0213_0231_1230

Counting sequence:
1, 1, 2, 6, 21, 78, 297, 1143, 4419, 17119, 66386, 257621, 1000407, 3887666, 15119991, 58856167, 229312425, 894263633, 3490636794, 13637575699, 53327459013, 208703945330, 817447047177, 3204204114421, 12568821046236, 49336156718513, 193783005926727, 761604774463568, 2994967479372029, 11783895012282379, 46388108590594260, 182697735039838809, 719874637117325802, 2837704181906385666, 11190615995197720956, 44147720499157334808, 174229398915698404922, 687836948000495816068, 2716395922928980405834, 10730927282346069377334, 42404467986978767933578, 167613954002981830545390, 662715765854724686382992, 2620944156755233329768006, 10368039767717767687795726, 41024127717647498635012032, 162360936174924185828822210, 642716076816621460127496798, 2544769090690334272941546454, 10077796803945361447429952214, 39917875205456561382658915916, 158143239427450258714329269706, 626631484839752124391099297102, 2483414972209008076187600268156, 9843718089637615394686101500246, 39024671824444605101232356232318, 154734492032134309425807923847952, 613621369787692344358807023080118, 2433756297320638576969479330946994, 9654168640414209051504333055905144, 38301154482621721163732179694414686, 151972898263036495698952003008986346, 603081379064033435447736254749034432, 2393533240091955428828456246899452966, 9500689708533764224463629419287108341, 37715629736597562018151216856438600982, 149739593842088351233909697765832385805, 594565303287877655570880136881155772183, 2361069241746109691769562949141514279094, 9376976375915620991565027672777374199849, 37244366971348096451856169046662601560253, 147945187278628406695906484580273349769610, 587736142227150818215526191483984747820827, 2335092848237507743084651539044720650105221, 9278228019628158013321924980414750882336836, 36869227269002237518401198042636065388899547, 146521093001302570579545815861788044123629545, 582334386623529370895050732357535484270408588, 2314621490488061155963638109816145126031097739, 9200721963884653747693934443494822069537305229, 36576097567001231165835235215818461005892114918, 145413774066038822491111269845644925486509790155, 578156826630648055606028226221361642793623445147, 2298883398958384806133746599794024056168408033626, 9141525456688494345818621706582125058573994105831, 36353827189241995629510780746452846626011432422127, 144580811633772993653581260170602426085830646150390, 575042010838229286548107572462617998614702436887643, 2287263758771528218264714690530055673923083051494959, 9098296100957042143362640229774907916781193501450734, 36193487509475633258877157393063700548228581822027643, 143988156163085645429127098598292306692034070467442795, 572860031491061513872307549176163947467705055181371742, 2279266719233479864212134425782114152767714961393192231, 9069140446835037203331373652888342359866047358740352951, 36087845187862212486136570145874580546622485593030543022, 143608163621690726024150812696117560331351255057391766465, 571505197886492798900599718657992316783632847704632735623, 2274488039132007291654458479917396962581769125699694685204, 9052511785885194928498879200584277103161680013195146953489, 36030980027440815627271336601224188190749313361361257494923

Generating function in Maple syntax:
-((-4*x+1)^(1/2)+1)/(-4*x+1)^(1/2)*(-1+3*x)/(2*x^2-6*x+2)

Generating function in latex syntax:
-\frac{\left(\sqrt{-4 x +1}+1\right) \left(-1+3 x \right)}{\sqrt{-4 x +1}\, \left(2 x^{2}-6 x +2\right)}

Generating function in sympy syntax:
(3*x - 1)*(-sqrt(1 - 4*x) - 1)/(sqrt(1 - 4*x)*(2*x**2 - 6*x + 2))

Implicit equation for the generating function in Maple syntax:
(4*x-1)*(x^2-3*x+1)^2*F(x)^2+(4*x-1)*(-1+3*x)*(x^2-3*x+1)*F(x)+x*(-1+3*x)^2 = 0

Implicit equation for the generating function in latex syntax:
\left(4 x -1\right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)^{2}+\left(4 x -1\right) \left(-1+3 x \right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+x \left(-1+3 x \right)^{2} = 0

Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(n+4) = -6*(3+2*n)/(n+4)*a(n)+(74+43*n)/(n+4)*a(n+1)-2*(41+17*n)/(n+4)*a(n+2)+2*(16+5*n)/(n+4)*a(n+3), n >= 4

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(n +4\right) = -\frac{6 \left(3+2 n \right) a \! \left(n \right)}{n +4}+\frac{\left(74+43 n \right) a \! \left(n +1\right)}{n +4}-\frac{2 \left(41+17 n \right) a \! \left(n +2\right)}{n +4}+\frac{2 \left(16+5 n \right) a \! \left(n +3\right)}{n +4}, \quad n \geq 4

Specification 1
Strategy pack name: col_placements_tracked_fusion
Tree: http://permpal.com/tree/308/
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,1]
F[5,x,k[0]] = F[1,x]+F[6,x,k[0]]+F[8,x,k[0]]
F[6,x,k[0]] = F[3,x]*F[7,x,k[0]]
F[7,x,k[0]] = 1/(-1+k[0])*(F[5,x,k[0]]*k[0]-F[5,x,1])
F[8,x,k[0]] = F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]*F[11,x,k[0]]
F[10,x,k[0]] = k[0]*x
F[11,x,k[0]] = F[12,x,k[0]]+F[5,x,k[0]]
F[12,x,k[0]] = F[10,x,k[0]]*F[13,x]*F[21,x,k[0]]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]*F[3,x]
F[15,x] = F[1,x]+F[16,x]+F[20,x]
F[16,x] = F[17,x]
F[17,x] = F[15,x]*F[18,x]*F[3,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[18,x]*F[3,x]
F[20,x] = F[15,x]*F[3,x]
F[21,x,k[0]] = F[22,x,k[0]]+F[25,x,k[0]]
F[22,x,k[0]] = F[1,x]+F[23,x,k[0]]
F[23,x,k[0]] = F[24,x,k[0]]
F[24,x,k[0]] = F[10,x,k[0]]*F[22,x,k[0]]
F[25,x,k[0]] = F[26,x,k[0]]+F[29,x,k[0]]
F[26,x,k[0]] = F[27,x,k[0]]
F[27,x,k[0]] = F[10,x,k[0]]*F[28,x,k[0]]
F[28,x,k[0]] = F[1,x]+F[26,x,k[0]]
F[29,x,k[0]] = F[30,x]+F[31,x,k[0]]+F[33,x,k[0]]
F[30,x] = 0
F[31,x,k[0]] = F[10,x,k[0]]*F[32,x,k[0]]
F[32,x,k[0]] = F[23,x,k[0]]+F[29,x,k[0]]
F[33,x,k[0]] = F[10,x,k[0]]*F[25,x,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_{4}\! \left(x \right)
F_{3}\! \left(x \right) = x
F_{4}\! \left(x \right) = F_{5}\! \left(x , 1\right)
F_{5}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{6}\! \left(x , y\right)+F_{8}\! \left(x , y\right)
F_{6}\! \left(x , y\right) = F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = \frac{y F_{5}\! \left(x , y\right)-F_{5}\! \left(x , 1\right)}{-1+y}
F_{8}\! \left(x , y\right) = F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{11}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = y x
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right)+F_{5}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{13}\! \left(x \right) F_{21}\! \left(x , y\right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right) F_{3}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{3}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{18}\! \left(x \right) F_{3}\! \left(x \right)
F_{20}\! \left(x \right) = F_{15}\! \left(x \right) F_{3}\! \left(x \right)
F_{21}\! \left(x , y\right) = F_{22}\! \left(x , y\right)+F_{25}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{23}\! \left(x , y\right)
F_{23}\! \left(x , y\right) = F_{24}\! \left(x , y\right)
F_{24}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{22}\! \left(x , y\right)
F_{25}\! \left(x , y\right) = F_{26}\! \left(x , y\right)+F_{29}\! \left(x , y\right)
F_{26}\! \left(x , y\right) = F_{27}\! \left(x , y\right)
F_{27}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{28}\! \left(x , y\right)
F_{28}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{26}\! \left(x , y\right)
F_{29}\! \left(x , y\right) = F_{30}\! \left(x \right)+F_{31}\! \left(x , y\right)+F_{33}\! \left(x , y\right)
F_{30}\! \left(x \right) = 0
F_{31}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{32}\! \left(x , y\right)
F_{32}\! \left(x , y\right) = F_{23}\! \left(x , y\right)+F_{29}\! \left(x , y\right)
F_{33}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{25}\! \left(x , y\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, 1))
Eq(F_5(x, k_0), F_1(x) + F_6(x, k_0) + F_8(x, k_0))
Eq(F_6(x, k_0), F_3(x)*F_7(x, k_0))
Eq(F_7(x, k_0), (k_0*F_5(x, k_0) - F_5(x, 1))/(k_0 - 1))
Eq(F_8(x, k_0), F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0)*F_11(x, k_0))
Eq(F_10(x, k_0), k_0*x)
Eq(F_11(x, k_0), F_12(x, k_0) + F_5(x, k_0))
Eq(F_12(x, k_0), F_10(x, k_0)*F_13(x)*F_21(x, k_0))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x)*F_3(x))
Eq(F_15(x), F_1(x) + F_16(x) + F_20(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_15(x)*F_18(x)*F_3(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_18(x)*F_3(x))
Eq(F_20(x), F_15(x)*F_3(x))
Eq(F_21(x, k_0), F_22(x, k_0) + F_25(x, k_0))
Eq(F_22(x, k_0), F_1(x) + F_23(x, k_0))
Eq(F_23(x, k_0), F_24(x, k_0))
Eq(F_24(x, k_0), F_10(x, k_0)*F_22(x, k_0))
Eq(F_25(x, k_0), F_26(x, k_0) + F_29(x, k_0))
Eq(F_26(x, k_0), F_27(x, k_0))
Eq(F_27(x, k_0), F_10(x, k_0)*F_28(x, k_0))
Eq(F_28(x, k_0), F_1(x) + F_26(x, k_0))
Eq(F_29(x, k_0), F_30(x) + F_31(x, k_0) + F_33(x, k_0))
Eq(F_30(x), 0)
Eq(F_31(x, k_0), F_10(x, k_0)*F_32(x, k_0))
Eq(F_32(x, k_0), F_23(x, k_0) + F_29(x, k_0))
Eq(F_33(x, k_0), F_10(x, k_0)*F_25(x, k_0))
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": false, "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.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": "col_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, 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": [1, 2, 3, 0], "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": [1, 0, 2], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 2]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 2], [1, 0], [1, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 0], [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, 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": [1, 2, 3, 0], "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"}}, {"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], "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": [1, 0, 2], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 1]]}, 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