0231_0321_1032_1203_3021
Counting sequence:
1, 1, 2, 6, 19, 54, 144, 375, 972, 2524, 6573, 17156, 44842, 117301, 306974, 803514, 2103439, 5506650, 14416332, 37742139, 98809848, 258687136, 677251257, 1773066296, 4641947254, 12152775049, 31816377434, 83296356750, 218072692267, 570921719454, 1494692465448, 3913155676191, 10244774562372, 26821168010116, 70218729467109, 183835020390284, 481286331702754, 1260023974716925, 3298785592446902, 8636332802622594, 22610212815419623, 59194305643634946, 154972704115483812, 405723806702815011, 1062198715992959664, 2780872341276062344, 7280418307835225649, 19060382582229612800, 49900729438853610862, 130641805734331217809, 342024687764140040498, 895432257558088901526, 2344272084910126661827, 6137383997172291081606, 16067879906606746580544, 42066255722647948657479, 110130887261337099389244, 288326406061363349507500, 754848330922752949130397, 1976218586706895497880724, 5173807429197933544508698, 13545203700886905135642181, 35461803673462781862414542, 92840207319501440451598026, 243058818285041539492375999, 636336247535623178025526314, 1665949924321827994584199164, 4361513525429860805727067275, 11418590651967754422596998632, 29894258430473402462063924464, 78264184639452452963594770473, 204898295487883956428720382536, 536430701824199416322566372582, 1404393809984714292538978730521, 3676750728129943461294369814154, 9625858374405116091344130706974, 25200824395085404812738022301659, 65976614810851098346869936192750, 172729020037467890227871786271192, 452210445301552572336745422615279, 1183902315867189826782364481568948, 3099496502300016908010348022085716, 8114587191032860897248679584682197, 21244265070798565783735690731954716, 55618208021362836453958392611175634, 145610358993289943578139487101565709, 381212868958506994280460068693514854, 998028247882231039263240718978972050, 2612871874688186123509262088243394327, 6840587376182327331264545545751203794, 17908890253858795870284374549010209748, 46886083385394060279588578101279417971, 122749359902323384968481359754828036512, 321361996321576094625855501163204683736, 841336629062404898909085143734786006689, 2202647890865638602101399930041153328144, 5766607043534510907395114646388673969374, 15097173239737894120083944009124868571425, 39524912675679171452856717380985931736162, 103477564787299620238486208133832926628134, 270907781686219689262601907020512848139123
Generating function in Maple syntax:
(3*x^5-2*x^4+6*x^3-9*x^2+5*x-1)/(x^2-3*x+1)/(x-1)^3
Generating function in latex syntax:
\frac{3 x^{5}-2 x^{4}+6 x^{3}-9 x^{2}+5 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(3*x**5 - 2*x**4 + 6*x**3 - 9*x**2 + 5*x - 1)/((x - 1)**3*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^3*F(x)-3*x^5+2*x^4-6*x^3+9*x^2-5*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)-3 x^{5}+2 x^{4}-6 x^{3}+9 x^{2}-5 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/10*(31*5^(1/2)-65)*(3/2-1/2*5^(1/2))^(-n)+1/10*(-31*5^(1/2)-65)*(3/2+1/2*5^(1/2))^(-n)+n^2-7*n+11)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(31 \sqrt{5}-65\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(-31 \sqrt{5}-65\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+n^{2}-7 n +11 & \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) = 54
a(n+2) = -n^2-a(n)+3*a(n+1)+5*n-3, 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) = 19
a \! \left(5\right) = 54
a \! \left(n +2\right) = -n^{2}-a \! \left(n \right)+3 a \! \left(n +1\right)+5 n -3, \quad n \geq 6
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20763/
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[19,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[10,x] = x
F[11,x] = F[12,x]+F[13,x]+F[17,x]
F[12,x] = 0
F[13,x] = F[10,x]*F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[10,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[10,x]*F[18,x]
F[18,x] = F[9,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[12,x]+F[21,x]+F[41,x]
F[21,x] = F[10,x]*F[22,x]
F[22,x] = F[23,x]+F[28,x]
F[23,x] = F[14,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[10,x]*F[26,x]
F[26,x] = F[10,x]+F[27,x]
F[27,x] = F[25,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[12,x]+F[21,x]+F[30,x]
F[30,x] = F[10,x]*F[31,x]
F[31,x] = F[2,x]+F[29,x]
F[32,x] = F[33,x]
F[33,x] = F[10,x]*F[34,x]
F[34,x] = F[35,x]+F[40,x]
F[35,x] = F[36,x]
F[36,x] = F[10,x]*F[37,x]
F[37,x] = F[38,x]+F[39,x]
F[38,x] = F[10,x]
F[39,x] = F[35,x]
F[40,x] = F[33,x]
F[41,x] = F[10,x]*F[42,x]
F[42,x] = F[19,x]+F[43,x]
F[43,x] = F[35,x]+F[44,x]
F[44,x] = F[45,x]
F[45,x] = F[10,x]*F[46,x]
F[46,x] = F[43,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_{19}\! \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_{10}\! \left(x \right) = x
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{17}\! \left(x \right)
F_{12}\! \left(x \right) = 0
F_{13}\! \left(x \right) = F_{10}\! \left(x \right) F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{10}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right) F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{9}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{12}\! \left(x \right)+F_{21}\! \left(x \right)+F_{41}\! \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_{28}\! \left(x \right)
F_{23}\! \left(x \right) = F_{14}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{10}\! \left(x \right) F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{10}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{25}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{12}\! \left(x \right)+F_{21}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{10}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{10}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{40}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{10}\! \left(x \right) F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{39}\! \left(x \right)
F_{38}\! \left(x \right) = F_{10}\! \left(x \right)
F_{39}\! \left(x \right) = F_{35}\! \left(x \right)
F_{40}\! \left(x \right) = F_{33}\! \left(x \right)
F_{41}\! \left(x \right) = F_{10}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{19}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{35}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{10}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{43}\! \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_19(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))
Eq(F_10(x), x)
Eq(F_11(x), F_12(x) + F_13(x) + F_17(x))
Eq(F_12(x), 0)
Eq(F_13(x), F_10(x)*F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_10(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_10(x)*F_18(x))
Eq(F_18(x), F_9(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_12(x) + F_21(x) + F_41(x))
Eq(F_21(x), F_10(x)*F_22(x))
Eq(F_22(x), F_23(x) + F_28(x))
Eq(F_23(x), F_14(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_10(x)*F_26(x))
Eq(F_26(x), F_10(x) + F_27(x))
Eq(F_27(x), F_25(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_12(x) + F_21(x) + F_30(x))
Eq(F_30(x), F_10(x)*F_31(x))
Eq(F_31(x), F_2(x) + F_29(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_10(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_40(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_10(x)*F_37(x))
Eq(F_37(x), F_38(x) + F_39(x))
Eq(F_38(x), F_10(x))
Eq(F_39(x), F_35(x))
Eq(F_40(x), F_33(x))
Eq(F_41(x), F_10(x)*F_42(x))
Eq(F_42(x), F_19(x) + F_43(x))
Eq(F_43(x), F_35(x) + F_44(x))
Eq(F_44(x), F_45(x))
Eq(F_45(x), F_10(x)*F_46(x))
Eq(F_46(x), F_43(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": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "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": [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, 2, 0, 3], "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, 2, 3, 1], "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, 2, 0, 3], "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, 2, 3, 1], "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, 2, 0, 3], "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": true, "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, 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