0132_1320_3021
Counting sequence:
1, 1, 2, 6, 21, 75, 262, 896, 3033, 10261, 34906, 119771, 415012, 1452361, 5130997, 18286959, 65698315, 237737996, 865865473, 3171845029, 11679264623, 43203976160, 160482694787, 598338596111, 2238318015341, 8398753558740, 31601402932427, 119204300012841, 450692347027373, 1707619054223202, 6482666884435455, 24655044634129524, 93927259816551992, 358394735473678126, 1369528727929732386, 5240588858923364750, 20079465244886219377, 77029014428636135901, 295839832178785412768, 1137448128263682508429, 4377780851428802629698, 16865558933107639306925, 65035488676522762992937, 251005526477737113425586, 969574251939514567062802, 3748234844232061274853704, 14501223284135346501051380, 56143551342298678334652466, 217519781470547960208055153, 843312241764623609388155061, 3271574111220249245470985766, 12699672446898119205008089081, 49327038834218101549234087502, 191700981619993367792963206381, 745419881142150604776157372731, 2900053842359370374632769793740, 11288369340563196274686850747092, 43961093180415323358573686459296, 171281464776255374026042085646808, 667651375697683176383199376254936, 2603633636657760226349041935455181, 10157675563668213067777461719283371, 39644874880222794501720640074271562, 154793324149680896573896926357402636, 604622566327092595123108411818762757, 2362534608230883238709900117915698749, 9234825696624030518255527859289791570, 36110304334477100309427404662312381471, 141247561261854521209495221563234609348, 552680033963381867423470772155047904389, 2163243644028141400437291685182021451413, 8469776605926583352327460123005623131740, 33171836714168797400636206864099357593962, 129955442122238386964931400965132436542886, 509264389233494928949277812285408548355132, 1996239623386418569436590174861437071323874, 7827072102428307067031357476677842865126495, 30697301549981779384212071117469907795326911, 120423790705429720394872285172204119373762908, 472533664518507364335308326719338559111758919, 1854636618446898845634117879931497480100422408, 7280947084915837111167020856571952151354017325, 28590209551940748357675602612687748243358639687, 112290952672453049000277557980328628448914727836, 441131217239253127256023430548129409238086503942, 1733341499888789932683544465129198925147105722914, 6812265950139093733363232476923421907185775268850, 26778611038080868595445906270062138022920307952276, 105286198080535572829332833897883401337521503165093, 414037620336000817675512761230137350150828503157339, 1628512947119820069078697818771387219304330352318088, 6406544253006754511119635766774290088811272054036605, 25207853167224323500572829299803127259595353577701212, 99203176139022852927804761785908054117916655330959411, 390473267697240408948832105519738460316495982088660387, 1537203733893972571433465871981293477656240394193600644, 6052633542435775275210224310893509438687270381401166160, 23835739840731019920086343824597774280699512306152407216, 93882085337099491393202752077628604214006088260013303728, 369832630825594088356966051587891179649318971825915481404, 1457118092641355571041226391968767859242704983553397111973
Generating function in Maple syntax:
1/2*(2*(x-1/2)*(x-1)^4*(1-4*x)^(1/2)-2*x^6+8*x^5-3*x^4-8*x^3+12*x^2-6*x+1)/x/(x^2-3*x+1)/(x-1)^4
Generating function in latex syntax:
\frac{2 \left(x -\frac{1}{2}\right) \left(x -1\right)^{4} \sqrt{1-4 x}-2 x^{6}+8 x^{5}-3 x^{4}-8 x^{3}+12 x^{2}-6 x +1}{2 x \left(x^{2}-3 x +1\right) \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-x**6 + 4*x**5 - 3*x**4/2 - 4*x**3 + 6*x**2 - 3*x + sqrt(1 - 4*x)*(x - 1)**4*(2*x - 1)/2 + 1/2)/(x*(x - 1)**4*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
x*(x^2-3*x+1)^2*(x-1)^8*F(x)^2+(x^2-3*x+1)*(2*x^6-8*x^5+3*x^4+8*x^3-12*x^2+6*x-1)*(x-1)^4*F(x)+x^11-4*x^10-18*x^9+150*x^8-422*x^7+684*x^6-720*x^5+511*x^4-243*x^3+74*x^2-13*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{8} F \! \left(x \right)^{2}+\left(x^{2}-3 x +1\right) \left(2 x^{6}-8 x^{5}+3 x^{4}+8 x^{3}-12 x^{2}+6 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{11}-4 x^{10}-18 x^{9}+150 x^{8}-422 x^{7}+684 x^{6}-720 x^{5}+511 x^{4}-243 x^{3}+74 x^{2}-13 x +1 = 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) = 262
a(7) = 896
a(8) = 3033
a(n+4) = -4*(2*n+3)/(n+5)*a(n)+2*(15*n+28)/(n+5)*a(n+1)-(27*n+77)/(n+5)*a(n+2)+(35+9*n)/(n+5)*a(n+3)+1/2*(n+1)*(n^3-6*n^2-7*n+10)/(n+5), n >= 9
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) = 262
a \! \left(7\right) = 896
a \! \left(8\right) = 3033
a \! \left(n +4\right) = -\frac{4 \left(2 n +3\right) a \! \left(n \right)}{n +5}+\frac{2 \left(15 n +28\right) a \! \left(n +1\right)}{n +5}-\frac{\left(27 n +77\right) a \! \left(n +2\right)}{n +5}+\frac{\left(35+9 n \right) a \! \left(n +3\right)}{n +5}+\frac{\left(n +1\right) \left(n^{3}-6 n^{2}-7 n +10\right)}{2 n +10}, \quad n \geq 9
Specification 1
Strategy pack name: insertion_point_placements
Tree: http://permpal.com/tree/301/
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[8,x]
F[4,x] = F[5,x]+F[9,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[5,x]^2*F[8,x]
F[8,x] = x
F[9,x] = F[10,x]+F[2,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[8,x]
F[12,x] = F[13,x]+F[34,x]
F[13,x] = F[14,x]+F[33,x]
F[14,x] = F[15,x]+F[6,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[8,x]
F[17,x] = F[18,x]+F[30,x]
F[18,x] = F[14,x]+F[19,x]
F[19,x] = F[20,x]*F[23,x]
F[20,x] = F[21,x]
F[21,x] = F[22,x]*F[8,x]
F[22,x] = F[1,x]+F[20,x]
F[23,x] = F[22,x]+F[24,x]
F[24,x] = F[25,x]+F[6,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[8,x]
F[27,x] = F[28,x]+F[30,x]
F[28,x] = F[19,x]+F[29,x]
F[29,x] = F[20,x]+F[25,x]
F[30,x] = F[25,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[6,x]^2*F[20,x]
F[33,x] = F[23,x]*F[6,x]
F[34,x] = F[10,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[5,x]^2*F[37,x]*F[6,x]*F[8,x]
F[37,x] = F[38,x]+F[41,x]
F[38,x] = F[39,x]
F[39,x] = F[40,x]*F[8,x]
F[40,x] = F[1,x]+F[38,x]
F[41,x] = F[40,x]*F[6,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_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{5}\! \left(x \right) = F_{1}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{2}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{34}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{33}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{6}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{8}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{30}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{23}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{8}\! \left(x \right)
F_{22}\! \left(x \right) = F_{1}\! \left(x \right)+F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{22}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{6}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{8}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{19}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{20}\! \left(x \right)+F_{25}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{6} \left(x \right)^{2} F_{20}\! \left(x \right)
F_{33}\! \left(x \right) = F_{23}\! \left(x \right) F_{6}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{5} \left(x \right)^{2} F_{37}\! \left(x \right) F_{6}\! \left(x \right) F_{8}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right) F_{8}\! \left(x \right)
F_{40}\! \left(x \right) = F_{1}\! \left(x \right)+F_{38}\! \left(x \right)
F_{41}\! \left(x \right) = F_{40}\! \left(x \right) F_{6}\! \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)*F_8(x))
Eq(F_4(x), F_5(x) + F_9(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_5(x)**2*F_8(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_2(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_8(x))
Eq(F_12(x), F_13(x) + F_34(x))
Eq(F_13(x), F_14(x) + F_33(x))
Eq(F_14(x), F_15(x) + F_6(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_8(x))
Eq(F_17(x), F_18(x) + F_30(x))
Eq(F_18(x), F_14(x) + F_19(x))
Eq(F_19(x), F_20(x)*F_23(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_22(x)*F_8(x))
Eq(F_22(x), F_1(x) + F_20(x))
Eq(F_23(x), F_22(x) + F_24(x))
Eq(F_24(x), F_25(x) + F_6(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_8(x))
Eq(F_27(x), F_28(x) + F_30(x))
Eq(F_28(x), F_19(x) + F_29(x))
Eq(F_29(x), F_20(x) + F_25(x))
Eq(F_30(x), F_25(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_20(x)*F_6(x)**2)
Eq(F_33(x), F_23(x)*F_6(x))
Eq(F_34(x), F_10(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_37(x)*F_5(x)**2*F_6(x)*F_8(x))
Eq(F_37(x), F_38(x) + F_41(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_40(x)*F_8(x))
Eq(F_40(x), F_1(x) + F_38(x))
Eq(F_41(x), F_40(x)*F_6(x))
Pack JSON:
{"expansion_strats": [[{"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}, {"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": true, "maxreqlen": 1, "one_cell_only": true, "strategy_class": "CellInsertionFactory"}], "iterative": false, "name": "insertion_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": [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, 1, 3, 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": [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": [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_insertion", "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": true, "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], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], 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