0231_0312_1203_2013
Counting sequence:
1, 1, 2, 6, 20, 66, 215, 702, 2319, 7772, 26415, 90926, 316573, 1113531, 3952955, 14148867, 51018461, 185182978, 676147042, 2481879320, 9153447623, 33903716995, 126063161075, 470380725447, 1760731074449, 6609938152688, 24880315918762, 93880277514273, 355033639917025, 1345449299354488, 5108600587308699, 19431861528909903, 74037514440501888, 282531325160444942, 1079731377802467068, 4131998879045479896, 15833049987043279309, 60743046470162499174, 233306373941999538189, 897072599606109128562, 3452825473710624544639, 13302852045898902313159, 51299933039991976249023, 198002704340958506913720, 764873364587424707141559, 2957025962217651753762303, 11440689016173441883607088, 44296150487044208554770194, 171625904241176087217746716, 665410891304925464739397665, 2581518013077784475748833527, 10021375065248772184714636881, 38925640659821676340170456975, 151283142417892248495509853499, 588277470648497026969263881014, 2288768864379904967946172471716, 8909253583489943309238209305643, 34697041554342542193995996542139, 135191011443722645647329164208700, 526987520016287547416284374203211, 2055148693993328190912786730199628, 8018076198242642859312492970012244, 31295006327557700479890097468016523, 122194614233127463051655511047442541, 477304734450664430913389702313644286, 1865094482429540452286527049073799307, 7290584623986665252906562000746555910, 28508576402309381633894827432514743163, 111515648600360094045985187062155404441, 436353897238533995436011711245803432349, 1707971061320517280478777233724531472702, 6687392048906934356436309749121479985005, 26191711669389126210334537272774367204305, 102611999546005981161738486331270214839776, 402120390970235228693436807381714443094320, 1576283625618618388931715277722074586195168, 6180587167885832465487551767588970974685381, 24240363077235706413221447969288681929715617, 95095408492735794209134262192027258357158143, 373154054319427337931042018791423171018562857, 1464610770389156080040795989140621968606934641, 5749883995276846433096882050858013371160940136, 22578557833053324858882376659482774337820957720, 88681110039361830611445720692462835458763818441, 348386636085570359264123267779954269519210082589, 1368941918259556822960437294101710622150978646800, 5380213738672488230651598523788481449390834880783, 21149635604868142067872176827098093288960630678644, 83155897745122765943423030286004641319714657179070, 327015287348399243893397031996076990161484307721109, 1286251746462146115693699252293984317456280474460058, 5060168480774699430774456642633100945461155638919686, 19910550626953799456940010157609648308720533082522774, 78357240360915244174178053583398734118855561753949212, 308425936717294159488617823612333603586126165067191071, 1214218748782370799090730265042439640437367214691292132, 4780966422335448053070172177064177219991915855268249627, 18828063457491388635262748109900263409131918360120318138, 74159252705581937550243378762969323718709547422424996461, 292141559923153531494059546818611652256761644513077406715, 1151034311503088911815457150066371272950056320016914779003
Generating function in Maple syntax:
-4*(x-1/2)*(x-1)^2*((x-1/2)*(x-1)^2*(1-4*x)^(1/2)-2*x^3+5/2*x^2-2*x+1/2)/(8*x^7-34*x^6+72*x^5-80*x^4+50*x^3-16*x^2+2*x)
Generating function in latex syntax:
-\frac{4 \left(x -\frac{1}{2}\right) \left(x -1\right)^{2} \left(\left(x -\frac{1}{2}\right) \left(x -1\right)^{2} \sqrt{1-4 x}-2 x^{3}+\frac{5 x^{2}}{2}-2 x +\frac{1}{2}\right)}{8 x^{7}-34 x^{6}+72 x^{5}-80 x^{4}+50 x^{3}-16 x^{2}+2 x}
Generating function in sympy syntax:
(2 - 4*x)*(x - 1)**2*(-2*x**3 + 5*x**2/2 - 2*x + sqrt(1 - 4*x)*(x - 1)**2*(x - 1/2) + 1/2)/(8*x**7 - 34*x**6 + 72*x**5 - 80*x**4 + 50*x**3 - 16*x**2 + 2*x)
Implicit equation for the generating function in Maple syntax:
x*(4*x^6-17*x^5+36*x^4-40*x^3+25*x^2-8*x+1)*F(x)^2-(2*x-1)*(4*x^3-5*x^2+4*x-1)*(x-1)^2*F(x)+(2*x-1)^2*(x-1)^4 = 0
Implicit equation for the generating function in latex syntax:
x \left(4 x^{6}-17 x^{5}+36 x^{4}-40 x^{3}+25 x^{2}-8 x +1\right) F \! \left(x \right)^{2}-\left(2 x -1\right) \left(4 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(2 x -1\right)^{2} \left(x -1\right)^{4} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 66
a(6) = 215
a(7) = 702
a(8) = 2319
a(n+9) = 16*(1+2*n)/(n+10)*a(n)-4*(79+48*n)/(n+10)*a(1+n)+2*(705+277*n)/(n+10)*a(n+2)-(3333+947*n)/(n+10)*a(n+3)+(4658+1029*n)/(n+10)*a(n+4)-2*(2031+365*n)/(n+10)*a(n+5)+5*(445+67*n)/(n+10)*a(n+6)-(736+95*n)/(n+10)*a(n+7)+(133+15*n)/(n+10)*a(n+8), n >= 9
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) = 66
a \! \left(6\right) = 215
a \! \left(7\right) = 702
a \! \left(8\right) = 2319
a \! \left(n +9\right) = \frac{16 \left(1+2 n \right) a \! \left(n \right)}{n +10}-\frac{4 \left(79+48 n \right) a \! \left(1+n \right)}{n +10}+\frac{2 \left(705+277 n \right) a \! \left(n +2\right)}{n +10}-\frac{\left(3333+947 n \right) a \! \left(n +3\right)}{n +10}+\frac{\left(4658+1029 n \right) a \! \left(n +4\right)}{n +10}-\frac{2 \left(2031+365 n \right) a \! \left(n +5\right)}{n +10}+\frac{5 \left(445+67 n \right) a \! \left(n +6\right)}{n +10}-\frac{\left(736+95 n \right) a \! \left(n +7\right)}{n +10}+\frac{\left(133+15 n \right) a \! \left(n +8\right)}{n +10}, \quad n \geq 9
Specification 1
Strategy pack name: point_and_row_placements_tracked_fusion
Tree: http://permpal.com/tree/21406/
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[16,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[16,x]*F[7,x]
F[7,x] = F[17,x]+F[8,x]
F[8,x] = F[9,x,1]
F[9,x,k[0]] = F[10,x,k[0]]+F[14,x,k[0]]
F[10,x,k[0]] = F[0,x]+F[11,x,k[0]]
F[11,x,k[0]] = F[12,x,k[0]]
F[12,x,k[0]] = F[13,x,k[0]]*F[9,x,k[0]]
F[13,x,k[0]] = k[0]*x
F[14,x,k[0]] = F[15,x,k[0]]*F[16,x]
F[15,x,k[0]] = 1/(-1+k[0])*(F[9,x,k[0]]*k[0]-F[9,x,1])
F[16,x] = x
F[17,x] = F[18,x]
F[18,x] = F[19,x]^2*F[0,x]*F[21,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[16,x]*F[19,x]
F[21,x] = F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[16,x]*F[19,x]*F[24,x]
F[24,x] = F[1,x]+F[22,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_{16}\! \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_{16}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{17}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{9}\! \left(x , 1\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right)+F_{14}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{0}\! \left(x \right)+F_{11}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = y x
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{16}\! \left(x \right)
F_{15}\! \left(x , y\right) = \frac{y F_{9}\! \left(x , y\right)-F_{9}\! \left(x , 1\right)}{-1+y}
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19} \left(x \right)^{2} F_{0}\! \left(x \right) F_{21}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{16}\! \left(x \right) F_{19}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \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_16(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_16(x)*F_7(x))
Eq(F_7(x), F_17(x) + F_8(x))
Eq(F_8(x), F_9(x, 1))
Eq(F_9(x, k_0), F_10(x, k_0) + F_14(x, k_0))
Eq(F_10(x, k_0), F_0(x) + F_11(x, k_0))
Eq(F_11(x, k_0), F_12(x, k_0))
Eq(F_12(x, k_0), F_13(x, k_0)*F_9(x, k_0))
Eq(F_13(x, k_0), k_0*x)
Eq(F_14(x, k_0), F_15(x, k_0)*F_16(x))
Eq(F_15(x, k_0), (k_0*F_9(x, k_0) - F_9(x, 1))/(k_0 - 1))
Eq(F_16(x), x)
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_0(x)*F_19(x)**2*F_21(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_16(x)*F_19(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_16(x)*F_19(x)*F_24(x))
Eq(F_24(x), F_1(x) + F_22(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"}, {"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.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_and_row_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, 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, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "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": [[1, 1]]}, {"patt": [0], "pos": [[2, 0]]}, {"patt": [0, 1], "pos": [[1, 0], [1, 0]]}, {"patt": [1, 0], "pos": [[1, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 1], [0, 1], [2, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 0, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}], "requirements": [[{"patt": [0], "pos": [[1, 0]]}]]}], "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, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 3, "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, 2], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"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, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [2, 0, 1, 3], "pos": [[0, 1], [0, 0], [0, 0], [0, 1]]}, {"patt": [2, 0, 1, 3], "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, 2], [0, 2]]}, {"patt": [0, 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