0132_0321_1032_1320_2301
Counting sequence:
1, 1, 2, 6, 19, 53, 139, 354, 890, 2228, 5579, 14007, 35301, 89348, 227124, 579750, 1485517, 3819473, 9850039, 25468518, 65998478, 171347288, 445553039, 1160068971, 3023643849, 7887793928, 20591503464, 53786053194, 140556940945, 367448562053, 960882775579, 2513320716522, 6575187059906, 17204187399548, 45020732140499, 117823649283615, 308379348749997, 807168368078348, 2112825107774364, 5530688479954158, 14477969021768533, 37900607245235513, 99218492594752639, 259743875422744686, 679990593685112054, 1780181726144224928, 4660460026747574039, 12201004840052008659, 31942158669222452625, 83624661927057306128, 218930173446461290704, 573162480712606038162, 1500550372554427412761, 3928474563201840667373, 10284844606603470102499, 26926000709813413823730, 70493038177446646050602, 184552870628146646321348, 483165078311034282158939, 1264941355498639669164423, 3311656934543454644388213, 8670025268791270064179988, 22698410369034259072655364, 59425188544488938051081526, 155577120100326664571756509, 407306100275357770039675553, 1066341035457637065084551239, 2791716710949648245861152374, 7308808497872125276938478430, 19134707565181618720123876328, 50095311725809025006353234079, 131151222594731068249937786427, 343358345875781451055787633241, 898923794372259922362727303448, 2353412995329995780564293005624, 6161315106615130727676537866970, 16130532152149019777723270295457, 42230281000376808873139527205973, 110560310140626434411248529268139, 289450647985903083568233915634938, 757791630908105062845751766000018, 1983924238844898734347704155672588, 5193981073688448671502897600794339, 13598018958041930887868405152586991, 35600075751475848297709836281366589, 93202208197253696796861511365606428, 244006548639603556036846254814227564, 638817437315357895923161850375801214, 1672445762484400574394691457512976485, 4378519848474361899471182777767430153, 11463113779573035642211729253407320079, 30010821483436074811094413470509346942, 78569350656963105854021655599867963846, 205697230459599591867049497143864249648, 538522340665509397959642033323814632999, 1409869791423038118397621617536860220835, 3691087033373348109926142088155528923937, 9663391308231540756609501663567655842848, 25299086890380439730045473893624104194464, 66233869361008302533184575915059601932578, 173402521188801901987566433478267820012073
Generating function in Maple syntax:
(2*x^8-10*x^7+14*x^6-11*x^5+25*x^4-37*x^3+25*x^2-8*x+1)/(x^2-3*x+1)/(2*x-1)^2/(x-1)^2
Generating function in latex syntax:
\frac{2 x^{8}-10 x^{7}+14 x^{6}-11 x^{5}+25 x^{4}-37 x^{3}+25 x^{2}-8 x +1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{2}}
Generating function in sympy syntax:
(2*x**8 - 10*x**7 + 14*x**6 - 11*x**5 + 25*x**4 - 37*x**3 + 25*x**2 - 8*x + 1)/((x - 1)**2*(2*x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
-(x^2-3*x+1)*(2*x-1)^2*(x-1)^2*F(x)+2*x^8-10*x^7+14*x^6-11*x^5+25*x^4-37*x^3+25*x^2-8*x+1 = 0
Implicit equation for the generating function in latex syntax:
-\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)+2 x^{8}-10 x^{7}+14 x^{6}-11 x^{5}+25 x^{4}-37 x^{3}+25 x^{2}-8 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/40*(-4*5^(1/2)+20)*(3/2-1/2*5^(1/2))^(-n)+1/40*(4*5^(1/2)+20)*(3/2+1/2*5^(1/2))^(-n)+1/40*(5*n+5)*2^n-n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(-4 \sqrt{5}+20\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{40}+\frac{\left(4 \sqrt{5}+20\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{40}+\frac{\left(5 n +5\right) 2^{n}}{40}-\\n & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(5) = 53
a(6) = 139
a(7) = 354
a(8) = 890
a(n+4) = -4*a(n)+16*a(n+1)-17*a(n+2)+7*a(n+3)-1+n, 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) = 19
a \! \left(5\right) = 53
a \! \left(6\right) = 139
a \! \left(7\right) = 354
a \! \left(8\right) = 890
a \! \left(n +4\right) = -4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-1+n, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/21750/
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[19,x]+F[8,x]
F[8,x] = F[18,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[13,x]*F[9,x]
F[13,x] = x
F[14,x] = F[11,x]*F[15,x]
F[15,x] = F[1,x]+F[16,x]
F[16,x] = F[17,x]
F[17,x] = F[13,x]*F[15,x]
F[18,x] = F[15,x]*F[2,x]
F[19,x] = F[15,x]*F[20,x]
F[20,x] = F[21,x]
F[21,x] = F[13,x]*F[22,x]
F[22,x] = F[23,x]+F[28,x]
F[23,x] = F[24,x]+F[26,x]
F[24,x] = F[15,x]*F[25,x]
F[25,x] = F[1,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[13,x]*F[25,x]
F[28,x] = F[16,x]+F[29,x]
F[29,x] = F[30,x]+F[31,x]+F[33,x]
F[30,x] = 0
F[31,x] = F[13,x]*F[32,x]
F[32,x] = F[26,x]+F[29,x]
F[33,x] = F[13,x]*F[28,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_{19}\! \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_{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_{13}\! \left(x \right) F_{9}\! \left(x \right)
F_{13}\! \left(x \right) = x
F_{14}\! \left(x \right) = F_{11}\! \left(x \right) F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{13}\! \left(x \right) F_{15}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right) F_{2}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{13}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{26}\! \left(x \right)
F_{24}\! \left(x \right) = F_{15}\! \left(x \right) F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{13}\! \left(x \right) F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = 0
F_{31}\! \left(x \right) = F_{13}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{26}\! \left(x \right)+F_{29}\! \left(x \right)
F_{33}\! \left(x \right) = F_{13}\! \left(x \right) F_{28}\! \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_19(x) + F_8(x))
Eq(F_8(x), F_18(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_13(x)*F_9(x))
Eq(F_13(x), x)
Eq(F_14(x), F_11(x)*F_15(x))
Eq(F_15(x), F_1(x) + F_16(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_13(x)*F_15(x))
Eq(F_18(x), F_15(x)*F_2(x))
Eq(F_19(x), F_15(x)*F_20(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_13(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_28(x))
Eq(F_23(x), F_24(x) + F_26(x))
Eq(F_24(x), F_15(x)*F_25(x))
Eq(F_25(x), F_1(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_13(x)*F_25(x))
Eq(F_28(x), F_16(x) + F_29(x))
Eq(F_29(x), F_30(x) + F_31(x) + F_33(x))
Eq(F_30(x), 0)
Eq(F_31(x), F_13(x)*F_32(x))
Eq(F_32(x), F_26(x) + F_29(x))
Eq(F_33(x), F_13(x)*F_28(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, 0, 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": [2, 3, 0, 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, 0, 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": [2, 3, 0, 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, 0, 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": [2, 3, 0, 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], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 1], [0, 1]]}, {"patt": [0, 2, 1], "pos": [[0, 1], [0, 1], [0, 1]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 1]]}, {"patt": [1, 2, 0], "pos": [[0, 1], [0, 1], [0, 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, 1], [0, 0], [0, 0]]}, {"patt": [1, 0, 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": [1, 3, 2, 0], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "pos": [[0, 0], [0, 1], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 1], "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, 0], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 2], [0, 2]]}, {"patt": [0, 2, 1], "pos": [[0, 2], [0, 2], [0, 2]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [0, 2]]}, {"patt": [1, 2, 0], "pos": [[0, 2], [0, 2], [0, 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, 2], [0, 0], [0, 0]]}, {"patt": [1, 0, 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]]}, 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