0213_0231_1032_1302_3120
Counting sequence:
1, 1, 2, 6, 19, 54, 142, 354, 849, 1978, 4508, 10102, 22343, 48910, 106186, 228986, 491037, 1047986, 2227512, 4717742, 9960467, 20970342, 44038822, 92273106, 192936169, 402651114, 838858452, 1744827814, 3623875679, 7516189438, 15569252738, 32212250602, 66571988533, 137438948450, 283467836016, 584115546206, 1202590836267, 2473901155286, 5085241270622, 10445360455362, 21440476732417, 43980465101082, 90159953466892, 184717953454806, 378231999942519, 774056185940974, 1583296743983162, 3236962232157274, 6614661952684109, 13510798882094098, 27584547717625768, 56294995342111502, 114841790497926723, 234187180623243590, 477381560501249046, 972777519512002226, 1981583836042991897, 4035225266123936586, 8214565720323755332, 16717361816799250182, 34011184385901953167, 69175290276410784222, 140656423562035294962, 285924533142498012106, 581072438321850836069, 1180591620717411261634, 2398076729582241666272, 4869940435459321580734, 9887454823508319618139, 20070057552195992107958, 40730410914750689915598, 82641413450218791184802, 167644010141872405028913, 340010386766614455326330, 689465506498968201137276, 1397820478929414983189046, 2833419889721787128149927, 5742397643169488579783886, 11635911013790805806473642, 23574053482485268906694202, 47752569874777852400814717, 96714065569170333976411762, 195845982777569926302315032, 396527668833598369303537006, 802726744224113772004808819, 1624796301562061610805005094, 3288278229351791355200699782, 6653927711158918977582690194, 13462597927228510489527869769, 27234680864278366047780623018, 55088331748199422233010914228, 111414603535684224740921062502, 225305087149939210031640487103, 455561934457019941162877588670, 921027389228322924524948292706, 1861861819085211933448282698666, 3763337719427556035693337502357, 7605903601369376408980219089186, 15370263527767281493147526217552, 31057439705591620336669228379422, 62748704711297355374086808509067
Generating function in Maple syntax:
-(4*x^7-6*x^6+9*x^5-20*x^4+28*x^3-20*x^2+7*x-1)/(2*x-1)^2/(x-1)^4
Generating function in latex syntax:
-\frac{4 x^{7}-6 x^{6}+9 x^{5}-20 x^{4}+28 x^{3}-20 x^{2}+7 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}
Generating function in sympy syntax:
(-4*x**7 + 6*x**6 - 9*x**5 + 20*x**4 - 28*x**3 + 20*x**2 - 7*x + 1)/((x - 1)**4*(2*x - 1)**2)
Implicit equation for the generating function in Maple syntax:
(2*x-1)^2*(x-1)^4*F(x)+4*x^7-6*x^6+9*x^5-20*x^4+28*x^3-20*x^2+7*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+4 x^{7}-6 x^{6}+9 x^{5}-20 x^{4}+28 x^{3}-20 x^{2}+7 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,5+2^(-1+n)*n-2^(-1+n)-23/6*n+n^2-1/6*n^3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 5+2^{-1+n} n -2^{-1+n}-\frac{23 n}{6}+n^{2}-\frac{n^{3}}{6} & \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(6) = 142
a(7) = 354
a(n+2) = -1/6*n^3+2*n^2-4*a(n)+4*a(n+1)-47/6*n+12, n >= 8
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(6\right) = 142
a \! \left(7\right) = 354
a \! \left(n +2\right) = -\frac{n^{3}}{6}+2 n^{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{47 n}{6}+12, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/21852/
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[11,x]*F[4,x]
F[4,x] = F[21,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[11,x]*F[8,x]
F[8,x] = F[5,x]+F[9,x]
F[9,x] = F[10,x]
F[10,x] = F[11,x]*F[12,x]*F[16,x]
F[11,x] = x
F[12,x] = F[13,x]+F[18,x]
F[13,x] = F[1,x]+F[14,x]
F[14,x] = F[15,x]
F[15,x] = F[11,x]*F[13,x]*F[16,x]
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[11,x]*F[16,x]
F[18,x] = F[19,x]
F[19,x] = F[11,x]*F[16,x]*F[20,x]
F[20,x] = F[1,x]+F[18,x]
F[21,x] = F[22,x]
F[22,x] = F[11,x]*F[23,x]
F[23,x] = F[24,x]+F[42,x]
F[24,x] = F[25,x]+F[37,x]
F[25,x] = F[26,x]*F[30,x]
F[26,x] = F[1,x]+F[27,x]
F[27,x] = F[28,x]
F[28,x] = F[11,x]*F[29,x]
F[29,x] = F[26,x]+F[27,x]
F[30,x] = F[31,x]+F[32,x]
F[31,x] = F[13,x]*F[16,x]
F[32,x] = F[33,x]
F[33,x] = F[11,x]*F[16,x]*F[20,x]*F[34,x]
F[34,x] = F[35,x]
F[35,x] = F[11,x]*F[36,x]
F[36,x] = F[1,x]+F[35,x]
F[37,x] = F[38,x]
F[38,x] = F[11,x]*F[39,x]
F[39,x] = F[37,x]+F[40,x]
F[40,x] = F[41,x]
F[41,x] = F[11,x]*F[20,x]*F[27,x]*F[36,x]
F[42,x] = F[43,x]*F[6,x]
F[43,x] = F[29,x]+F[44,x]
F[44,x] = F[16,x]*F[34,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_{11}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{21}\! \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_{11}\! \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_{10}\! \left(x \right) = F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{11}\! \left(x \right) = x
F_{12}\! \left(x \right) = F_{13}\! \left(x \right)+F_{18}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{11}\! \left(x \right) F_{16}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{1}\! \left(x \right)+F_{18}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{11}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{42}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{37}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{30}\! \left(x \right)
F_{26}\! \left(x \right) = F_{1}\! \left(x \right)+F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{11}\! \left(x \right) F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)+F_{32}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right) F_{16}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{20}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{11}\! \left(x \right) F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{1}\! \left(x \right)+F_{35}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{11}\! \left(x \right) F_{39}\! \left(x \right)
F_{39}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{11}\! \left(x \right) F_{20}\! \left(x \right) F_{27}\! \left(x \right) F_{36}\! \left(x \right)
F_{42}\! \left(x \right) = F_{43}\! \left(x \right) F_{6}\! \left(x \right)
F_{43}\! \left(x \right) = F_{29}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{16}\! \left(x \right) F_{34}\! \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_11(x)*F_4(x))
Eq(F_4(x), F_21(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_11(x)*F_8(x))
Eq(F_8(x), F_5(x) + F_9(x))
Eq(F_9(x), F_10(x))
Eq(F_10(x), F_11(x)*F_12(x)*F_16(x))
Eq(F_11(x), x)
Eq(F_12(x), F_13(x) + F_18(x))
Eq(F_13(x), F_1(x) + F_14(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_11(x)*F_13(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_11(x)*F_16(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_11(x)*F_16(x)*F_20(x))
Eq(F_20(x), F_1(x) + F_18(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_11(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_42(x))
Eq(F_24(x), F_25(x) + F_37(x))
Eq(F_25(x), F_26(x)*F_30(x))
Eq(F_26(x), F_1(x) + F_27(x))
Eq(F_27(x), F_28(x))
Eq(F_28(x), F_11(x)*F_29(x))
Eq(F_29(x), F_26(x) + F_27(x))
Eq(F_30(x), F_31(x) + F_32(x))
Eq(F_31(x), F_13(x)*F_16(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_11(x)*F_16(x)*F_20(x)*F_34(x))
Eq(F_34(x), F_35(x))
Eq(F_35(x), F_11(x)*F_36(x))
Eq(F_36(x), F_1(x) + F_35(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_11(x)*F_39(x))
Eq(F_39(x), F_37(x) + F_40(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_11(x)*F_20(x)*F_27(x)*F_36(x))
Eq(F_42(x), F_43(x)*F_6(x))
Eq(F_43(x), F_29(x) + F_44(x))
Eq(F_44(x), F_16(x)*F_34(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": [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": [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": [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": [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": [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": [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": "RequirementInsertionStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [1, 0, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 0], "pos": [[1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[0, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 0, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [0, 0], [1, 0]]}, {"patt": [3, 1, 2, 0], "pos": [[0, 0], [0, 0], [1, 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", 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