0132_0231_0312_1023_1203_3012
Counting sequence:
1, 1, 2, 6, 18, 51, 152, 477, 1557, 5228, 17920, 62395, 219969, 783433, 2814269, 10183972, 37088221, 135826093, 499897140, 1847976842, 6858596899, 25546432146, 95463674513, 357795415180, 1344657763761, 5066102543036, 19130987649153, 72398145125756, 274522296527989, 1042865707323154, 3968498096244951, 15125949886454629, 57739637594836715, 220719442767585736, 844862474467295344, 3238002761601270599, 12424640046180891985, 47728516486253575689, 183541338856594718473, 706525751546138025201, 2722314643054803435963, 10498918282104126553767, 40525450122250950231017, 156556338486531596992317, 605280438207727091921947, 2341911853486528475574357, 9067720382262355693339607, 35133872268126308691484927, 136220268087775866199490547, 528486032805912302462762429, 2051588847777791453921805613, 7968962606219110060393099683, 30971204996825177956815124391, 120434256626876353014937367239, 468563945402179616870503827303, 1823922377144708024755634449583, 7103210474273466403468538740891, 27676200611634111842067847830003, 107883418715593220911414227909695, 420719576052085298445747163475863, 1641399163186676657769866709659787, 6406386731059512850378358869609799, 25014039591964309048270817201705875, 97705902983569261757546742138445716, 381785006804545939903115608625133689, 1492358209320615310184603435489423989, 5835508480314207488157195985074112740, 22826074459986154340814232600374980354, 89315397871733629076497283909123233843, 349590721805568441124338796516964308482, 1368762061171579768804220518494281016521, 5360764578289704580934492151113171889760, 21001610445732520151465435022691504683849, 82300423510411557357086098914086477088188, 322605533076304731735107676205576923412717, 1264907068021382676659883626684819978555736, 4960887912610250902537778929068421211362325, 19461272754999245991914839346340704608046974, 76364453729040639057012804044374513314525423, 299720679088574478634245282980575993457763242, 1176643719659190059368944085482585172502649681, 4620335712906697517281475404436223398947916330, 18146794381801664590273060159902968902768199509, 71288878617742661897071883782892694354973545538, 280115316189895556506412797472110657451178870537, 1100887511225970006784660243599553714935358520356, 4327508561111825288720332616355198049418635741535, 17014523308628890270570590326381311627062364385794, 66909305758963429522587682498496718217081988294353, 263169916501877766807093175460849510462384687540852, 1035302113228207885039198479019097371502808506250691, 4073589995588705737787363381665190820826509571450950, 16031167938128536692854928286403695289384400811282669, 63099944503127601934629112471007749025644517309621388, 248408894014095088920345572463728620943309880007097307, 978088362456952728482915694531270443455039576588364676, 3851769706993233643546233430663501097622304909921997645, 15170934661055534963864543814389738819586979182900733164, 59763048443826162375170728737721600682807523733030959433, 235461629894618304909185491322524346129281699214214134892, 927840253679242704203647086315674212243115993545861281341
Generating function in Maple syntax:
1/2/x*(-1+(1-4*x)^(1/2))*(x^5-x^4-x^3-x^2+2*x-1)/(x-1)^2
Generating function in latex syntax:
\frac{\left(-1+\sqrt{1-4 x}\right) \left(x^{5}-x^{4}-x^{3}-x^{2}+2 x -1\right)}{2 x \left(x -1\right)^{2}}
Generating function in sympy syntax:
(sqrt(1 - 4*x) - 1)*(x**5 - x**4 - x**3 - x**2 + 2*x - 1)/(2*x*(x - 1)**2)
Implicit equation for the generating function in Maple syntax:
x*(x-1)^4*F(x)^2+(x^5-x^4-x^3-x^2+2*x-1)*(x-1)^2*F(x)+(x^5-x^4-x^3-x^2+2*x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{4} F \! \left(x \right)^{2}+\left(x^{5}-x^{4}-x^{3}-x^{2}+2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(x^{5}-x^{4}-x^{3}-x^{2}+2 x -1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 18
a(5) = 51
a(6) = 152
a(7) = 477
a(8) = 1557
a(9) = 5228
a(10) = 17920
a(n+7) = 2*(-5+2*n)/(8+n)*a(n)-(-17+9*n)/(8+n)*a(1+n)+(-13+2*n)/(8+n)*a(n+2)-9/(8+n)*a(n+3)+3*(19+4*n)/(8+n)*a(n+4)-3*(28+5*n)/(8+n)*a(n+5)+(47+7*n)/(8+n)*a(n+6)+1/(8+n), 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) = 18
a \! \left(5\right) = 51
a \! \left(6\right) = 152
a \! \left(7\right) = 477
a \! \left(8\right) = 1557
a \! \left(9\right) = 5228
a \! \left(10\right) = 17920
a \! \left(n +7\right) = \frac{2 \left(-5+2 n \right) a \! \left(n \right)}{8+n}-\frac{\left(-17+9 n \right) a \! \left(1+n \right)}{8+n}+\frac{\left(-13+2 n \right) a \! \left(n +2\right)}{8+n}-\frac{9 a \! \left(n +3\right)}{8+n}+\frac{3 \left(19+4 n \right) a \! \left(n +4\right)}{8+n}-\frac{3 \left(28+5 n \right) a \! \left(n +5\right)}{8+n}+\frac{\left(47+7 n \right) a \! \left(n +6\right)}{8+n}+\frac{1}{8+n}, \quad n \geq 11
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://www.permpal.com/tree/22286/
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[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[17,x]*F[7,x]
F[7,x] = F[18,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[15,x,k[0]] = F[16,x,k[0]]*F[17,x]
F[16,x,k[0]] = (F[9,x,k[0]]*k[0]-F[9,x,1])/(-1+k[0])
F[17,x] = x
F[18,x] = F[19,x]
F[19,x] = F[20,x]+F[30,x]
F[20,x] = F[21,x]+F[25,x]
F[21,x] = F[17,x]*F[22,x]
F[22,x] = F[1,x]+F[23,x]
F[23,x] = F[24,x]
F[24,x] = F[17,x]*F[22,x]
F[25,x] = F[26,x]
F[26,x] = F[17,x]*F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[17,x]*F[29,x]
F[29,x] = F[1,x]+F[28,x]
F[30,x] = F[17,x]*F[22,x]*F[23,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_{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_{17}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{18}\! \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_{15}\! \left(x , y\right) = F_{16}\! \left(x , y\right) F_{17}\! \left(x \right)
F_{16}\! \left(x , y\right) = \frac{y F_{9}\! \left(x , y\right)-F_{9}\! \left(x , 1\right)}{-1+y}
F_{17}\! \left(x \right) = x
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{30}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{21}\! \left(x \right) = F_{17}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{17}\! \left(x \right) F_{22}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{17}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{17}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{1}\! \left(x \right)+F_{28}\! \left(x \right)
F_{30}\! \left(x \right) = F_{17}\! \left(x \right) F_{22}\! \left(x \right) F_{23}\! \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_17(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_17(x)*F_7(x))
Eq(F_7(x), F_18(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))
Eq(F_15(x, k_0), F_16(x, k_0)*F_17(x))
Eq(F_16(x, k_0), (-k_0*F_9(x, k_0) + F_9(x, 1))/(1 - k_0))
Eq(F_17(x), x)
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x) + F_30(x))
Eq(F_20(x), F_21(x) + F_25(x))
Eq(F_21(x), F_17(x)*F_22(x))
Eq(F_22(x), F_1(x) + F_23(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_17(x)*F_22(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_17(x)*F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_17(x)*F_29(x))
Eq(F_29(x), F_1(x) + F_28(x))
Eq(F_30(x), F_17(x)*F_22(x)*F_23(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": 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": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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, 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, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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, 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, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "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"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [2, 0, 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], [0, 0], [0, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], 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