01342_01432_10342_10432_13042_13402_13420_14032_14302_14320
Counting sequence:
1, 1, 2, 6, 24, 110, 540, 2758, 14448, 77022, 415860, 2267078, 12452616, 68814798, 382168332, 2131181190, 11926192608, 66939251646, 376688820708, 2124542494342, 12006377078520, 67971210660654, 385409625059580, 2188442069645126, 12442365010054224, 70823083773243550, 403558158870752340, 2301748449902225478, 13140012140411446248, 75074007837858496142, 429252183272935308780, 2456068986419787174150, 14062169043699659573184, 80561679305123803131774, 461797710883272708458436, 2648539086822876131514630, 15197707001049164988588504, 87247666294882232571188718, 501096964738856630395438044, 2879194861405325192112891846, 16549760148463390209688937520, 95164404039866968681172802654, 547407926939895625347033875508, 3149876568473117566605935932614, 18130703479683106615964757108936, 104392122148688127017169666508110, 601238121820773686211067797140172, 3463739293794634153072283608143750, 19959900462006096346344676605940128, 115048503955200826031786522747349822, 663297375427680115468458771568251300, 3825038839904955184406472000506992518, 22062744248151975589542520192279488696, 127284470057710991760156647488993542318, 734479776556149628499432483504633526972, 4239057860008728736101278986275359554630, 24470377348813468592274176985217842151440, 141283268406334337705057057002139520897054, 815861874698682796054936006510206665334804, 4712112341910215630528263085276686072520006, 27219812081631011741187640027402975410016680, 157261579640014052571575983202101502049336334, 908711095252826346706332584277827966082419628, 5251610328223632446653606815083209999758586374, 30354320343707495548303553714683243826802335616, 175472022805508253819835167905641357158337199870, 1014501497577250575330957948376128088393395478404, 5866149458038211268296651697879987409520344392198, 33924030292995346847458904513640247813130181137304, 196206772465688375475262391146250415838986454510446, 1134935547347174325627271482977140595328862102197660, 6565647285209006547126357904053149701797962513470662, 37986702840001784026942710561860860827185377410359280, 219802168317744827438484632176206519903646675105098718, 1271971398474726575765010813496589132599435401768812532, 7361502340141197283674237292956512387201682976306282950, 42608680499811957854938964824828959852301767212063186568, 246644295257451810689956873957368992199827052780826575566, 1427855659913313125052060471572664893158835615480530536588, 8266786297940842239401878713588174326632673678957119520902, 47866013195484749899595901358760028234863976483129284784480, 277175572771031633972245280821547265972821817482137891228350, 1605161933426738586355450973426266995338116933469047460098404, 9296469212488551632802903531202198902365986500393286486825094, 53845773871620436838745000038537948680990556421479602014429816, 311902435685717446465916431937525114265357179169678970709456430, 1806835635389814330562332187410705185291751049749610973879506044, 10467680979455266085826630434417018508292884228568748654436340038, 60647583207022098418755171328339137050481761247299232887296361424, 351404222925291162520521851437339906245628985361592681339675479966, 2036245803381361701121854120455434392209332004511436543026631844820, 11800013216036928280761775028642334894260086986225618683829296874054, 68385368346627467879899389095608792910227624077068529267892986367848, 396343422049437983297784764235536451310495995520469868501373330733454, 2297244761412859648136599064752589607866706246076382142267941744863596, 13315866712387272657550641359788819335763951617560717734280961122575110, 77189386002997673280912262605707470625596082221883625474521645225232192, 447477447970574951795508394122390256810979924681900358222067406704602750, 2594236690896820096906787869191807763721826027254355670971359385064195908, 15040850593602508164375898627790251235693107376780176039362877097072007942, 87208545881506095261277811762483228737162764664557562435295226474129888600
Generating function in Maple syntax:
((-2*x^3+9*x^2-x)*(x^2-6*x+1)^(1/2)-2*x^4+15*x^3-18*x^2-9*x+2)/(2*x^2-12*x+2)
Generating function in latex syntax:
\frac{\left(-2 x^{3}+9 x^{2}-x \right) \sqrt{x^{2}-6 x +1}-2 x^{4}+15 x^{3}-18 x^{2}-9 x +2}{2 x^{2}-12 x +2}
Generating function in sympy syntax:
(-2*x**4 + 15*x**3 - 18*x**2 - 9*x + sqrt(x**2 - 6*x + 1)*(-2*x**3 + 9*x**2 - x) + 2)/(2*x**2 - 12*x + 2)
Implicit equation for the generating function in Maple syntax:
(x^2-6*x+1)*F(x)^2+(x-2)*(2*x+1)*(x^2-6*x+1)*F(x)-2*x^4+3*x^3-17*x^2-3*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-6 x +1\right) F \! \left(x \right)^{2}+\left(x -2\right) \left(2 x +1\right) \left(x^{2}-6 x +1\right) F \! \left(x \right)-2 x^{4}+3 x^{3}-17 x^{2}-3 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(n+4) = -2*(n-2)/(3+n)*a(n)+3*(8+5*n)/(3+n)*a(3+n)+3*(-6+7*n)/(3+n)*a(n+1)-3*(9+19*n)/(3+n)*a(n+2), 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) = 24
a \! \left(n +4\right) = -\frac{2 \left(n -2\right) a \! \left(n \right)}{3+n}+\frac{3 \left(8+5 n \right) a \! \left(3+n \right)}{3+n}+\frac{3 \left(-6+7 n \right) a \! \left(n +1\right)}{3+n}-\frac{3 \left(9+19 n \right) a \! \left(n +2\right)}{3+n}, \quad n \geq 5
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/24220/
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[17,x]*F[4,x]
F[4,x] = F[5,x,1]
F[5,x,k[0]] = (F[6,x,k[0]]*k[0]-F[6,x,1])/(-1+k[0])
F[6,x,k[0]] = F[1,x]+F[7,x,k[0]]
F[7,x,k[0]] = F[8,x,k[0]]
F[8,x,k[0]] = F[16,x,k[0]]*F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]+F[6,x,k[0]]
F[10,x,k[0]] = F[11,x,k[0]]
F[11,x,k[0]] = F[12,x,k[0]]*F[16,x,k[0]]
F[12,x,k[0]] = F[13,x,k[0]]+F[14,x,k[0]]
F[13,x,k[0]] = F[6,x,k[0]]*F[9,x,k[0]]
F[14,x,k[0]] = F[15,x,k[0]]
F[15,x,k[0]] = F[12,x,k[0]]*F[16,x,k[0]]*F[9,x,k[0]]
F[16,x,k[0]] = k[0]*x
F[17,x] = 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_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x , 1\right)
F_{5}\! \left(x , y\right) = \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}
F_{6}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{7}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = F_{16}\! \left(x , y\right) F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right)+F_{6}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{16}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right)+F_{14}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = F_{6}\! \left(x , y\right) F_{9}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right)
F_{15}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{9}\! \left(x , y\right)
F_{16}\! \left(x , y\right) = y x
F_{17}\! \left(x \right) = x
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_17(x)*F_4(x))
Eq(F_4(x), F_5(x, 1))
Eq(F_5(x, k_0), (-k_0*F_6(x, k_0) + F_6(x, 1))/(1 - k_0))
Eq(F_6(x, k_0), F_1(x) + F_7(x, k_0))
Eq(F_7(x, k_0), F_8(x, k_0))
Eq(F_8(x, k_0), F_16(x, k_0)*F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0) + F_6(x, k_0))
Eq(F_10(x, k_0), F_11(x, k_0))
Eq(F_11(x, k_0), F_12(x, k_0)*F_16(x, k_0))
Eq(F_12(x, k_0), F_13(x, k_0) + F_14(x, k_0))
Eq(F_13(x, k_0), F_6(x, k_0)*F_9(x, k_0))
Eq(F_14(x, k_0), F_15(x, k_0))
Eq(F_15(x, k_0), F_12(x, k_0)*F_16(x, k_0)*F_9(x, k_0))
Eq(F_16(x, k_0), k_0*x)
Eq(F_17(x), 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.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_placements_tracked_fusion", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 4, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 4, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 3, 2, 0], "pos": [[0, 0], [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, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 4, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 4, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 3, 2, 0], "pos": [[0, 0], [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, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 4, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 4, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 4, 3, 2, 0], "pos": [[0, 0], [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"}}, {"children": [{"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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", 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