0213_1320_3021
Counting sequence:
1, 1, 2, 6, 21, 75, 267, 951, 3407, 12309, 44867, 164891, 610347, 2273020, 8508804, 31991549, 120734511, 457129176, 1735730619, 6607223257, 25207606841, 96365703918, 369070368271, 1415863217868, 5439991258764, 20930861647331, 80638367290921, 311043531047557, 1201127506963082, 4643153806744397, 17966516485441448, 69584936104368016, 269738722680839474, 1046469093647657335, 4062981072790425330, 15786301410198525857, 61378544354472545667, 238802082341913765823, 929674411069107625509, 3621444725808411968970, 14114938413778195256426, 55044134270766614106455, 214767025560030418194476, 838376183268793563702340, 3274281035223463080172744, 12793523374612220927746738, 50009626893719711972851485, 195568464063906692545668237, 765101016009543912408356302, 2994379699230476262042552895, 11723488824007778269850884389, 45915880636790370547935489709, 179895150507864312622669941331, 705052150524196428861309919630, 2764160829866787290846519417660, 10840291225704325531536009201927, 42525526587089120586843597358096, 166872710032129451295857291285270, 655003733496499608894015148633738, 2571705504551747201896411136873494, 10099829538129039724003097559212511, 39675163340776437110506423613195963, 155894916172624254920665588784506234, 612703784600496511898846612340135841, 2408637957135256391932886667893282410, 9470915185623463658345336480298540041, 37248523102791847653418295735688326824, 146527859533975947760097040097049173377, 576531247435878658190402401904109429414, 2268895834592640811496371719069386180228, 8930854029646619573463876917678620933481, 35160570103400516518704590009700906378395, 138452612721675957273050573193896964245665, 545288908416523846492859776837628222134270, 2147981037200744905280026041643092892222607, 8462731821642088859286941552235752111915120, 33347649477756987154163522140900573235596600, 131429427552922949959790538546288020177314013, 518073089127278219771233367712042007541817719, 2042484794018719898764099222640372758749485225, 8053680950528423194228172252036300960930525982, 31761153479277994266305130457797754841207311577, 125274560467998939276415687021254334794397411018, 494188749560684454943439797726209520571289644828, 1949776287008475812323355826121351952983848786537, 7693737094556034391828054272477191248330429758261, 30363318499611943185131433960332240438002814924113, 119844805471804074923104959537701479591149780421784, 473092508892088297632643593692820605491674667858033, 1867792529212388232849721502659493385448136048668944, 7375063582518126751067020175216098389563284209836632, 29124359109167339262988649529349519436919274131148255, 115026894481381749809100358243164358624847778338482307, 454353406870782984264186479162675265073640034719277201, 1794892977702769544424143360494630160352656567270809244, 7091412233144792360582396824484192883668464253837161733, 28020466898966010353356360738273049999362927905578547928, 110730063602668632499969630188175152209307791900807909028, 437625267239202620683828276066817949223076166273572515895, 1729756706685663751380876385348722949607100872852049296323, 6837740472626139526114785774294828324803332580744842491633
Generating function in Maple syntax:
1/4*((x^5-2*x^4-2*x^3+8*x^2-5*x+1)*(1-4*x)^(1/2)+2*x^6-13*x^5+34*x^4-42*x^3+28*x^2-9*x+1)/(x^3-4*x^2+5*x-1)/(x-1)^2/(x-1/2)
Generating function in latex syntax:
\frac{\left(x^{5}-2 x^{4}-2 x^{3}+8 x^{2}-5 x +1\right) \sqrt{1-4 x}+2 x^{6}-13 x^{5}+34 x^{4}-42 x^{3}+28 x^{2}-9 x +1}{4 \left(x^{3}-4 x^{2}+5 x -1\right) \left(x -1\right)^{2} \left(x -\frac{1}{2}\right)}
Generating function in sympy syntax:
(x**6/2 - 13*x**5/4 + 17*x**4/2 - 21*x**3/2 + 7*x**2 - 9*x/4 + sqrt(1 - 4*x)*(x**5 - 2*x**4 - 2*x**3 + 8*x**2 - 5*x + 1)/4 + 1/4)/((x - 1)**2*(x - 1/2)*(x**3 - 4*x**2 + 5*x - 1))
Implicit equation for the generating function in Maple syntax:
(x^3-4*x^2+5*x-1)*(2*x-1)^2*(x-1)^4*F(x)^2-(x^5-6*x^4+14*x^3-14*x^2+7*x-1)*(2*x-1)^2*(x-1)^2*F(x)+x*(x^8-8*x^7+35*x^6-81*x^5+107*x^4-81*x^3+36*x^2-9*x+1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{3}-4 x^{2}+5 x -1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)^{2}-\left(x^{5}-6 x^{4}+14 x^{3}-14 x^{2}+7 x -1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)+x \left(x^{8}-8 x^{7}+35 x^{6}-81 x^{5}+107 x^{4}-81 x^{3}+36 x^{2}-9 x +1\right) = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 21
a(5) = 75
a(6) = 267
a(7) = 951
a(8) = 3407
a(9) = 12309
a(10) = 44867
a(11) = 164891
a(12) = 610347
a(n+11) = -4*(1+2*n)/(n+11)*a(n)+2*(29+31*n)/(n+11)*a(n+1)-(178+179*n)/(n+11)*a(n+2)+(-325+157*n)/(n+11)*a(n+3)+(3086+347*n)/(n+11)*a(n+4)-(7849+1158*n)/(n+11)*a(n+5)+(10461+1498*n)/(n+11)*a(n+6)-(8231+1084*n)/(n+11)*a(n+7)+5*(787+94*n)/(n+11)*a(n+8)-(1117+121*n)/(n+11)*a(n+9)+(172+17*n)/(n+11)*a(n+10)-2/(n+11), n >= 13
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) = 75
a \! \left(6\right) = 267
a \! \left(7\right) = 951
a \! \left(8\right) = 3407
a \! \left(9\right) = 12309
a \! \left(10\right) = 44867
a \! \left(11\right) = 164891
a \! \left(12\right) = 610347
a \! \left(n +11\right) = -\frac{4 \left(1+2 n \right) a \! \left(n \right)}{n +11}+\frac{2 \left(29+31 n \right) a \! \left(n +1\right)}{n +11}-\frac{\left(178+179 n \right) a \! \left(n +2\right)}{n +11}+\frac{\left(-325+157 n \right) a \! \left(n +3\right)}{n +11}+\frac{\left(3086+347 n \right) a \! \left(n +4\right)}{n +11}-\frac{\left(7849+1158 n \right) a \! \left(n +5\right)}{n +11}+\frac{\left(10461+1498 n \right) a \! \left(n +6\right)}{n +11}-\frac{\left(8231+1084 n \right) a \! \left(n +7\right)}{n +11}+\frac{5 \left(787+94 n \right) a \! \left(n +8\right)}{n +11}-\frac{\left(1117+121 n \right) a \! \left(n +9\right)}{n +11}+\frac{\left(172+17 n \right) a \! \left(n +10\right)}{n +11}-\frac{2}{n +11}, \quad n \geq 13
Specification 1
Strategy pack name: row_and_col_placements
Tree: http://permpal.com/tree/338/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[11,x]*F[3,x]
F[3,x] = F[1,x]+F[23,x]+F[4,x]
F[4,x] = F[11,x]*F[5,x]
F[5,x] = F[3,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[14,x]^2*F[11,x]*F[8,x]
F[8,x] = F[18,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]+F[12,x]
F[10,x] = F[11,x]*F[8,x]
F[11,x] = x
F[12,x] = F[11,x]*F[13,x]
F[13,x] = F[14,x]+F[16,x]
F[14,x] = F[1,x]+F[15,x]
F[15,x] = F[11,x]*F[13,x]
F[16,x] = F[17,x]
F[17,x] = F[11,x]*F[13,x]*F[14,x]
F[18,x] = F[19,x]
F[19,x] = F[11,x]*F[14,x]*F[20,x]*F[8,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[11,x]*F[20,x]
F[23,x] = F[11,x]*F[24,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[14,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[14,x]^2*F[11,x]*F[13,x]
F[28,x] = F[29,x]
F[29,x] = F[11,x]*F[30,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[14,x]^3
F[32,x] = F[20,x]^2*F[33,x]*F[34,x]
F[33,x] = F[15,x]
F[34,x] = F[1,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[11,x]*F[20,x]*F[34,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_{11}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \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_{3}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{14} \left(x \right)^{2} F_{11}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{18}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{12}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{8}\! \left(x \right)
F_{11}\! \left(x \right) = x
F_{12}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{16}\! \left(x \right)
F_{14}\! \left(x \right) = F_{1}\! \left(x \right)+F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{14}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{20}\! \left(x \right) F_{8}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \left(x \right) F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{14}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{14} \left(x \right)^{2} F_{11}\! \left(x \right) F_{13}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{11}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{14} \left(x \right)^{3}
F_{32}\! \left(x \right) = F_{20} \left(x \right)^{2} F_{33}\! \left(x \right) F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{15}\! \left(x \right)
F_{34}\! \left(x \right) = F_{1}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{11}\! \left(x \right) F_{20}\! \left(x \right) F_{34}\! \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_11(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_23(x) + F_4(x))
Eq(F_4(x), F_11(x)*F_5(x))
Eq(F_5(x), F_3(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_11(x)*F_14(x)**2*F_8(x))
Eq(F_8(x), F_18(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x) + F_12(x))
Eq(F_10(x), F_11(x)*F_8(x))
Eq(F_11(x), x)
Eq(F_12(x), F_11(x)*F_13(x))
Eq(F_13(x), F_14(x) + F_16(x))
Eq(F_14(x), F_1(x) + F_15(x))
Eq(F_15(x), F_11(x)*F_13(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_11(x)*F_13(x)*F_14(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_11(x)*F_14(x)*F_20(x)*F_8(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_11(x)*F_20(x))
Eq(F_23(x), F_11(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_14(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_11(x)*F_13(x)*F_14(x)**2)
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_11(x)*F_30(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_14(x)**3)
Eq(F_32(x), F_20(x)**2*F_33(x)*F_34(x))
Eq(F_33(x), F_15(x))
Eq(F_34(x), F_1(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_11(x)*F_20(x)*F_34(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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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, 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, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [2, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0], [2, 0]]}], "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, 2, 1, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "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": 1, "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": 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