0213_0231_0312_0321_1032_1302_2031_3120
Counting sequence:
1, 1, 2, 6, 16, 40, 97, 227, 518, 1162, 2575, 5653, 12316, 26660, 57389, 122935, 262210, 557134, 1179739, 2490473, 5243000, 11010184, 23068825, 48234667, 100663486, 209715410, 436207847, 905969917, 1879048468, 3892314412, 8053064005, 16642998623, 34359738746, 70866960790, 146028888499, 300647711185, 618475291120, 1271310320144, 2611340116529, 5360119186003, 10995116278390, 22539988370074, 46179488367295, 94557999989477, 193514046489356, 395824186000180, 809240558043997, 1653665488176007, 3377699720528818, 6896136929412062, 14073748835533835, 28710447624487993, 58546795155817576, 119345390125319320, 243194379878008009, 495395959010755835, 1008806316530992430, 2053641430080947554, 4179340454199821719, 8502796096475497933, 17293822569102706180, 35164105890508834364, 71481133285624514165, 145268109580462720687, 295147905179352827626, 599519182395560429350, 1217485108864830408547, 2471863705877079918497, 5017514388048998041568, 10182602728687672494112, 20660353362554697812065, 41911002535468101273763, 85002596691653613848806, 172366376624742050302250, 349455119732353745815919, 708354972430446782056885, 1435599410792372144966140, 2908977753447701451639364, 5893513370621317226695309, 11938142468694463100226263, 24178516392292583494126370, 48961495694392481575603054, 99131917208399592325909435, 200681686056028443001228297, 406199075390515402701278296, 822069557337947838800202920, 1663481927789729744395701497, 3365649481807127622381997387, 6808670216069591511945186718, 13772082937049855558252760562, 27853650883921056185230298695, 56326271787484802507910155933, 113890483614254985290719432436, 230256847307080731131237109580, 465465454771302983362070712229, 940834429856889008923334414335, 1901475900342344102245054812250, 3842565881941820373286881595574, 7764359926397905084167307137299, 15687176177824338843521702170993, 31691265005705735037417580138960
Generating function in Maple syntax:
-(4*x^8-4*x^7-x^6+4*x^5+3*x^4-14*x^3+14*x^2-6*x+1)/(x-1)^3/(2*x-1)^2
Generating function in latex syntax:
-\frac{4 x^{8}-4 x^{7}-x^{6}+4 x^{5}+3 x^{4}-14 x^{3}+14 x^{2}-6 x +1}{\left(x -1\right)^{3} \left(2 x -1\right)^{2}}
Generating function in sympy syntax:
(-4*x**8 + 4*x**7 + x**6 - 4*x**5 - 3*x**4 + 14*x**3 - 14*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(x-1)^3*(2*x-1)^2*F(x)+4*x^8-4*x^7-x^6+4*x^5+3*x^4-14*x^3+14*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(x -1\right)^{3} \left(2 x -1\right)^{2} F \! \left(x \right)+4 x^{8}-4 x^{7}-x^{6}+4 x^{5}+3 x^{4}-14 x^{3}+14 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,n = 3,6,10-9/2*n+1/2*n^2+1/4*2^n*n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ 10-\frac{9 n}{2}+\frac{n^{2}}{2}+\frac{2^{n} n}{4} & \mathit{\text{otherwise}} \end{array}\right.
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 16
a(5) = 40
a(6) = 97
a(7) = 227
a(8) = 518
a(n+2) = 1/2*n^2-4*a(n)+4*a(n+1)-13/2*n+19, 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) = 16
a \! \left(5\right) = 40
a \! \left(6\right) = 97
a \! \left(7\right) = 227
a \! \left(8\right) = 518
a \! \left(n +2\right) = \frac{n^{2}}{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{13 n}{2}+19, \quad n \geq 9
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/23483/
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[4,x]*F[8,x]
F[4,x] = F[33,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[8,x]*F[9,x]
F[8,x] = x
F[9,x] = F[10,x]+F[15,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[8,x]
F[13,x] = F[10,x]+F[14,x]
F[14,x] = F[8,x]
F[15,x] = F[16,x]+F[19,x]
F[16,x] = F[17,x]
F[17,x] = F[18,x]*F[8,x]
F[18,x] = F[1,x]+F[16,x]
F[19,x] = F[20,x]+F[21,x]+F[29,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[8,x]
F[22,x] = F[23,x]+F[26,x]
F[23,x] = F[24,x]
F[24,x] = F[25,x]*F[8,x]
F[25,x] = F[1,x]+F[23,x]
F[26,x] = F[20,x]+F[21,x]+F[27,x]
F[27,x] = F[28,x]*F[8,x]
F[28,x] = F[16,x]+F[26,x]
F[29,x] = F[30,x]*F[8,x]
F[30,x] = F[15,x]+F[31,x]
F[31,x] = F[32,x]
F[32,x] = F[16,x]*F[8,x]
F[33,x] = F[34,x]
F[34,x] = F[35,x]*F[8,x]
F[35,x] = F[36,x]+F[45,x]
F[36,x] = F[37,x]+F[44,x]
F[37,x] = F[38,x]*F[41,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[18,x]*F[5,x]
F[40,x] = F[21,x]
F[41,x] = F[1,x]+F[42,x]
F[42,x] = F[43,x]
F[43,x] = F[25,x]*F[41,x]*F[8,x]
F[44,x] = F[25,x]*F[5,x]*F[8,x]
F[45,x] = F[46,x]
F[46,x] = F[25,x]*F[40,x]*F[41,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_{4}\! \left(x \right) F_{8}\! \left(x \right)
F_{4}\! \left(x \right) = F_{33}\! \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_{8}\! \left(x \right) F_{9}\! \left(x \right)
F_{8}\! \left(x \right) = x
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{15}\! \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_{8}\! \left(x \right)
F_{13}\! \left(x \right) = F_{10}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{8}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)+F_{19}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right) F_{8}\! \left(x \right)
F_{18}\! \left(x \right) = F_{1}\! \left(x \right)+F_{16}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{29}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{8}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{26}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{8}\! \left(x \right)
F_{25}\! \left(x \right) = F_{1}\! \left(x \right)+F_{23}\! \left(x \right)
F_{26}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right) F_{8}\! \left(x \right)
F_{28}\! \left(x \right) = F_{16}\! \left(x \right)+F_{26}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{8}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{16}\! \left(x \right) F_{8}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{8}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{45}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)+F_{44}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{41}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{18}\! \left(x \right) F_{5}\! \left(x \right)
F_{40}\! \left(x \right) = F_{21}\! \left(x \right)
F_{41}\! \left(x \right) = F_{1}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{25}\! \left(x \right) F_{41}\! \left(x \right) F_{8}\! \left(x \right)
F_{44}\! \left(x \right) = F_{25}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{25}\! \left(x \right) F_{40}\! \left(x \right) F_{41}\! \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_4(x)*F_8(x))
Eq(F_4(x), F_33(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_8(x)*F_9(x))
Eq(F_8(x), x)
Eq(F_9(x), F_10(x) + F_15(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_8(x))
Eq(F_13(x), F_10(x) + F_14(x))
Eq(F_14(x), F_8(x))
Eq(F_15(x), F_16(x) + F_19(x))
Eq(F_16(x), F_17(x))
Eq(F_17(x), F_18(x)*F_8(x))
Eq(F_18(x), F_1(x) + F_16(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_29(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_8(x))
Eq(F_22(x), F_23(x) + F_26(x))
Eq(F_23(x), F_24(x))
Eq(F_24(x), F_25(x)*F_8(x))
Eq(F_25(x), F_1(x) + F_23(x))
Eq(F_26(x), F_20(x) + F_21(x) + F_27(x))
Eq(F_27(x), F_28(x)*F_8(x))
Eq(F_28(x), F_16(x) + F_26(x))
Eq(F_29(x), F_30(x)*F_8(x))
Eq(F_30(x), F_15(x) + F_31(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_16(x)*F_8(x))
Eq(F_33(x), F_34(x))
Eq(F_34(x), F_35(x)*F_8(x))
Eq(F_35(x), F_36(x) + F_45(x))
Eq(F_36(x), F_37(x) + F_44(x))
Eq(F_37(x), F_38(x)*F_41(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_18(x)*F_5(x))
Eq(F_40(x), F_21(x))
Eq(F_41(x), F_1(x) + F_42(x))
Eq(F_42(x), F_43(x))
Eq(F_43(x), F_25(x)*F_41(x)*F_8(x))
Eq(F_44(x), F_25(x)*F_5(x)*F_8(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_25(x)*F_40(x)*F_41(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, 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": [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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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, 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": [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, 0, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 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, 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": [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, 0, 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