0132_0213_0231_3021
Counting sequence:
1, 1, 2, 6, 20, 68, 232, 796, 2755, 9630, 33990, 121048, 434546, 1571048, 5715649, 20910232, 76879100, 283916568, 1052724781, 3917560484, 14626836447, 54776510470, 205701953393, 774437552820, 2922491145133, 11052566116798, 41884039255729, 159018957724680, 604796970224127, 2303994328039054, 8790623073685094, 33587972705959976, 128509968720971083, 492317109642808750, 1888326736535664114, 7251118347111472808, 27874191080732177258, 107261729076601711288, 413152313106517016966, 1592860464440198139012, 6146504268018221096492, 23737974545291692137932, 91750297639569049088722, 354898448382281903504132, 1373785668386064754824582, 5321554575446397888787792, 20627660133752926054903712, 80009462881326354137183532, 310528479725054994300086554, 1205923147470048186285742424, 4685823955266209969407854720, 18217595058235323274265881564, 70864051826139234942035759168, 275792457053945525048435562780, 1073871490965112400795130436912, 4183388950692629610701650179668, 16304333435365298437167012898980, 63572597295350018408490150283492, 247983358173386049191874531265544, 967731064948286070046024482490932, 3777988028938641498457923576387040, 14754846143331701839090004939572796, 57646338761959080947831209763693517, 225302846435721574303584415035494752, 880875903168852051827557367891210876, 3445175075379746953306648897483431912, 13478823774666218083189940606108178373, 52751235207979392877026142140400604084, 206513857375152065825322438432494864895, 808719377367286315074567822761999376406, 3167925086924103673786715955142912097717, 12413002155485538983567180000063159151828, 48651949354798013951389599492593252359189, 190740052693413746646394650965537726482190, 747994950359926357890317870582119897040901, 2934050263045196013926018023967540971369432, 11511864381598243733676850165436278775108123, 45178336668716169441736064463437252621635014, 177344856844888908135656489797591268492605915, 696318888117569526621965689534850569950495316, 2734616906873694969842042051117212168845565747, 10741901543484612283199710955301071592464531298, 42204610213818107610612045489035373925240057599, 165855719779760793355412803890641111917189365400, 651914595996145865387949129611741772315575647721, 2562940829574661762830938722663927793973483587042, 10077947258944831857631579842399362584964658019123, 39635950683817634378445571917454096288250316186004, 155915147173880688949595331241123985939080933618665, 613433326414199703899906866452586012443177108667406, 2413930195485197792792238377777007418244570586951887, 9500766469076308155874053308395263366444393405733768, 37399650176630181544777656911632832307888223213099649, 147248139217638949616067349375433376117549753783600330, 579834345209783483367185642656167457100797279901162861, 2283644080935158002352514377339527321368789820558618292, 8995427744435166255805447790187479797226610594223440373, 35439105041513419000378944767998723320202786348775518454, 139640000918864320231608707258106084433708593436103223825, 550302894264267325741506885957295706571462090021289089856, 2168988916631907119587610752796818928663752546399696146407
Generating function in Maple syntax:
1/2*((-x^4+6*x^3-7*x^2+4*x-1)*(1-4*x)^(1/2)+7*x^4-18*x^3+15*x^2-6*x+1)/x^2/(x-1)^2
Generating function in latex syntax:
\frac{\left(-x^{4}+6 x^{3}-7 x^{2}+4 x -1\right) \sqrt{1-4 x}+7 x^{4}-18 x^{3}+15 x^{2}-6 x +1}{2 x^{2} \left(x -1\right)^{2}}
Generating function in sympy syntax:
(7*x**4/2 - 9*x**3 + 15*x**2/2 - 3*x + sqrt(1 - 4*x)*(-x**4 + 6*x**3 - 7*x**2 + 4*x - 1)/2 + 1/2)/(x**2*(x - 1)**2)
Implicit equation for the generating function in Maple syntax:
x^2*(x-1)^4*F(x)^2-(7*x^4-18*x^3+15*x^2-6*x+1)*(x-1)^2*F(x)+x^7-10*x^5+29*x^4-34*x^3+21*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
x^{2} \left(x -1\right)^{4} F \! \left(x \right)^{2}-\left(7 x^{4}-18 x^{3}+15 x^{2}-6 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{7}-10 x^{5}+29 x^{4}-34 x^{3}+21 x^{2}-7 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 68
a(6) = 232
a(7) = 796
a(n+6) = -2*(2*n-1)/(8+n)*a(n)-(59*n+128)/(8+n)*a(2+n)+(31+29*n)/(8+n)*a(n+1)+(205+57*n)/(8+n)*a(n+3)-(31*n+158)/(8+n)*a(n+4)+(59+9*n)/(8+n)*a(n+5)+4/(8+n), 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) = 20
a \! \left(5\right) = 68
a \! \left(6\right) = 232
a \! \left(7\right) = 796
a \! \left(n +6\right) = -\frac{2 \left(2 n -1\right) a \! \left(n \right)}{8+n}-\frac{\left(59 n +128\right) a \! \left(2+n \right)}{8+n}+\frac{\left(31+29 n \right) a \! \left(n +1\right)}{8+n}+\frac{\left(205+57 n \right) a \! \left(n +3\right)}{8+n}-\frac{\left(31 n +158\right) a \! \left(n +4\right)}{8+n}+\frac{\left(59+9 n \right) a \! \left(n +5\right)}{8+n}+\frac{4}{8+n}, \quad n \geq 8
Specification 1
Strategy pack name: row_placements
Tree: http://permpal.com/tree/21173/
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[9,x]
F[3,x] = F[1,x]+F[15,x]+F[4,x]
F[4,x] = F[5,x]
F[5,x] = F[10,x]*F[6,x]*F[9,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[6,x]^2*F[9,x]
F[9,x] = x
F[10,x] = F[1,x]+F[11,x]+F[4,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[6,x]*F[9,x]
F[13,x] = F[1,x]+F[11,x]+F[14,x]
F[14,x] = F[13,x]*F[9,x]
F[15,x] = F[16,x]*F[9,x]
F[16,x] = F[1,x]+F[15,x]+F[17,x]+F[19,x]
F[17,x] = F[18,x]*F[9,x]
F[18,x] = F[1,x]+F[11,x]+F[17,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[6,x]^2*F[13,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_{9}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{10}\! \left(x \right) F_{6}\! \left(x \right) F_{9}\! \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_{6} \left(x \right)^{2} F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{4}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{6}\! \left(x \right) F_{9}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{9}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)+F_{17}\! \left(x \right)+F_{19}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{9}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{17}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{6} \left(x \right)^{2} F_{13}\! \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)*F_9(x))
Eq(F_3(x), F_1(x) + F_15(x) + F_4(x))
Eq(F_4(x), F_5(x))
Eq(F_5(x), F_10(x)*F_6(x)*F_9(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_6(x)**2*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_1(x) + F_11(x) + F_4(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_6(x)*F_9(x))
Eq(F_13(x), F_1(x) + F_11(x) + F_14(x))
Eq(F_14(x), F_13(x)*F_9(x))
Eq(F_15(x), F_16(x)*F_9(x))
Eq(F_16(x), F_1(x) + F_15(x) + F_17(x) + F_19(x))
Eq(F_17(x), F_18(x)*F_9(x))
Eq(F_18(x), F_1(x) + F_11(x) + F_17(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_13(x)*F_6(x)**2*F_9(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "row_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": [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], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"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], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}], "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": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 1, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 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": [[2, 1]]}, {"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], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"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]]}]]}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"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], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0], "pos": [[3, 1]]}, {"patt": [0, 1], "pos": [[0, 0], [2, 0]]}, {"patt": [0, 1], "pos": [[0, 0], [3, 0]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[2, 0], [2, 0], [3, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": 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