0132_1302_1320_2301
Counting sequence:
1, 1, 2, 6, 20, 65, 202, 605, 1764, 5046, 14238, 39775, 110290, 304087, 834722, 2283294, 6227960, 16947581, 46026586, 124787825, 337826496, 913373766, 2466589362, 6654071611, 17933314210, 48289162795, 129922166462, 349289633070, 938378852684, 2519297433881, 6759365746858, 18124771973765, 48572745942300, 130100344118742, 348290051701062, 931946081096599, 2492510941670770, 6663290134115071, 17805489701321306, 47559831941851710, 126985766080118816, 338925955755203381, 904265534721795322, 2411761909806582041, 6430219445839217400, 17138600720715214950, 45665491946131084122, 121637889718054840435, 323908294187153145154, 862287293996014827091, 2294894303910858146678, 6106017324176361457230, 16242081790352688236660, 43193378142695334022097, 114838577677539837283402, 305250777661730303734829, 811198494079663300816980, 2155270489114290898756086, 5725105570087892559193326, 15204598191057607951046095, 40371852243927317608941586, 107175896149228094572690471, 284467485500195452733156882, 754896570055709221498526046, 2002920603330625094146784840, 5313290363917050741009479405, 14092527477087802268272854682, 37371597900143924765060261825, 99088606714622658917560709424, 262685937847869055810909921926, 696278013076353856799542237698, 1845282804371581603027871866795, 4889665702467040964140812582370, 12954865506734806426165922812027, 34318289125243932092096876125742, 90898925585450202805138286842734, 240731900468237394927749162749340, 637458090604756191870649864741001, 1687772902846493625825912662279146, 4468090897612838879276726989467605, 11827060097450345284145558354012556, 31302540037283933868462350702027286, 82838351634278113280103754298875542, 219196439082030177043325976618207751, 579944946640166148285828479105060530, 1534236419704631836411316048571891535, 4058364388039030633025176609987229642, 10734058952232528903741310990247076990, 28387819016534118516239156977153925744, 75068215533137759491792596447966596261, 198489273342230024809092205003420431418, 524778124335682234133617627431204116585, 1387308213431546085326386586191520824680, 3667157337416395355831643558840986095782, 9692733149421990636352937084248092647082, 25616739341202713301715674487405831411939, 67696007214636257099918072243460768606850, 178881152093693341222607710003748842902691, 472638536098935701369860009100213112577766, 1248697847509562669987010871364603411790990, 3298754093316527866926819051526714977249476
Generating function in Maple syntax:
-(5*x^5-18*x^4+32*x^3-24*x^2+8*x-1)/(2*x-1)/(x-1)/(x^2-3*x+1)^2
Generating function in latex syntax:
-\frac{5 x^{5}-18 x^{4}+32 x^{3}-24 x^{2}+8 x -1}{\left(2 x -1\right) \left(x -1\right) \left(x^{2}-3 x +1\right)^{2}}
Generating function in sympy syntax:
(-5*x**5 + 18*x**4 - 32*x**3 + 24*x**2 - 8*x + 1)/((x - 1)*(2*x - 1)*(x**2 - 3*x + 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x-1)*(x^2-3*x+1)^2*F(x)+5*x^5-18*x^4+32*x^3-24*x^2+8*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x -1\right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)+5 x^{5}-18 x^{4}+32 x^{3}-24 x^{2}+8 x -1 = 0
Explicit closed form in Maple syntax:
1/25*((5*n+6)*5^(1/2)-10*n)*(3/2-1/2*5^(1/2))^(-n)+1/25*((-5*n-6)*5^(1/2)-10*n)*(3/2+1/2*5^(1/2))^(-n)-2^n+2
Explicit closed form in latex syntax:
\frac{\left(\left(5 n +6\right) \sqrt{5}-10 n \right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{25}+\frac{\left(\left(-5 n -6\right) \sqrt{5}-10 n \right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{25}-2^{n}+2
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 65
a(n+5) = 2*a(n)-13*a(n+1)+28*a(n+2)-23*a(n+3)+8*a(n+4)-2, n >= 6
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) = 65
a \! \left(n +5\right) = 2 a \! \left(n \right)-13 a \! \left(n +1\right)+28 a \! \left(n +2\right)-23 a \! \left(n +3\right)+8 a \! \left(n +4\right)-2, \quad n \geq 6
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/21290/
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[9,x]
F[3,x] = F[0,x]+F[4,x]
F[4,x] = F[5,x]
F[5,x] = F[6,x]*F[9,x]
F[6,x] = F[24,x]+F[3,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[10,x]*F[13,x]*F[9,x]
F[9,x] = x
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[9,x]
F[13,x] = F[14,x]+F[22,x]
F[14,x] = F[10,x]*F[15,x]
F[15,x] = F[16,x]+F[21,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[18,x]*F[9,x]
F[18,x] = F[16,x]+F[19,x]
F[19,x] = F[20,x]
F[20,x] = F[10,x]*F[18,x]*F[9,x]
F[21,x] = F[17,x]
F[22,x] = F[23,x]
F[23,x] = F[10,x]^2*F[21,x]*F[9,x]
F[24,x] = F[10,x]*F[25,x]*F[9,x]
F[25,x] = F[26,x]+F[30,x]
F[26,x] = F[1,x]+F[27,x]+F[29,x]
F[27,x] = F[28,x]
F[28,x] = F[10,x]*F[16,x]*F[9,x]
F[29,x] = F[25,x]*F[9,x]
F[30,x] = F[31,x]
F[31,x] = F[10,x]*F[25,x]*F[9,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_{9}\! \left(x \right)
F_{3}\! \left(x \right) = F_{0}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x \right) F_{9}\! \left(x \right)
F_{6}\! \left(x \right) = F_{24}\! \left(x \right)+F_{3}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = x
F_{10}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{10}\! \left(x \right) F_{9}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{22}\! \left(x \right)
F_{14}\! \left(x \right) = F_{10}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{21}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{9}\! \left(x \right)
F_{18}\! \left(x \right) = F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right) F_{9}\! \left(x \right)
F_{21}\! \left(x \right) = F_{17}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{10} \left(x \right)^{2} F_{21}\! \left(x \right) F_{9}\! \left(x \right)
F_{24}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right) F_{9}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{30}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{9}\! \left(x \right)
F_{29}\! \left(x \right) = F_{25}\! \left(x \right) F_{9}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{10}\! \left(x \right) F_{25}\! \left(x \right) F_{9}\! \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)*F_9(x))
Eq(F_3(x), F_0(x) + F_4(x))
Eq(F_4(x), F_5(x))
Eq(F_5(x), F_6(x)*F_9(x))
Eq(F_6(x), F_24(x) + F_3(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_10(x)*F_13(x)*F_9(x))
Eq(F_9(x), x)
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_9(x))
Eq(F_13(x), F_14(x) + F_22(x))
Eq(F_14(x), F_10(x)*F_15(x))
Eq(F_15(x), F_16(x) + F_21(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_18(x)*F_9(x))
Eq(F_18(x), F_16(x) + F_19(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_10(x)*F_18(x)*F_9(x))
Eq(F_21(x), F_17(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_10(x)**2*F_21(x)*F_9(x))
Eq(F_24(x), F_10(x)*F_25(x)*F_9(x))
Eq(F_25(x), F_26(x) + F_30(x))
Eq(F_26(x), F_1(x) + F_27(x) + F_29(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_10(x)*F_16(x)*F_9(x))
Eq(F_29(x), F_25(x)*F_9(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_10(x)*F_25(x)*F_9(x))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "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": "row_and_col_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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 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, 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, 0], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 2], [0, 2], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}], "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": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 0, "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], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 0], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 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