0231_0321_3021
Counting sequence:
1, 1, 2, 6, 21, 79, 309, 1237, 5026, 20626, 85242, 354080, 1476368, 6173634, 25873744, 108628550, 456710589, 1922354351, 8098984433, 34147706833, 144068881455, 608151037123, 2568318694867, 10850577045131, 45856273670841, 193850277807569, 819669810565949, 3466587141136257, 14663715160666221, 62037527980971481, 262497329542451713, 1110830621569613797, 4701289519349672290, 19898799583649216610, 84231153659973540850, 356574677613631895224, 1509581429398659769188, 6391278766119594919158, 27060848306272358920724, 114581651832372928591634, 485184211693559015580650, 2054538498790235604021104, 8700341490234756869254986, 36844379296682978034330188, 156033490006612232138440238, 660807372046699476522875680, 2798603595449522402837756478, 11852676493285936020897679312, 50199476698792791666364671616, 212612567317913316817402210226, 900502658724414829246638161592, 3814053702647430708767305588190, 16154510946695871177432972458228, 68423542728024116462523183757834, 289815493895519256332025549247884, 1227556688670850183090820221029394, 5199541864605389040209060865694312, 22023779530937207123069650942891094, 93287098656571768016904161565849792, 395142757853229527235198569420539022, 1673743632763171842423261591097312592, 7089671414894875132042278489371992726, 30030692165478276626029428765537305352, 127205669174758102984733269605831405766, 538826966026354232410621885281077941981, 2282410533197256989387760142848124820015, 9668067121539154026333139901354253352641, 40953109608072678809110951214499768048929, 173474375870929024216565333733438136772807, 734826663390677953373781679025845731498907, 3112687420316156252258617905426957615850635, 13185210339251172164800959323943591131322547, 55852097236703389605669424430509366059134173, 236587998326831209459544491343579334680303189, 1002182169581444757822373953049617780844102929, 4245230856084095957180971798645407520846319645, 17982769505180703448752402226895661780690036329, 76174992307465121162042765288826409697413534381, 322677579833116181461942032812438413793232301293, 1366865091169907330045394529884039274473615824953, 5790058668375863104250675985435351765808763315335, 24526788873845430758822758685272303551329014739083, 103895994655917837778664515803437048767024983856707, 440105982632282457047903240833050883609147848126491, 1864301146701102362280811691485085673572577380397371, 7897236854682067400315639208976509403230163114965915, 33452959131877097261789198351732720920119849587263351, 141707936693749401603027082128445346608874374698311039, 600280240821341586477022022531334015704468572214330847, 2542811356665528183291500323166463741177494507138920727, 10771456725338829505867694785450537091154219740465735355, 45628367271830126557561589097015254439423303193231679947, 193283860451888917966889822030831635489035394239622724943, 818759619876040370291100916279441757991248023924450296319, 3468305864043473280100009308077360605386759390010692170219, 14691918265012774865617549751547603508016832016150170016795, 62235723931164959785752216341255112193435893281232165993449, 263633810563887132821972715957757156830435928493985038164641, 1116767016900047590877152277697423500314642772678636324935613, 4730686331875366449811178930058825915781532475165400595742593, 20039451136441959612682618878970082638013058611364716465898325
Generating function in Maple syntax:
(-5*(-4*x+1)^(1/2)*x-2*x^2+(-4*x+1)^(1/2)+13*x-3)/(2*x^2+8*x-2)
Generating function in latex syntax:
\frac{-5 \sqrt{-4 x +1}\, x -2 x^{2}+\sqrt{-4 x +1}+13 x -3}{2 x^{2}+8 x -2}
Generating function in sympy syntax:
(-2*x**2 - 5*x*sqrt(1 - 4*x) + 13*x + sqrt(1 - 4*x) - 3)/(2*x**2 + 8*x - 2)
Implicit equation for the generating function in Maple syntax:
(x^2+4*x-1)*F(x)^2+(2*x^2-13*x+3)*F(x)+x^2+8*x-2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}+4 x -1\right) F \! \left(x \right)^{2}+\left(2 x^{2}-13 x +3\right) F \! \left(x \right)+x^{2}+8 x -2 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(n+4) = 10*(1+2*n)/(n+4)*a(n)+(20+71*n)/(n+4)*a(n+1)-(86+55*n)/(n+4)*a(n+2)+(36+13*n)/(n+4)*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) = 21
a \! \left(n +4\right) = \frac{10 \left(1+2 n \right) a \! \left(n \right)}{n +4}+\frac{\left(20+71 n \right) a \! \left(n +1\right)}{n +4}-\frac{\left(86+55 n \right) a \! \left(n +2\right)}{n +4}+\frac{\left(36+13 n \right) a \! \left(n +3\right)}{n +4}, \quad n \geq 5
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/351/
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[4,x]
F[3,x] = x
F[4,x] = F[5,x]+F[8,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[5,x]^2*F[3,x]
F[8,x] = F[9,x]
F[9,x] = F[10,x]*F[3,x]
F[10,x] = F[11,x]+F[23,x]+F[4,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[3,x]
F[13,x] = F[10,x]+F[14,x]+F[21,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[3,x]*F[4,x]*F[5,x]
F[16,x] = F[1,x]+F[17,x]+F[19,x]
F[17,x] = F[18,x]
F[18,x] = F[16,x]*F[3,x]*F[5,x]
F[19,x] = F[20,x]
F[20,x] = F[16,x]*F[3,x]*F[5,x]
F[21,x] = F[22,x]
F[22,x] = F[13,x]*F[3,x]*F[5,x]
F[23,x] = F[10,x]*F[3,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_{4}\! \left(x \right)
F_{3}\! \left(x \right) = x
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)+F_{8}\! \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_{5} \left(x \right)^{2} F_{3}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{3}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{23}\! \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_{3}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \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_{3}\! \left(x \right) F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)+F_{19}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right) F_{3}\! \left(x \right) F_{5}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right) F_{3}\! \left(x \right) F_{5}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{13}\! \left(x \right) F_{3}\! \left(x \right) F_{5}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10}\! \left(x \right) F_{3}\! \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_4(x))
Eq(F_3(x), x)
Eq(F_4(x), F_5(x) + F_8(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_3(x)*F_5(x)**2)
Eq(F_8(x), F_9(x))
Eq(F_9(x), F_10(x)*F_3(x))
Eq(F_10(x), F_11(x) + F_23(x) + F_4(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_3(x))
Eq(F_13(x), F_10(x) + F_14(x) + F_21(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_3(x)*F_4(x)*F_5(x))
Eq(F_16(x), F_1(x) + F_17(x) + F_19(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_16(x)*F_3(x)*F_5(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_16(x)*F_3(x)*F_5(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_13(x)*F_3(x)*F_5(x))
Eq(F_23(x), F_10(x)*F_3(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "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_and_col_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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 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, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[1, 2], [1, 0], [1, 2], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 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": 2, "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], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 1], [0, 0]]}], "requirements": []}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[1, 2], [1, 0], [1, 2], [1, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 1]], [[1, 0], [1, 2]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0], "pos": [[2, 2]]}, {"patt": [0, 1], "pos": [[1, 2], [1, 2]]}, {"patt": [1, 0], "pos": [[0, 1], [2, 1]]}, {"patt": [1, 0], "pos": [[1, 2], [1, 2]]}, {"patt": [1, 0], "pos": [[2, 1], [2, 1]]}, {"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": [[0, 0], [0, 1], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [2, 1], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 1], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 1], [0, 1], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 0], [0, 1], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 2]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": 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