0132_0231_3021
Counting sequence:
1, 1, 2, 6, 21, 77, 286, 1067, 3992, 14976, 56338, 212517, 803758, 3047409, 11580777, 44103581, 168294630, 643372977, 2463727945, 9449450566, 36295744476, 139603479110, 537635985211, 2072983564340, 8001769489951, 30919255604475, 119590765611790, 462985137030000, 1793967226232814, 6956954436339567, 26999880609908018, 104863633780813270, 407560457639314069, 1585069061862899857, 6168497396748006646, 24019981695057395790, 93587102006869571752, 364836385988918599095, 1423012255413170176456, 5553128247423468619097, 21680843811580009579705, 84686755487080945990642, 330938729257240918587486, 1293793360088450900637566, 5060114527010726692763347, 19798266949453727434203656, 77492394730419755437229257, 303424114904625819352550757, 1188487356868774404239290662, 4656795395245488410716417176, 18252518843197491651039533348, 71564347049448827454157187337, 280675383677713821629270476968, 1101136892744992417195405759339, 4321192005457176832268013686863, 16962394506002887423287510680346, 66602168604884351848541076067796, 261579460316018336099234625908758, 1027612963019914538706984021414551, 4037968575323398775452128971661982, 15870866045350842669003602630810823, 62393548828556460273759075335353977, 245345043704815149551296106341525374, 964963123966980794369159742842245259, 3796097637901653486973459017759798888, 14936703823262917736718554072721934152, 58784186653691585142212638969581624466, 231394116771881409323913142184026804609, 911020110550252577716329810587150055582, 3587445816782137200586623699091212434789, 14129358233592962471845935957802733977225, 55659238909491041411093617347356534495300, 219294604705674128913508846441203275581138, 864157038003404157583175516652812562377964, 3405882709498234726196095237656590275661127, 13425711491321084903303922883029020406781116, 52931464811232618972338768647281766290960335, 208717049210226638321619283566038426162370837, 823128873936057691976660811898314527934183960, 3246700144583137116918740942130064376572761316, 12807947698348607422980209255852259713175668726, 50533405390024969773080116810117734006760277175, 199405861857714404631121438466907772015106588286, 786966659290902566800121062571492370344371343455, 3106222541757008874509486228096870217040078744885, 12262116142542279833383071008454649969233940169590, 48412080789145167891053302875127614699673644830664, 191159733600395602566745622435622703611668102363344, 754905064832616462389094073254029278737872357273153, 2981538779169267589360008506121617205100492665104512, 11777137983642152393082376294341428509597141489661291, 46525307276879366934207332258542137442548413362963559, 183817967506457262555262999125441753812399696343562142, 726331631106962245217072606625123679095700002551662764, 2870313519051313895476861956037627398109064068853277610, 11344104400376640150973961248647744951264562025725645885, 44839084400793160651004462319075434250778158605604569998, 177250744380621416058425456463849586467519943376974967809, 700750475494692491982382456682142935577160271018746164937, 2770651798410215104576941603456174448550807297513264554006, 10955771021675882165529704497521073160026182141509433699790
Generating function in Maple syntax:
1/2*((-x^3+4*x^2-4*x+1)*(1-4*x)^(1/2)-4*x^4+15*x^3-20*x^2+8*x-1)/x/(x^3-4*x^2+5*x-1)/(x-1)
Generating function in latex syntax:
\frac{\left(-x^{3}+4 x^{2}-4 x +1\right) \sqrt{1-4 x}-4 x^{4}+15 x^{3}-20 x^{2}+8 x -1}{2 x \left(x^{3}-4 x^{2}+5 x -1\right) \left(x -1\right)}
Generating function in sympy syntax:
(-2*x**4 + 15*x**3/2 - 10*x**2 + 4*x + sqrt(1 - 4*x)*(-x**3 + 4*x**2 - 4*x + 1)/2 - 1/2)/(x*(x - 1)*(x**3 - 4*x**2 + 5*x - 1))
Implicit equation for the generating function in Maple syntax:
x*(x^3-4*x^2+5*x-1)*(x-1)^2*F(x)^2+(x-1)*(4*x^4-15*x^3+20*x^2-8*x+1)*F(x)+4*x^4-13*x^3+16*x^2-7*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{3}-4 x^{2}+5 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)^{2}+\left(x -1\right) \left(4 x^{4}-15 x^{3}+20 x^{2}-8 x +1\right) F \! \left(x \right)+4 x^{4}-13 x^{3}+16 x^{2}-7 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 77
a(6) = 286
a(7) = 1067
a(n+6) = -2*(3+2*n)/(7+n)*a(n)+(69+29*n)/(7+n)*a(1+n)-(247+79*n)/(7+n)*a(n+2)+2*(193+49*n)/(7+n)*a(n+3)-2*(129+26*n)/(7+n)*a(n+4)+12*(n+6)/(7+n)*a(n+5)-3*(n+3)/(7+n), n >= 8
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(5\right) = 77
a \! \left(6\right) = 286
a \! \left(7\right) = 1067
a \! \left(n +6\right) = -\frac{2 \left(3+2 n \right) a \! \left(n \right)}{7+n}+\frac{\left(69+29 n \right) a \! \left(1+n \right)}{7+n}-\frac{\left(247+79 n \right) a \! \left(n +2\right)}{7+n}+\frac{2 \left(193+49 n \right) a \! \left(n +3\right)}{7+n}-\frac{2 \left(129+26 n \right) a \! \left(n +4\right)}{7+n}+\frac{12 \left(n +6\right) a \! \left(n +5\right)}{7+n}-\frac{3 \left(n +3\right)}{7+n}, \quad n \geq 8
Specification 1
Strategy pack name: point_and_row_placements
Tree: http://permpal.com/tree/273/
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[4,x]
F[4,x] = F[11,x]*F[5,x]
F[5,x] = F[0,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[11,x]*F[8,x]
F[8,x] = F[25,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]+F[22,x]+F[24,x]
F[10,x] = F[11,x]*F[12,x]
F[11,x] = x
F[12,x] = F[1,x]+F[10,x]+F[13,x]+F[17,x]
F[13,x] = F[14,x]
F[14,x] = F[19,x]^2*F[11,x]*F[15,x]
F[15,x] = F[1,x]+F[16,x]+F[17,x]
F[16,x] = F[11,x]*F[15,x]
F[17,x] = F[18,x]
F[18,x] = F[11,x]*F[15,x]*F[19,x]
F[19,x] = F[1,x]+F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[19,x]^2*F[11,x]
F[22,x] = F[23,x]
F[23,x] = F[11,x]*F[19,x]*F[9,x]
F[24,x] = F[11,x]*F[9,x]
F[25,x] = F[26,x]
F[26,x] = F[15,x]*F[19,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_{3}\! \left(x \right) = F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{11}\! \left(x \right) F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{0}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{11}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{25}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{12}\! \left(x \right)
F_{11}\! \left(x \right) = x
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{19} \left(x \right)^{2} F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)
F_{16}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{19} \left(x \right)^{2} F_{11}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right) F_{19}\! \left(x \right) F_{9}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right) F_{9}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{15}\! \left(x \right) F_{19}\! \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))
Eq(F_3(x), F_4(x))
Eq(F_4(x), F_11(x)*F_5(x))
Eq(F_5(x), F_0(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_11(x)*F_8(x))
Eq(F_8(x), F_25(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x) + F_22(x) + F_24(x))
Eq(F_10(x), F_11(x)*F_12(x))
Eq(F_11(x), x)
Eq(F_12(x), F_1(x) + F_10(x) + F_13(x) + F_17(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_11(x)*F_15(x)*F_19(x)**2)
Eq(F_15(x), F_1(x) + F_16(x) + F_17(x))
Eq(F_16(x), F_11(x)*F_15(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_11(x)*F_15(x)*F_19(x))
Eq(F_19(x), F_1(x) + F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_11(x)*F_19(x)**2)
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_11(x)*F_19(x)*F_9(x))
Eq(F_24(x), F_11(x)*F_9(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_15(x)*F_19(x)*F_3(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.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "point_and_row_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, 2, 3, 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": [[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, 2, 1], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [1, 2, 0], "pos": [[2, 1], [2, 1], [2, 1]]}, {"patt": [2, 1, 0], "pos": [[0, 1], [2, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 1], [0, 1], [2, 1]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 1], [0, 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, 2, 3, 1], "pos": [[0, 1], [0, 1], [2, 1], [2, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1], [2, 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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 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": 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, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 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], [1, 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], [0, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 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