0231_0321_2013_2301_3021
Counting sequence:
1, 1, 2, 6, 19, 53, 138, 348, 865, 2135, 5248, 12862, 31447, 76729, 186886, 454512, 1103997, 2678763, 6494140, 15732402, 38089811, 92173933, 222961698, 539145604, 1303349721, 3150038975, 7611815864, 18390447462, 44426264719, 107310085217, 259180652286, 625939824600, 1511597171701, 3650207909075, 8814160472692, 21282823820890, 51388398048139, 124076799785365, 299576357356186, 723298233973356, 1746310264255185, 4216193640389415, 10179247300846576, 24575787753708942, 59333021831518535, 143246229463255561, 345834276851050230, 834932375351398720, 2015734211925934701, 4866471167947443899, 11748817285309175852, 28364387213542504194, 68478154662347603395, 165321822438144551357, 399124051338450389010, 963574428714672697428, 2326281915967050522313, 5616156275047283221391, 13558630494858635926312, 32733489322358592999094, 79025753254763897777407, 190785284062262540262577, 460596897840041281722862, 1111980232663849710551976, 2684559669010749916517349, 6481104182371367970971043, 15646777257125522713231588, 37774677143366487106982058, 91196168437346644346295035, 220167087805036070637774565, 531530491621371375298252938, 1283228366195684000587102012, 3097987814308549735178104257, 7479205175404404188354609399, 18056400526300599546709925216, 43592010950372086151419668702, 105240431871777737588839685047, 254072893583393492807679888473, 613386256817496586161361166310, 1480845482776250391044725634832, 3575077373485724820079459268701, 8631000531979154934860937843083, 20837079041906944497115922302108, 50305159824718863543721957147474, 121447401109196310813818186003251, 293199966878814763629875027972365, 707847344538232394990601639590914, 1708894675298092667445145102446372, 4125636733820043957549025435074489, 9960168220309433037879463153783647, 24045973329181414943980486105025176, 58052115188157272747185504088607750, 140150204324465980081041631731795311, 338352525075029272194649042451314945, 816855256950404603041100266432665822, 1972063043927598635418370674913135480, 4760981354709122188160883815451922389, 11494025773152883640306222704202959475, 27749032940628970725905498020629807956, 66992091733638987606381556339006516986, 161733216566363270967197285885730733675
Generating function in Maple syntax:
-(2*x^7+4*x^6-2*x^5-10*x^3+13*x^2-6*x+1)/(2*x-1)/(x^2+2*x-1)/(x-1)^3
Generating function in latex syntax:
-\frac{2 x^{7}+4 x^{6}-2 x^{5}-10 x^{3}+13 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{3}}
Generating function in sympy syntax:
(-2*x**7 - 4*x**6 + 2*x**5 + 10*x**3 - 13*x**2 + 6*x - 1)/((x - 1)**3*(2*x - 1)*(x**2 + 2*x - 1))
Implicit equation for the generating function in Maple syntax:
(2*x-1)*(x^2+2*x-1)*(x-1)^3*F(x)+2*x^7+4*x^6-2*x^5-10*x^3+13*x^2-6*x+1 = 0
Implicit equation for the generating function in latex syntax:
\left(2 x -1\right) \left(x^{2}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+2 x^{7}+4 x^{6}-2 x^{5}-10 x^{3}+13 x^{2}-6 x +1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,1/4*(2-2^(1/2))*(-1-2^(1/2))^(-n)+1/4*(2+2^(1/2))*(2^(1/2)-1)^(-n)+1/2*n^2-7/2*n-2^(-1+n)+4)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(2-\sqrt{2}\right) \left(-1-\sqrt{2}\right)^{-n}}{4}+\frac{\left(2+\sqrt{2}\right) \left(\sqrt{2}-1\right)^{-n}}{4}+\frac{n^{2}}{2}-\frac{7 n}{2}-2^{-1+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) = 19
a(5) = 53
a(6) = 138
a(7) = 348
a(n+3) = n^2-2*a(n)-3*a(n+1)+4*a(n+2)-9*n+13, 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) = 53
a \! \left(6\right) = 138
a \! \left(7\right) = 348
a \! \left(n +3\right) = n^{2}-2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right)-9 n +13, \quad n \geq 8
Specification 1
Strategy pack name: point_placements
Tree: http://www.permpal.com/tree/20939/
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[5,x]
F[4,x] = x
F[5,x] = F[17,x]+F[6,x]
F[6,x] = F[1,x]+F[7,x]
F[7,x] = F[8,x]
F[8,x] = F[4,x]*F[9,x]
F[9,x] = F[10,x]+F[6,x]
F[10,x] = F[11,x]+F[14,x]
F[11,x] = F[12,x]
F[12,x] = F[13,x]*F[4,x]
F[13,x] = F[1,x]+F[11,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[7,x]
F[17,x] = F[18,x]+F[36,x]
F[18,x] = F[19,x]
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[13,x]+F[21,x]
F[21,x] = F[18,x]+F[22,x]
F[22,x] = F[23,x]+F[24,x]+F[28,x]
F[23,x] = 0
F[24,x] = F[25,x]*F[4,x]
F[25,x] = F[26,x]+F[27,x]
F[26,x] = F[11,x]
F[27,x] = F[22,x]
F[28,x] = F[29,x]*F[4,x]
F[29,x] = F[30,x]+F[33,x]
F[30,x] = F[31,x]
F[31,x] = F[32,x]*F[4,x]
F[32,x] = F[1,x]+F[30,x]
F[33,x] = F[23,x]+F[28,x]+F[34,x]
F[34,x] = F[35,x]*F[4,x]
F[35,x] = F[11,x]+F[33,x]
F[36,x] = F[23,x]+F[37,x]+F[52,x]
F[37,x] = F[38,x]*F[4,x]
F[38,x] = F[39,x]+F[43,x]
F[39,x] = F[40,x]+F[7,x]
F[40,x] = F[41,x]
F[41,x] = F[4,x]*F[42,x]
F[42,x] = F[4,x]
F[43,x] = F[36,x]+F[44,x]
F[44,x] = 2*F[23,x]+F[45,x]+F[49,x]
F[45,x] = F[4,x]*F[46,x]
F[46,x] = F[47,x]+F[48,x]
F[47,x] = F[40,x]
F[48,x] = F[44,x]
F[49,x] = F[4,x]*F[50,x]
F[50,x] = F[51,x]
F[51,x] = F[30,x]*F[4,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[58,x]
F[54,x] = F[30,x]+F[55,x]
F[55,x] = F[23,x]+F[52,x]+F[56,x]
F[56,x] = F[4,x]*F[57,x]
F[57,x] = F[55,x]+F[7,x]
F[58,x] = F[33,x]+F[59,x]
F[59,x] = F[60,x]
F[60,x] = F[4,x]*F[61,x]
F[61,x] = F[55,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_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{17}\! \left(x \right)+F_{6}\! \left(x \right)
F_{6}\! \left(x \right) = F_{1}\! \left(x \right)+F_{7}\! \left(x \right)
F_{7}\! \left(x \right) = F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{4}\! \left(x \right) F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{10}\! \left(x \right)+F_{6}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)+F_{14}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)
F_{12}\! \left(x \right) = F_{13}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{1}\! \left(x \right)+F_{11}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right) F_{4}\! \left(x \right)
F_{16}\! \left(x \right) = F_{7}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{36}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{13}\! \left(x \right)+F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{18}\! \left(x \right)+F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)
F_{23}\! \left(x \right) = 0
F_{24}\! \left(x \right) = F_{25}\! \left(x \right) F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)
F_{27}\! \left(x \right) = F_{22}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right) F_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{33}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{1}\! \left(x \right)+F_{30}\! \left(x \right)
F_{33}\! \left(x \right) = F_{23}\! \left(x \right)+F_{28}\! \left(x \right)+F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{11}\! \left(x \right)+F_{33}\! \left(x \right)
F_{36}\! \left(x \right) = F_{23}\! \left(x \right)+F_{37}\! \left(x \right)+F_{52}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{43}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{7}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right) F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right)
F_{43}\! \left(x \right) = F_{36}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = 2 F_{23}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)
F_{45}\! \left(x \right) = F_{4}\! \left(x \right) F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{48}\! \left(x \right)
F_{47}\! \left(x \right) = F_{40}\! \left(x \right)
F_{48}\! \left(x \right) = F_{44}\! \left(x \right)
F_{49}\! \left(x \right) = F_{4}\! \left(x \right) F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{54}\! \left(x \right)+F_{58}\! \left(x \right)
F_{54}\! \left(x \right) = F_{30}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{23}\! \left(x \right)+F_{52}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{55}\! \left(x \right)+F_{7}\! \left(x \right)
F_{58}\! \left(x \right) = F_{33}\! \left(x \right)+F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{4}\! \left(x \right) F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{55}\! \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_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_17(x) + F_6(x))
Eq(F_6(x), F_1(x) + F_7(x))
Eq(F_7(x), F_8(x))
Eq(F_8(x), F_4(x)*F_9(x))
Eq(F_9(x), F_10(x) + F_6(x))
Eq(F_10(x), F_11(x) + F_14(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_13(x)*F_4(x))
Eq(F_13(x), F_1(x) + F_11(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_7(x))
Eq(F_17(x), F_18(x) + F_36(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_13(x) + F_21(x))
Eq(F_21(x), F_18(x) + F_22(x))
Eq(F_22(x), F_23(x) + F_24(x) + F_28(x))
Eq(F_23(x), 0)
Eq(F_24(x), F_25(x)*F_4(x))
Eq(F_25(x), F_26(x) + F_27(x))
Eq(F_26(x), F_11(x))
Eq(F_27(x), F_22(x))
Eq(F_28(x), F_29(x)*F_4(x))
Eq(F_29(x), F_30(x) + F_33(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_32(x)*F_4(x))
Eq(F_32(x), F_1(x) + F_30(x))
Eq(F_33(x), F_23(x) + F_28(x) + F_34(x))
Eq(F_34(x), F_35(x)*F_4(x))
Eq(F_35(x), F_11(x) + F_33(x))
Eq(F_36(x), F_23(x) + F_37(x) + F_52(x))
Eq(F_37(x), F_38(x)*F_4(x))
Eq(F_38(x), F_39(x) + F_43(x))
Eq(F_39(x), F_40(x) + F_7(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_4(x)*F_42(x))
Eq(F_42(x), F_4(x))
Eq(F_43(x), F_36(x) + F_44(x))
Eq(F_44(x), 2*F_23(x) + F_45(x) + F_49(x))
Eq(F_45(x), F_4(x)*F_46(x))
Eq(F_46(x), F_47(x) + F_48(x))
Eq(F_47(x), F_40(x))
Eq(F_48(x), F_44(x))
Eq(F_49(x), F_4(x)*F_50(x))
Eq(F_50(x), F_51(x))
Eq(F_51(x), F_30(x)*F_4(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_58(x))
Eq(F_54(x), F_30(x) + F_55(x))
Eq(F_55(x), F_23(x) + F_52(x) + F_56(x))
Eq(F_56(x), F_4(x)*F_57(x))
Eq(F_57(x), F_55(x) + F_7(x))
Eq(F_58(x), F_33(x) + F_59(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_4(x)*F_61(x))
Eq(F_61(x), F_55(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:
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