0132_0321_1302_3021
Counting sequence:
1, 1, 2, 6, 20, 66, 216, 713, 2392, 8157, 28209, 98706, 348862, 1243818, 4468895, 16165947, 58834514, 215284061, 791586584, 2923366585, 10838915938, 40331930379, 150568947010, 563799206628, 2116946072843, 7968879001631, 30067895596107, 113697904030945, 430802840467693, 1635389293234951, 6219077341175606, 23688844384369962, 90371260697134455, 345258837179790827, 1320840144400625185, 5059561257861907207, 19404459670279590476, 74505410196246935749, 286381749102520936874, 1101919491688805901810, 4244034866510179002728, 16361072253553417091126, 63129013483178539452640, 243788206397218889028989, 942206481091622402454289, 3644295314869928741436089, 14105890714980971821296624, 54637808885090012939890326, 211777116267753245144516588, 821383087799087320526062000, 3187734570585885147445394679, 12378772567716316423665716382, 48097447903101807150652672360, 186984690896534078086856668251, 727311937165512512259648922282, 2830463730528261928523313836960, 11020687990731157640819295145060, 42930554366447514878358592860127, 167310731263538222903120352136407, 652339700956325925368301128884462, 2544544657024506251536822367417263, 9929478950397858086528844398960655, 38762976377182568562003819111474478, 151382783755052314209378864992240517, 591424446289665512303466649042001182, 2311428251804799888341291794096684735, 9036808931453194517558993386589504966, 35342617783858256867872470964592668120, 138269648068595407608760220460125142194, 541122161211213188377137995448812260381, 2118361498500454970641858256168620001597, 8295398653627495714445285218536537812618, 32494000765504736766651999547361835298517, 127319318231108736368071003888612196465391, 499007664886095994093178788092664066951232, 1956314437402373496435431513765331060965258, 7671592063933329742066320791328404200225451, 30091561829466350374409482843913385605320420, 118062900823991174919267073376187612816631417, 463328353830887332946733573954889021355240619, 1818730437762255088382891531827072979506744332, 7140838991877881109588438040587937106344825807, 28043299523279219072599138043487227528856109908, 110155338471859141300729917861832989815073152143, 432789033884467139384510408167275136944773363353, 1700744125650794446244685502955745452740855919429, 6684848896120362393635553981986852746794720797070, 26280402916325076932532468266207510158513933538115, 103337570967907897952933719579219677552362100694983, 406413707867707875721729305827533422045698838900085, 1598675961655630695282343943266484944733960740655879, 6289740540657767719176397565321044742290732018879246, 24750470281404037605931837156464724276459816262204388, 97411655788087189640910877214504127729294738006169621, 383454202601835865163196920245543890924493973920631201, 1509696323836386008200764284041337089473259809444192578, 5944805838852180890029086696435072722351009333796014800, 23412955744804552920060491377694212266146082715599243344, 92223980235263337079117497498094413223961545856346925577, 363328216789115236720413572718819822431694194411076089792, 1431596682198741188542291423740867924401091166454868175633
Generating function in Maple syntax:
-1/2*(-x^3+2*x^2+(x^3-4*x^2+3*x-1)*(1-4*x)^(1/2)-3*x+1)/x/(x^6-8*x^5+23*x^4-29*x^3+20*x^2-7*x+1)*(x-1)^3
Generating function in latex syntax:
-\frac{\left(-x^{3}+2 x^{2}+\left(x^{3}-4 x^{2}+3 x -1\right) \sqrt{1-4 x}-3 x +1\right) \left(x -1\right)^{3}}{2 x \left(x^{6}-8 x^{5}+23 x^{4}-29 x^{3}+20 x^{2}-7 x +1\right)}
Generating function in sympy syntax:
(x - 1)**3*(x**3/2 - x**2 + 3*x/2 - sqrt(1 - 4*x)*(x**3 - 4*x**2 + 3*x - 1)/2 - 1/2)/(x*(x**6 - 8*x**5 + 23*x**4 - 29*x**3 + 20*x**2 - 7*x + 1))
Implicit equation for the generating function in Maple syntax:
x*(x^6-8*x^5+23*x^4-29*x^3+20*x^2-7*x+1)*F(x)^2-(x^3-2*x^2+3*x-1)*(x-1)^3*F(x)+(x-1)^6 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{6}-8 x^{5}+23 x^{4}-29 x^{3}+20 x^{2}-7 x +1\right) F \! \left(x \right)^{2}-\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(x -1\right)^{6} = 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) = 216
a(7) = 713
a(8) = 2392
a(9) = 8157
a(10) = 28209
a(n+11) = 2*(1+2*n)/(n+12)*a(n)-(76+53*n)/(n+12)*a(1+n)+(706+293*n)/(n+12)*a(n+2)-2*(1528+443*n)/(n+12)*a(n+3)+8*(928+205*n)/(n+12)*a(n+4)-(11180+1999*n)/(n+12)*a(n+5)+10*(1112+167*n)/(n+12)*a(n+6)-(7476+967*n)/(n+12)*a(n+7)+(3374+383*n)/(n+12)*a(n+8)-3*(326+33*n)/(n+12)*a(n+9)+(164+15*n)/(n+12)*a(n+10), n >= 11
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) = 216
a \! \left(7\right) = 713
a \! \left(8\right) = 2392
a \! \left(9\right) = 8157
a \! \left(10\right) = 28209
a \! \left(n +11\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +12}-\frac{\left(76+53 n \right) a \! \left(1+n \right)}{n +12}+\frac{\left(706+293 n \right) a \! \left(n +2\right)}{n +12}-\frac{2 \left(1528+443 n \right) a \! \left(n +3\right)}{n +12}+\frac{8 \left(928+205 n \right) a \! \left(n +4\right)}{n +12}-\frac{\left(11180+1999 n \right) a \! \left(n +5\right)}{n +12}+\frac{10 \left(1112+167 n \right) a \! \left(n +6\right)}{n +12}-\frac{\left(7476+967 n \right) a \! \left(n +7\right)}{n +12}+\frac{\left(3374+383 n \right) a \! \left(n +8\right)}{n +12}-\frac{3 \left(326+33 n \right) a \! \left(n +9\right)}{n +12}+\frac{\left(164+15 n \right) a \! \left(n +10\right)}{n +12}, \quad n \geq 11
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21263/
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[12,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[12,x]*F[7,x]
F[7,x] = F[13,x]+F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[9,x]^2*F[12,x]
F[12,x] = x
F[13,x] = F[14,x]*F[18,x]
F[14,x] = F[15,x]
F[15,x] = F[0,x]*F[12,x]*F[16,x]*F[18,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[12,x]*F[16,x]
F[18,x] = F[1,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[12,x]*F[18,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_{12}\! \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_{12}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{13}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right) F_{18}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{0}\! \left(x \right) F_{12}\! \left(x \right) F_{16}\! \left(x \right) F_{18}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right) F_{18}\! \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_12(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_12(x)*F_7(x))
Eq(F_7(x), F_13(x) + F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x)**2)
Eq(F_12(x), x)
Eq(F_13(x), F_14(x)*F_18(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_0(x)*F_12(x)*F_16(x)*F_18(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_12(x)*F_16(x))
Eq(F_18(x), F_1(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_12(x)*F_18(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.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[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, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "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_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "strategy_class": "RequirementInsertionStrategy"}}, {"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, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 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], [2, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 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], [2, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [2, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 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}}, {"children": [{"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, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"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], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": 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