0231_1203_1302
Counting sequence:
1, 1, 2, 6, 21, 77, 286, 1067, 3993, 14992, 56488, 213600, 810449, 3084733, 11774727, 45061101, 172844990, 664377328, 2558511483, 9869553455, 38131001344, 147526844729, 571511352982, 2216637220718, 8606777052693, 33452404848298, 130143807289744, 506758281194538, 1974849699700206, 7701941830782075, 30059290472048512, 117395306838614221, 458774376859230391, 1793941809972945579, 7018818519182675308, 27475967490686797535, 107612617286070421997, 421680435728404978776, 1653116448510572923712, 6483561257643354868450, 25439296119771404540033, 99854952539847603283407, 392102386409927126721243, 1540236965250493158816795, 6052392321647575831811577, 23790950557319364623197844, 93548283508201501893478228, 367954086328961840347329077, 1447704817285191745611007485, 5697575632357323146604439884, 22429482730358722488432614669, 88320830866510349448007382302, 347870541034250018104776944392, 1370499529008861304724639287213, 5400614516849606092701196958192, 21286635864174623429355653170988, 83920320970022583674467688169928, 330917970958646859099636607497468, 1305159816092923710244194335235365, 5148661250982352709950614966746124, 20314658333703461441488827155012739, 80169048081101152643299160860982754, 316434242524152864248524964322094589, 1249215653872718277827800582234101957, 4932491178919187197646964069626941866, 19479061590629759752924431748339505798, 76937932164940086273567725516809538244, 303935754892600359669568074957860812679, 1200853431917116055673739100930938061943, 4745296358425508432708997611754278277035, 18754266302450746372312864609914846003761, 74130778493495724814185700812788685132111, 293060440280411141749106745761147811353062, 1158709215973502274972626022144608130639870, 4581934567190285505788890010651236622210947, 18120870244192634929000729949763548826372503, 71674323951566065412014805089875283942714832, 283531449849275661907863302090460982168801629, 1121736084238928924100060215334499748014232880, 4438444018198999002069302889594239845009552538, 17563870874253838680142664846223075305695769519, 69511718962573120990083133113725453878703590594, 275133222454670142959747303116001362772395891378, 1089116161468104150764308186419354815710568399464, 4311719653127270786003728583387548972863756978288, 17071472820793649691317244203560233477862580699483, 67598129679004200175896519173729383108481431192434, 267695249428851742255700216712203726183608734250331, 1060200623813507542220586174766703701591992108167580, 4199291220985716921399895202998525205693781864761141, 16634266215325964516246158341692322590958532338674881, 65897692930125613718239086063547372414895279743403233, 261080779576167629836235998283653784907073554920381156, 1034467765378444852342129577327873944950549364624963589, 4099167463709471217904150506516994540158561810065746855, 16244647239511753855424429530169028588167914689671753920, 64381359550795224743780684387218535943703926004211162698, 255178785840203393016493855321179834142510531035557576018, 1011493051749473213714831026463800160375454129058579172375, 4009724397553917967533585349362665610681206558496589209673, 15896401037069688634244138761309753050072073371274686558501
Generating function in Maple syntax:
-4*(x-1/2)*((x-1/2)*(-4*x+1)^(1/2)+x^3-4*x^2+3*x-1/2)/(2*x^5-16*x^4+44*x^3-42*x^2+16*x-2)
Generating function in latex syntax:
-\frac{4 \left(x -\frac{1}{2}\right) \left(\left(x -\frac{1}{2}\right) \sqrt{-4 x +1}+x^{3}-4 x^{2}+3 x -\frac{1}{2}\right)}{2 x^{5}-16 x^{4}+44 x^{3}-42 x^{2}+16 x -2}
Generating function in sympy syntax:
(2 - 4*x)*(x**3 - 4*x**2 + 3*x + sqrt(1 - 4*x)*(x - 1/2) - 1/2)/(2*x**5 - 16*x**4 + 44*x**3 - 42*x**2 + 16*x - 2)
Implicit equation for the generating function in Maple syntax:
(x^5-8*x^4+22*x^3-21*x^2+8*x-1)*F(x)^2+(2*x-1)*(2*x^3-8*x^2+6*x-1)*F(x)+x*(2*x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}-8 x^{4}+22 x^{3}-21 x^{2}+8 x -1\right) F \! \left(x \right)^{2}+\left(2 x -1\right) \left(2 x^{3}-8 x^{2}+6 x -1\right) F \! \left(x \right)+x \left(2 x -1\right)^{2} = 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(n+7) = 4*(2*n+5)/(n+7)*a(n)-10*(7*n+19)/(n+7)*a(n+1)+3*(75*n+229)/(n+7)*a(n+2)-4*(77*n+284)/(n+7)*a(n+3)+4*(53*n+236)/(n+7)*a(n+4)-11*(7*n+37)/(n+7)*a(n+5)+2*(7*n+43)/(n+7)*a(n+6), n >= 7
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(n +7\right) = \frac{4 \left(2 n +5\right) a \! \left(n \right)}{n +7}-\frac{10 \left(7 n +19\right) a \! \left(n +1\right)}{n +7}+\frac{3 \left(75 n +229\right) a \! \left(n +2\right)}{n +7}-\frac{4 \left(77 n +284\right) a \! \left(n +3\right)}{n +7}+\frac{4 \left(53 n +236\right) a \! \left(n +4\right)}{n +7}-\frac{11 \left(7 n +37\right) a \! \left(n +5\right)}{n +7}+\frac{2 \left(7 n +43\right) a \! \left(n +6\right)}{n +7}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/360/
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[7,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[7,x]*F[8,x]
F[7,x] = x
F[8,x] = F[15,x]+F[9,x]
F[9,x] = F[10,x]+F[4,x]
F[10,x] = F[11,x]
F[11,x] = F[0,x]*F[12,x]*F[13,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]^2*F[7,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]+F[19,x]
F[17,x] = F[18,x]
F[18,x] = F[7,x]*F[8,x]
F[19,x] = F[20,x]
F[20,x] = F[13,x]^2*F[0,x]*F[12,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_{7}\! \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_{7}\! \left(x \right) F_{8}\! \left(x \right)
F_{7}\! \left(x \right) = x
F_{8}\! \left(x \right) = F_{15}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{4}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{0}\! \left(x \right) F_{12}\! \left(x \right) F_{13}\! \left(x \right)
F_{12}\! \left(x \right) = F_{1}\! \left(x \right)+F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{12} \left(x \right)^{2} F_{7}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{19}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{7}\! \left(x \right) F_{8}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13} \left(x \right)^{2} F_{0}\! \left(x \right) F_{12}\! \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_7(x))
Eq(F_4(x), F_0(x) + F_5(x))
Eq(F_5(x), F_6(x))
Eq(F_6(x), F_7(x)*F_8(x))
Eq(F_7(x), x)
Eq(F_8(x), F_15(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_4(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_0(x)*F_12(x)*F_13(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)**2*F_7(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x) + F_19(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_7(x)*F_8(x))
Eq(F_19(x), F_20(x))
Eq(F_20(x), F_0(x)*F_12(x)*F_13(x)**2)
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.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}], "iterative": false, "name": "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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 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, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 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, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 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, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 3, 0, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 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": 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