0213_1032_3120
Counting sequence:
1, 1, 2, 6, 21, 71, 218, 614, 1619, 4065, 9840, 23168, 53393, 120995, 270518, 598218, 1310943, 2851061, 6160652, 13238564, 28311869, 60293463, 127926642, 270533006, 570425771, 1199571401, 2516582888, 5268046344, 11005854249, 22951232075, 47781511790, 99321119378, 206158430903, 427349246685, 884763263748, 1829656068908, 3779571221333, 7799660610431, 16080357557162, 33122787787734, 68169720923139, 140187732542513, 288072046478432, 591537255744656, 1213860837065921, 2489294325286131, 5101733952881958, 10449758510384474, 21392098230011279, 43769358878508485, 89509042593990140, 182958734861927988, 373798769071752813, 763360136839300775, 1558245471070193378, 3179541336923571998, 6485183463413516123, 13222568505959778201, 26949540170185050072, 54907886656901089304, 111833385946864158809, 227701997159852279963, 463474444851952486622, 943089790768400828706, 1918461383665793370471, 3901486371589570169261, 7932099951695107197428, 16122454320422148115004, 32761417474908163672709, 66555852617944062233295, 135177740572143594244890, 274487551816798128048998, 557239244978618135219123, 1131006772647280028683265, 2295070110674647573859408, 4656253352109470180707488, 9444732965739290427395313, 19153918454519280986754371, 38836741955119962237439382, 78731294002402725002743274, 159578208189131051061218879, 323387656746913304233905813, 655237794231129012690751212, 1327400549936862833827385156, 2688651022822935284546539421, 5445001891544289802876620791, 11025403474885418073320329298, 22321606333364513081774837934, 45184811433916380033818038539, 91452820402207467808172806505, 185072035873164351097419076040, 374476861883827533156985082408, 757619304042652728238264029833, 1532569768635300780325115794155, 3099801858370592208347407061838, 6268928358941165712089165075378, 12676506002282294014967032058903, 25630310573364513211511467938941, 51815218284328876786177743525092, 104739630843857454298665102349644, 211697650238114310049949435303349
Generating function in Maple syntax:
(2*x^6-8*x^4+14*x^3-14*x^2+6*x-1)/(2*x-1)^2/(x-1)^3
Generating function in latex syntax:
\frac{2 x^{6}-8 x^{4}+14 x^{3}-14 x^{2}+6 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{3}}
Generating function in sympy syntax:
(2*x**6 - 8*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:
-(2*x-1)^2*(x-1)^3*F(x)+2*x^6-8*x^4+14*x^3-14*x^2+6*x-1 = 0
Implicit equation for the generating function in latex syntax:
-\left(2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+2 x^{6}-8 x^{4}+14 x^{3}-14 x^{2}+6 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,7+1/2*n^2+11/2*n+7/4*2^n*n-8*2^n)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 7+\frac{n^{2}}{2}+\frac{11 n}{2}+\frac{7 \,2^{n} n}{4}-8 \,2^{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) = 21
a(5) = 71
a(6) = 218
a(n+2) = -4*a(n)+4*a(n+1)+1/2*(n+8)*(-1+n), n >= 7
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) = 21
a \! \left(5\right) = 71
a \! \left(6\right) = 218
a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{\left(n +8\right) \left(-1+n \right)}{2}, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/326/
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[15,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[15,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[15,x]*F[16,x]
F[11,x] = F[12,x]+F[18,x]
F[12,x] = F[1,x]+F[13,x]
F[13,x] = F[14,x]
F[14,x] = F[12,x]*F[15,x]*F[16,x]
F[15,x] = x
F[16,x] = F[1,x]+F[17,x]
F[17,x] = F[15,x]*F[16,x]
F[18,x] = F[19,x]
F[19,x] = F[15,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[15,x]*F[23,x]
F[23,x] = F[24,x]+F[45,x]
F[24,x] = F[25,x]+F[26,x]
F[25,x] = F[16,x]*F[5,x]
F[26,x] = F[27,x]
F[27,x] = F[15,x]*F[16,x]*F[28,x]*F[35,x]
F[28,x] = F[29,x]+F[8,x]
F[29,x] = F[30,x]+F[32,x]
F[30,x] = F[16,x]*F[31,x]
F[31,x] = F[13,x]
F[32,x] = F[33,x]*F[34,x]
F[33,x] = F[17,x]
F[34,x] = F[18,x]
F[35,x] = F[36,x]+F[39,x]
F[36,x] = F[1,x]+F[37,x]
F[37,x] = F[38,x]
F[38,x] = F[15,x]*F[36,x]
F[39,x] = F[33,x]+F[40,x]
F[40,x] = F[41,x]+F[42,x]+F[44,x]
F[41,x] = 0
F[42,x] = F[15,x]*F[43,x]
F[43,x] = F[37,x]+F[40,x]
F[44,x] = F[15,x]*F[39,x]
F[45,x] = F[46,x]
F[46,x] = F[15,x]*F[16,x]*F[47,x]*F[55,x]
F[47,x] = F[48,x]+F[53,x]
F[48,x] = F[49,x]+F[51,x]
F[49,x] = F[12,x]+F[50,x]
F[50,x] = F[16,x]*F[34,x]
F[51,x] = F[52,x]
F[52,x] = F[15,x]*F[31,x]*F[33,x]*F[36,x]
F[53,x] = F[50,x]+F[54,x]
F[54,x] = F[31,x]*F[33,x]
F[55,x] = F[16,x]+F[56,x]
F[56,x] = F[37,x]+F[57,x]
F[57,x] = F[41,x]+F[58,x]+F[60,x]
F[58,x] = F[15,x]*F[59,x]
F[59,x] = F[33,x]+F[57,x]
F[60,x] = F[15,x]*F[56,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_{15}\! \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_{15}\! \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_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right)+F_{18}\! \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_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{15}\! \left(x \right) = x
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{17}\! \left(x \right)
F_{17}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{15}\! \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_{15}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{45}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)+F_{26}\! \left(x \right)
F_{25}\! \left(x \right) = F_{16}\! \left(x \right) F_{5}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)
F_{27}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{28}\! \left(x \right) F_{35}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{8}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)+F_{32}\! \left(x \right)
F_{30}\! \left(x \right) = F_{16}\! \left(x \right) F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{13}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{34}\! \left(x \right)
F_{33}\! \left(x \right) = F_{17}\! \left(x \right)
F_{34}\! \left(x \right) = F_{18}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{39}\! \left(x \right)
F_{36}\! \left(x \right) = F_{1}\! \left(x \right)+F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{15}\! \left(x \right) F_{36}\! \left(x \right)
F_{39}\! \left(x \right) = F_{33}\! \left(x \right)+F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = 0
F_{42}\! \left(x \right) = F_{15}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{37}\! \left(x \right)+F_{40}\! \left(x \right)
F_{44}\! \left(x \right) = F_{15}\! \left(x \right) F_{39}\! \left(x \right)
F_{45}\! \left(x \right) = F_{46}\! \left(x \right)
F_{46}\! \left(x \right) = F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{47}\! \left(x \right) F_{55}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{53}\! \left(x \right)
F_{48}\! \left(x \right) = F_{49}\! \left(x \right)+F_{51}\! \left(x \right)
F_{49}\! \left(x \right) = F_{12}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{16}\! \left(x \right) F_{34}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)
F_{52}\! \left(x \right) = F_{15}\! \left(x \right) F_{31}\! \left(x \right) F_{33}\! \left(x \right) F_{36}\! \left(x \right)
F_{53}\! \left(x \right) = F_{50}\! \left(x \right)+F_{54}\! \left(x \right)
F_{54}\! \left(x \right) = F_{31}\! \left(x \right) F_{33}\! \left(x \right)
F_{55}\! \left(x \right) = F_{16}\! \left(x \right)+F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{37}\! \left(x \right)+F_{57}\! \left(x \right)
F_{57}\! \left(x \right) = F_{41}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)
F_{58}\! \left(x \right) = F_{15}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{33}\! \left(x \right)+F_{57}\! \left(x \right)
F_{60}\! \left(x \right) = F_{15}\! \left(x \right) F_{56}\! \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_15(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_15(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_15(x)*F_16(x))
Eq(F_11(x), F_12(x) + F_18(x))
Eq(F_12(x), F_1(x) + F_13(x))
Eq(F_13(x), F_14(x))
Eq(F_14(x), F_12(x)*F_15(x)*F_16(x))
Eq(F_15(x), x)
Eq(F_16(x), F_1(x) + F_17(x))
Eq(F_17(x), F_15(x)*F_16(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_15(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_15(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_45(x))
Eq(F_24(x), F_25(x) + F_26(x))
Eq(F_25(x), F_16(x)*F_5(x))
Eq(F_26(x), F_27(x))
Eq(F_27(x), F_15(x)*F_16(x)*F_28(x)*F_35(x))
Eq(F_28(x), F_29(x) + F_8(x))
Eq(F_29(x), F_30(x) + F_32(x))
Eq(F_30(x), F_16(x)*F_31(x))
Eq(F_31(x), F_13(x))
Eq(F_32(x), F_33(x)*F_34(x))
Eq(F_33(x), F_17(x))
Eq(F_34(x), F_18(x))
Eq(F_35(x), F_36(x) + F_39(x))
Eq(F_36(x), F_1(x) + F_37(x))
Eq(F_37(x), F_38(x))
Eq(F_38(x), F_15(x)*F_36(x))
Eq(F_39(x), F_33(x) + F_40(x))
Eq(F_40(x), F_41(x) + F_42(x) + F_44(x))
Eq(F_41(x), 0)
Eq(F_42(x), F_15(x)*F_43(x))
Eq(F_43(x), F_37(x) + F_40(x))
Eq(F_44(x), F_15(x)*F_39(x))
Eq(F_45(x), F_46(x))
Eq(F_46(x), F_15(x)*F_16(x)*F_47(x)*F_55(x))
Eq(F_47(x), F_48(x) + F_53(x))
Eq(F_48(x), F_49(x) + F_51(x))
Eq(F_49(x), F_12(x) + F_50(x))
Eq(F_50(x), F_16(x)*F_34(x))
Eq(F_51(x), F_52(x))
Eq(F_52(x), F_15(x)*F_31(x)*F_33(x)*F_36(x))
Eq(F_53(x), F_50(x) + F_54(x))
Eq(F_54(x), F_31(x)*F_33(x))
Eq(F_55(x), F_16(x) + F_56(x))
Eq(F_56(x), F_37(x) + F_57(x))
Eq(F_57(x), F_41(x) + F_58(x) + F_60(x))
Eq(F_58(x), F_15(x)*F_59(x))
Eq(F_59(x), F_33(x) + F_57(x))
Eq(F_60(x), F_15(x)*F_56(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": [1, 0, 3, 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": [1, 0, 3, 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": 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