231716638188795806573916479008098778630, 341925623640877757500414131358804756774519, 1340621815303631193378633543133229525887437, 5257693084627216530035714644487958486275704, 20625074328849966648282710578066063106967160, 80929023493858972134278024611845426767326785, 317628026345332141253752855146556340665248649, 1246913781431237835511271837096510369974811322, 4896148484217969713163665288066034499918889924, 19229625555843958467839410107091876999750704785, 75541012710700871216187771026165230001791320765, 296816626658240060298601159163602276406177600784, 1166500406786797096894462111875313663703750539082, 4585330968676548649259732461290277109680473208439, 18027834267159817690141465030034393812334052249795, 70892700471541305865025820687096724394630289343930, 278832032652369747243054893427506146670702265180284, 1096895214684165960741576118169917466080849826117677, 4315858158877915582568707456646508415556601445440789, 16984274436434436530485945109127508576905358525159220, 66850240563761432323501995186108992886236894783500170, 263168236243730914012263489206475145801381402637551839, 1036184080040594078097929948332245660032112343463822277, 4080485050539291238822086372256823036589486125422569234, 16071507799077697323916600262610487574171957615567268310, 63309660441596130733554034794864118616000289755741789455, 249431034970775162790071940043978271377451731501150092609, 982871616712886854784009674871945836781601831626999512372
Generating function in Maple syntax:
1/2*((2*x^4-3*x^3+4*x^2-3*x+1)*(1-4*x)^(1/2)+2*x^5-6*x^4+5*x^3-4*x^2+3*x-1)/x/(x-1)^3
Generating function in latex syntax:
\frac{\left(2 x^{4}-3 x^{3}+4 x^{2}-3 x +1\right) \sqrt{1-4 x}+2 x^{5}-6 x^{4}+5 x^{3}-4 x^{2}+3 x -1}{2 x \left(x -1\right)^{3}}
Generating function in sympy syntax:
(x**5 - 3*x**4 + 5*x**3/2 - 2*x**2 + 3*x/2 + sqrt(1 - 4*x)*(2*x**4 - 3*x**3 + 4*x**2 - 3*x + 1)/2 - 1/2)/(x*(x - 1)**3)
Implicit equation for the generating function in Maple syntax:
x*(x-1)^6*F(x)^2-(2*x^5-6*x^4+5*x^3-4*x^2+3*x-1)*(x-1)^3*F(x)+x^9-2*x^8+x^7+9*x^6-21*x^5+27*x^4-25*x^3+16*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{6} F \! \left(x \right)^{2}-\left(2 x^{5}-6 x^{4}+5 x^{3}-4 x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{9}-2 x^{8}+x^{7}+9 x^{6}-21 x^{5}+27 x^{4}-25 x^{3}+16 x^{2}-6 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 54
a(6) = 165
a(7) = 520
a(8) = 1692
a(9) = 5657
a(n+6) = -4*(-1+2*n)/(7+n)*a(n)+22*n/(7+n)*a(1+n)-3*(19+11*n)/(7+n)*a(n+2)+(113+35*n)/(7+n)*a(n+3)-(105+23*n)/(7+n)*a(n+4)+2*(23+4*n)/(7+n)*a(n+5)+(n-7)/(7+n), n >= 10
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) = 18
a \! \left(5\right) = 54
a \! \left(6\right) = 165
a \! \left(7\right) = 520
a \! \left(8\right) = 1692
a \! \left(9\right) = 5657
a \! \left(n +6\right) = -\frac{4 \left(-1+2 n \right) a \! \left(n \right)}{7+n}+\frac{22 n a \! \left(1+n \right)}{7+n}-\frac{3 \left(19+11 n \right) a \! \left(n +2\right)}{7+n}+\frac{\left(113+35 n \right) a \! \left(n +3\right)}{7+n}-\frac{\left(105+23 n \right) a \! \left(n +4\right)}{7+n}+\frac{2 \left(23+4 n \right) a \! \left(n +5\right)}{7+n}+\frac{n -7}{7+n}, \quad n \geq 10
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22140/
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[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[12,x]*F[7,x]
F[7,x] = F[13,x]+F[8,x]
F[8,x] = F[9,x]^2*F[0,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[9,x]^2*F[12,x]
F[12,x] = x
F[13,x] = F[14,x]*F[16,x]
F[14,x] = F[15,x]+F[19,x]
F[15,x] = F[16,x]*F[17,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[16,x]
F[19,x] = F[12,x]*F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[10,x]*F[12,x]*F[16,x]*F[9,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_{0}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{12}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{13}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{0}\! \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_{9} \left(x \right)^{2} F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{16}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{12}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{16}\! \left(x \right) F_{9}\! \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_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_12(x)*F_7(x))
Eq(F_7(x), F_13(x) + F_8(x))
Eq(F_8(x), F_0(x)*F_9(x)**2)
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)**2)
Eq(F_12(x), x)
Eq(F_13(x), F_14(x)*F_16(x))
Eq(F_14(x), F_15(x) + F_19(x))
Eq(F_15(x), F_16(x)*F_17(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_16(x))
Eq(F_19(x), F_12(x)*F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_10(x)*F_12(x)*F_16(x)*F_9(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": 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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "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"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [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, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 1], [0, 0], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "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, 2]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [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, 2, 0], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 2], [0, 0], [0, 2]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [0, 2]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"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": 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