0132_0213_0321_1302_3021_3120
Counting sequence:
1, 1, 2, 6, 18, 48, 118, 276, 626, 1392, 3054, 6636, 14314, 30696, 65510, 139236, 294882, 622560, 1310686, 2752476, 5767130, 12058584, 25165782, 52428756, 109051858, 226492368, 469761998, 973078476, 2013265866, 4160749512, 8589934534, 17716740036, 36507221954, 75161927616, 154618822590, 317827579836, 652835028922, 1340029796280, 2748779069366, 5634997092276, 11544872091570, 23639499997104, 48378511622062, 98956046499756, 202310139510698, 413416372043688, 844424930131878, 1724034232352676, 3518437208883106, 7177611906121632, 14636698788954014, 29836347531329436, 60798594969501594, 123848989752688536, 252201579132747670, 513410357520236436, 1044835113549954962, 2125699024118874000, 4323455642275676046, 8791026472627208076, 17870283321406128010, 36317027395115679624, 73786976294838206342, 149879795598890106756, 304371277216207601538, 617965926469269979008, 1254378597012249509758, 2545650682171918122876, 5165088340638674452346, 10477750633867025317752, 21250649172913403461494, 43091594156185512574836, 87363779933088436453234, 177088743107611695513456, 358899852698093036240750, 727244438361925362909036, 1473378342655329306673002, 2984535617173615775055720, 6044629098073145873530726, 12240373923598120393899876, 24782979302099898081476450, 50170421514007110750306144, 101549768847628850675318622, 205517389334486959700049756, 415870481947432436098924378, 841412370451781905595498328, 1702167554017397877986295638, 3443020734262463889563189076, 6963412720980264046307573586, 14081567946871200626977537872, 28472620903563746322679856974, 57564211826770182782809276236, 116366363692825745840517676874, 235208607464222252230833602376, 475368975085586025561263701830, 960641470485455093321720397636, 1941089981599476271041826783042, 3921794044456084710880425541440, 7922816251426433759354395033406, 16004088827881396193895877967676, 32325090305819849738165931736890
Generating function in Maple syntax:
-(2*x^5+5*x^3-9*x^2+5*x-1)/(x-1)^2/(2*x-1)^2
Generating function in latex syntax:
-\frac{2 x^{5}+5 x^{3}-9 x^{2}+5 x -1}{\left(x -1\right)^{2} \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**5 - 5*x**3 + 9*x**2 - 5*x + 1)/((x - 1)**2*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x-1)^2*(2*x-1)^2*F(x)+2*x^5+5*x^3-9*x^2+5*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{2} \left(2 x -1\right)^{2} F \! \left(x \right)+2 x^{5}+5 x^{3}-9 x^{2}+5 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,2-2*n+1/4*2^n*n+2^(-1+n))
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2-2 n +\frac{2^{n} n}{4}+2^{-1+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) = 18
a(5) = 48
a(n+2) = -4*a(n)+4*a(n+1)-2*n+6, n >= 6
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) = 18
a \! \left(5\right) = 48
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)-2 n +6, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/22240/
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[10,x]*F[4,x]
F[4,x] = F[20,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[10,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]*F[11,x]*F[17,x]
F[10,x] = x
F[11,x] = F[12,x]+F[15,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[10,x]*F[12,x]
F[15,x] = F[16,x]
F[16,x] = F[10,x]*F[11,x]*F[12,x]
F[17,x] = F[1,x]+F[18,x]
F[18,x] = F[19,x]
F[19,x] = F[10,x]*F[12,x]*F[17,x]
F[20,x] = F[21,x]
F[21,x] = F[10,x]*F[22,x]
F[22,x] = F[23,x]+F[35,x]
F[23,x] = F[17,x]*F[24,x]
F[24,x] = F[25,x]+F[28,x]
F[25,x] = F[1,x]+F[26,x]
F[26,x] = F[27,x]
F[27,x] = F[10,x]*F[24,x]
F[28,x] = F[29,x]+F[33,x]
F[29,x] = F[12,x]*F[30,x]
F[30,x] = F[31,x]
F[31,x] = F[10,x]*F[32,x]
F[32,x] = F[1,x]+F[30,x]
F[33,x] = F[34,x]
F[34,x] = F[10,x]*F[11,x]*F[30,x]
F[35,x] = F[12,x]*F[36,x]
F[36,x] = F[30,x]+F[37,x]
F[37,x] = F[38,x]+F[39,x]+F[42,x]
F[38,x] = 0
F[39,x] = F[10,x]*F[40,x]
F[40,x] = F[13,x]+F[41,x]
F[41,x] = F[39,x]
F[42,x] = F[10,x]*F[43,x]
F[43,x] = F[44,x]+F[47,x]
F[44,x] = F[30,x]+F[45,x]
F[45,x] = F[38,x]+F[39,x]+F[46,x]
F[46,x] = F[10,x]*F[44,x]
F[47,x] = F[48,x]+F[51,x]
F[48,x] = F[49,x]
F[49,x] = F[10,x]*F[50,x]
F[50,x] = F[30,x]+F[48,x]
F[51,x] = 2*F[38,x]+F[52,x]+F[54,x]
F[52,x] = F[10,x]*F[53,x]
F[53,x] = F[45,x]+F[51,x]
F[54,x] = F[10,x]*F[47,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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{20}\! \left(x \right)+F_{5}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{5}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{17}\! \left(x \right)
F_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{15}\! \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_{10}\! \left(x \right) F_{12}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{12}\! \left(x \right)
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_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{10}\! \left(x \right) F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{35}\! \left(x \right)
F_{23}\! \left(x \right) = F_{17}\! \left(x \right) F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{28}\! \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_{10}\! \left(x \right) F_{24}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{33}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right) F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{10}\! \left(x \right) F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{30}\! \left(x \right)
F_{35}\! \left(x \right) = F_{12}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{30}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{42}\! \left(x \right)
F_{38}\! \left(x \right) = 0
F_{39}\! \left(x \right) = F_{10}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{13}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{39}\! \left(x \right)
F_{42}\! \left(x \right) = F_{10}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{47}\! \left(x \right)
F_{44}\! \left(x \right) = F_{30}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{10}\! \left(x \right) F_{44}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{51}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{10}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{30}\! \left(x \right)+F_{48}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{38}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{10}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{45}\! \left(x \right)+F_{51}\! \left(x \right)
F_{54}\! \left(x \right) = F_{10}\! \left(x \right) F_{47}\! \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_10(x)*F_4(x))
Eq(F_4(x), F_20(x) + F_5(x))
Eq(F_5(x), F_1(x) + F_6(x))
Eq(F_6(x), F_7(x))
Eq(F_7(x), F_10(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x)*F_11(x)*F_17(x))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_15(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_10(x)*F_12(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_10(x)*F_11(x)*F_12(x))
Eq(F_17(x), F_1(x) + F_18(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_10(x)*F_12(x)*F_17(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_10(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_35(x))
Eq(F_23(x), F_17(x)*F_24(x))
Eq(F_24(x), F_25(x) + F_28(x))
Eq(F_25(x), F_1(x) + F_26(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_10(x)*F_24(x))
Eq(F_28(x), F_29(x) + F_33(x))
Eq(F_29(x), F_12(x)*F_30(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_10(x)*F_32(x))
Eq(F_32(x), F_1(x) + F_30(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_10(x)*F_11(x)*F_30(x))
Eq(F_35(x), F_12(x)*F_36(x))
Eq(F_36(x), F_30(x) + F_37(x))
Eq(F_37(x), F_38(x) + F_39(x) + F_42(x))
Eq(F_38(x), 0)
Eq(F_39(x), F_10(x)*F_40(x))
Eq(F_40(x), F_13(x) + F_41(x))
Eq(F_41(x), F_39(x))
Eq(F_42(x), F_10(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_47(x))
Eq(F_44(x), F_30(x) + F_45(x))
Eq(F_45(x), F_38(x) + F_39(x) + F_46(x))
Eq(F_46(x), F_10(x)*F_44(x))
Eq(F_47(x), F_48(x) + F_51(x))
Eq(F_48(x), F_49(x))
Eq(F_49(x), F_10(x)*F_50(x))
Eq(F_50(x), F_30(x) + F_48(x))
Eq(F_51(x), 2*F_38(x) + F_52(x) + F_54(x))
Eq(F_52(x), F_10(x)*F_53(x))
Eq(F_53(x), F_45(x) + F_51(x))
Eq(F_54(x), F_10(x)*F_47(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, 2, 1, 3], "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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "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, 2, 1, 3], "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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "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, 2, 1, 3], "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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "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": 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