0213_0231_0312_1203
Counting sequence:
1, 1, 2, 6, 20, 68, 233, 805, 2807, 9879, 35073, 125513, 452389, 1641029, 5986994, 21954974, 80884424, 299233544, 1111219334, 4140813374, 15478839554, 58028869154, 218123355524, 821908275548, 3104046382352, 11747506651600, 44546351423300, 169227201341652, 643973514492300, 2454475583281868, 9369138618324169, 35813931416918549, 137081932075065263, 525353713400309807, 2015751124008149541, 7742978520054000509, 29774186072954835889, 114606440210718052561, 441564086366680450151, 1702840385183220958191, 6572504892268134874571, 25389097620896782565771, 98153717002173246307991, 379746878184999236074631, 1470264493076253923152271, 5696357496454931166691151, 22084473830579631273028001, 85674824558366902978160921, 332571273253645310076785693, 1291728440650571687591343005, 5019990795652566765574610249, 19519586400396559945255395449, 75939123964633742295524406305, 295583099081022345242026223969, 1151078279633934273339287081093, 4484705974261170028903985793461, 17480744975346274938049889777501, 68167299550918875163057766764253, 265935374100251252826704572507205, 1037896348441217324052190464811877, 4052322377833832889165725753950421, 15827804203709748413903022605791061, 61844179276390474550415490459624562, 241731648034663174453553763280411262, 945191881433949524368025572097182712, 3697036319692422921616961489458597112, 14465398172528314914987682691817256862, 56616872127575565714456218127404465462, 221664526639870969818868730255060873162, 868115630039721289195243320535173168362, 3400841337845262307634299160149779795512, 13326591878262451051725417340414289388512, 52236300463921029167075657836254276221612, 204806200044869594622195474090364872479852, 803208097853041764175883613060708735216392, 3150825719355588051846748377439648934658872, 12363150585438864804401337239737465205859872, 48522108743351554113668766536209704058695392, 190481702695551536648572308856405812844416422, 747941118292358009671637493802091734983245902, 2937512012386005948271613037298890005147672762, 11539516928026141818080270960202279274544597562, 45340788029572782856498041959039615144737335102, 178189356803407511469364401178870887279438203342, 700428157646891111385116132644010213161199046982, 2753800825358868173953738644727964754143743552262, 10828955512911245501993491949149035980653831318562, 42591586429343306278351839513769445156600185789682, 167549223561818274061312005320318729317784487294282, 659235507754171462942804374299177299507150833191882, 2594278539737466584501859445989443080556260333314082, 10211011895061346620794479264241087293022128101102082, 40197172785417983558000388917514049720963994752868882, 158268765232814157223991038379331474943305364426222482, 623255872174088354738765355587317625767974467684667332, 2454751480625186324597424156056685091803019648593553532, 9669794427942852990486560919955224534096961204735920032, 38097293785285059898640505834684015344691782927693020832, 150119408440311693996084645014287156632692002279720846172, 591621693941984491359615634297010081141446158615384660812, 2331925120646357396693534005617323073487777818072950249492
Generating function in Maple syntax:
-1/2*(2*x-1+(-4*x+1)^(1/2))/x^2*(2*x-1)/(x-1)
Generating function in latex syntax:
-\frac{\left(2 x -1+\sqrt{-4 x +1}\right) \left(2 x -1\right)}{2 x^{2} \left(x -1\right)}
Generating function in sympy syntax:
(2*x - 1)*(-x - sqrt(1 - 4*x)/2 + 1/2)/(x**2*(x - 1))
Implicit equation for the generating function in Maple syntax:
x^2*(x-1)^2*F(x)^2+(x-1)*(2*x-1)^2*F(x)+(2*x-1)^2 = 0
Implicit equation for the generating function in latex syntax:
x^{2} \left(x -1\right)^{2} F \! \left(x \right)^{2}+\left(x -1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+\left(2 x -1\right)^{2} = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(n+3) = 4*(3+2*n)/(5+n)*a(n)+(25+7*n)/(5+n)*a(2+n)-2*(16+7*n)/(5+n)*a(n+1), n >= 3
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(n +3\right) = \frac{4 \left(3+2 n \right) a \! \left(n \right)}{5+n}+\frac{\left(25+7 n \right) a \! \left(2+n \right)}{5+n}-\frac{2 \left(16+7 n \right) a \! \left(n +1\right)}{5+n}, \quad n \geq 3
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/21303/
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[16,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[16,x]*F[17,x]*F[7,x]
F[7,x] = F[8,x,1]
F[8,x,k[0]] = F[13,x,k[0]]+F[9,x,k[0]]
F[9,x,k[0]] = F[0,x]+F[10,x,k[0]]
F[10,x,k[0]] = F[11,x,k[0]]
F[11,x,k[0]] = F[12,x,k[0]]*F[8,x,k[0]]
F[12,x,k[0]] = k[0]*x
F[13,x,k[0]] = F[14,x,k[0]]
F[14,x,k[0]] = F[15,x,k[0]]*F[16,x]
F[15,x,k[0]] = 1/(-1+k[0])*(F[8,x,k[0]]*k[0]-F[8,x,1])
F[16,x] = x
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[16,x]*F[20,x]
F[20,x] = F[21,x]+F[24,x]
F[21,x] = F[1,x]+F[22,x]
F[22,x] = F[23,x]
F[23,x] = F[16,x]*F[21,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[26,x]
F[26,x] = F[16,x]*F[27,x]
F[27,x] = F[1,x]+F[25,x]
F[28,x] = F[29,x]+F[30,x]+F[32,x]
F[29,x] = 0
F[30,x] = F[16,x]*F[31,x]
F[31,x] = F[22,x]+F[28,x]
F[32,x] = F[16,x]*F[24,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_{16}\! \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_{16}\! \left(x \right) F_{17}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x , 1\right)
F_{8}\! \left(x , y\right) = F_{13}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{0}\! \left(x \right)+F_{10}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = y x
F_{13}\! \left(x , y\right) = F_{14}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right) F_{16}\! \left(x \right)
F_{15}\! \left(x , y\right) = \frac{y F_{8}\! \left(x , y\right)-F_{8}\! \left(x , 1\right)}{-1+y}
F_{16}\! \left(x \right) = x
F_{17}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{24}\! \left(x \right)
F_{21}\! \left(x \right) = F_{1}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{16}\! \left(x \right) F_{21}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{16}\! \left(x \right) F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{1}\! \left(x \right)+F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = 0
F_{30}\! \left(x \right) = F_{16}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{22}\! \left(x \right)+F_{28}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right) F_{24}\! \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_16(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_16(x)*F_17(x)*F_7(x))
Eq(F_7(x), F_8(x, 1))
Eq(F_8(x, k_0), F_13(x, k_0) + F_9(x, k_0))
Eq(F_9(x, k_0), F_0(x) + F_10(x, k_0))
Eq(F_10(x, k_0), F_11(x, k_0))
Eq(F_11(x, k_0), F_12(x, k_0)*F_8(x, k_0))
Eq(F_12(x, k_0), k_0*x)
Eq(F_13(x, k_0), F_14(x, k_0))
Eq(F_14(x, k_0), F_15(x, k_0)*F_16(x))
Eq(F_15(x, k_0), (k_0*F_8(x, k_0) - F_8(x, 1))/(k_0 - 1))
Eq(F_16(x), x)
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_16(x)*F_20(x))
Eq(F_20(x), F_21(x) + F_24(x))
Eq(F_21(x), F_1(x) + F_22(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_16(x)*F_21(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_16(x)*F_27(x))
Eq(F_27(x), F_1(x) + F_25(x))
Eq(F_28(x), F_29(x) + F_30(x) + F_32(x))
Eq(F_29(x), 0)
Eq(F_30(x), F_16(x)*F_31(x))
Eq(F_31(x), F_22(x) + F_28(x))
Eq(F_32(x), F_16(x)*F_24(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, 2, 1, 3], "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, 2, 0, 3], "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, 1, 3], "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, 2, 0, 3], "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, 1, 3], "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, 2, 0, 3], "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"}}, {"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, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 1], [0, 1], [0, 1], [0, 1]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 1], [0, 1], [0, 0], [0, 1]]}, {"patt": [1, 2, 0, 3], "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, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 2], [0, 2], [0, 0], [0, 2]]}, {"patt": [1, 2, 0, 3], "pos": [[0, 2], [0, 2], [0, 2], [0, 2]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, 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