0132_0321_1320_3021
Counting sequence:
1, 1, 2, 6, 20, 67, 223, 741, 2471, 8298, 28118, 96233, 332744, 1162148, 4097963, 14579577, 52296194, 188978326, 687461150, 2515802160, 9255958736, 34216550044, 127029001862, 473402339778, 1770322145612, 6640853541964, 24981556514500, 94217100523088, 356171596621968, 1349350352067816, 5122155965629108, 19479547213231309, 74207126356427334, 283140537136505767, 1081938399180317446, 4140054488024219608, 15862645379079773569, 60852396176969237232, 233712409602496772402, 898586842900387167367, 3458494194335531488988, 13324145233737876058176, 51380156385515101610989, 198305764274825008085260, 766021010531517110062680, 2961381483231067124766489, 11457252069057104277711436, 44359251415687262721267238, 171866707786745497708241517, 666331269838651781410350654, 2585040894114660279438573964, 10034877572036369585351904049, 38977458288842035544347087590, 151482234128844675837933790284, 589043262195219478570574112133, 2291717505384485776979577025150, 8920618386979492806041178104016, 34740885298986746392378355353449, 135360304038051945155601533629894, 527641754670357443479260505427596, 2057679020502857507714459621262441, 8027869982917190306456106626053506, 31332941340886916089912204133827192, 122341652867299391336377031870686846, 477875044485939110097653132385668796, 1867307916513913094205435795769059855, 7299180431756140704878573642899289407, 28541977485048897761056534740072736354, 111645509390469801197765799026938016493, 436859058093036861685555708955050205313, 1709937165647580419142592828526965129984, 6695048019125740100542819089160337268127, 26221538202721848589432625636259006016485, 102728253221082038290798821169037443942266, 402573710967261202433936994856964286872641, 1578052066809081521783713065914183675816845, 6187488891420591207549719645722627338831222, 24267309408610759613402047078119827808009969, 95200655837248816048850899853766494193198853, 373565286038794001529596789546253604691862268, 1466218158791920583374122496728921758416136609, 5756169047313522576975392596364708973002120747, 22603141464350113341020733054974371881507091610, 88777299569245637148887547904699596450666948801, 348763123003265520696745368437831730096853522211, 1370415954613407208095623939729732679585201464230, 5385986701941437172802275293480479202900767609217, 21172251716326687300370253258448287860206548981979, 83244523924115964737188035925506665031440153803598, 327362685593818295212871325771614432382065618538673, 1287613854073636750092102942534022654449020095676739, 5065510559160216410203641640000383659593500707448462, 19931507239793780714238981600265871320603550551093365, 78439472389100420734342587122592454946910921176520151, 308748687411977561003907367275593212253520519026787406, 1215485808090632718319625357113671506586984103643502435, 4785941818655007214243777925570947870780783332616168367, 18847604912544297473556026897895966053083518559988719684, 74236021052783193509655557304269763224677139562949865753, 292443208471676432691231156206489482325969291504095437183, 1152219839493208565713671956191677307963313620630036025264
Generating function in Maple syntax:
1/2*((-x^3+5*x^2-4*x+1)*(1-4*x)^(1/2)+3*x^3-5*x^2+4*x-1)/x/(x^2-3*x+1)/(x-1)
Generating function in latex syntax:
\frac{\left(-x^{3}+5 x^{2}-4 x +1\right) \sqrt{1-4 x}+3 x^{3}-5 x^{2}+4 x -1}{2 x \left(x^{2}-3 x +1\right) \left(x -1\right)}
Generating function in sympy syntax:
(3*x**3/2 - 5*x**2/2 + 2*x + sqrt(1 - 4*x)*(-x**3 + 5*x**2 - 4*x + 1)/2 - 1/2)/(x*(x - 1)*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
x*(x-1)^2*(x^2-3*x+1)^2*F(x)^2-(x-1)*(x^2-3*x+1)*(3*x^3-5*x^2+4*x-1)*F(x)+x^6-8*x^5+28*x^4-38*x^3+25*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
x \left(x -1\right)^{2} \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)^{2}-\left(x -1\right) \left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)+x^{6}-8 x^{5}+28 x^{4}-38 x^{3}+25 x^{2}-8 x +1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(5) = 67
a(6) = 223
a(n+7) = 2*(1+2*n)/(n+8)*a(n)-(60+37*n)/(n+8)*a(1+n)+11*(28+11*n)/(n+8)*a(n+2)-4*(157+45*n)/(n+8)*a(n+3)+2*(314+69*n)/(n+8)*a(n+4)-3*(108+19*n)/(n+8)*a(n+5)+2*(41+6*n)/(n+8)*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) = 20
a \! \left(5\right) = 67
a \! \left(6\right) = 223
a \! \left(n +7\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +8}-\frac{\left(60+37 n \right) a \! \left(1+n \right)}{n +8}+\frac{11 \left(28+11 n \right) a \! \left(n +2\right)}{n +8}-\frac{4 \left(157+45 n \right) a \! \left(n +3\right)}{n +8}+\frac{2 \left(314+69 n \right) a \! \left(n +4\right)}{n +8}-\frac{3 \left(108+19 n \right) a \! \left(n +5\right)}{n +8}+\frac{2 \left(41+6 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21266/
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[13,x]*F[4,x]
F[4,x] = F[0,x]+F[5,x]
F[5,x] = F[6,x]
F[6,x] = F[13,x]*F[7,x]
F[7,x] = F[17,x]+F[8,x]
F[8,x] = F[32,x]+F[9,x]
F[9,x] = F[10,x]+F[14,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]^2*F[13,x]
F[13,x] = x
F[14,x] = F[15,x]
F[15,x] = F[13,x]*F[16,x]
F[16,x] = F[17,x]+F[9,x]
F[17,x] = F[18,x]
F[18,x] = F[13,x]*F[19,x]*F[9,x]
F[19,x] = F[20,x]+F[23,x]
F[20,x] = F[1,x]+F[21,x]
F[21,x] = F[22,x]
F[22,x] = F[13,x]*F[20,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[25,x]
F[25,x] = F[13,x]*F[26,x]
F[26,x] = F[1,x]+F[24,x]
F[27,x] = F[28,x]+F[29,x]+F[31,x]
F[28,x] = 0
F[29,x] = F[13,x]*F[30,x]
F[30,x] = F[21,x]+F[27,x]
F[31,x] = F[13,x]*F[23,x]
F[32,x] = F[33,x]
F[33,x] = F[10,x]*F[13,x]*F[20,x]*F[8,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_{13}\! \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_{13}\! \left(x \right) F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{17}\! \left(x \right)+F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{32}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
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)^{2} F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{9}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{13}\! \left(x \right) F_{19}\! \left(x \right) F_{9}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{23}\! \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_{13}\! \left(x \right) F_{20}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{13}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = 0
F_{29}\! \left(x \right) = F_{13}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{21}\! \left(x \right)+F_{27}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right) F_{23}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{20}\! \left(x \right) F_{8}\! \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_13(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_13(x)*F_7(x))
Eq(F_7(x), F_17(x) + F_8(x))
Eq(F_8(x), F_32(x) + F_9(x))
Eq(F_9(x), F_10(x) + F_14(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)**2*F_13(x))
Eq(F_13(x), x)
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_13(x)*F_16(x))
Eq(F_16(x), F_17(x) + F_9(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_13(x)*F_19(x)*F_9(x))
Eq(F_19(x), F_20(x) + F_23(x))
Eq(F_20(x), F_1(x) + F_21(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_13(x)*F_20(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_13(x)*F_26(x))
Eq(F_26(x), F_1(x) + F_24(x))
Eq(F_27(x), F_28(x) + F_29(x) + F_31(x))
Eq(F_28(x), 0)
Eq(F_29(x), F_13(x)*F_30(x))
Eq(F_30(x), F_21(x) + F_27(x))
Eq(F_31(x), F_13(x)*F_23(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_10(x)*F_13(x)*F_20(x)*F_8(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.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, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "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": [0, 3, 2, 1], "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": [0, 3, 2, 1], "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": false, "strategy_class": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 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], 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