0123_0213_0231_2103_3012
Counting sequence:
1, 1, 2, 6, 19, 53, 139, 360, 934, 2432, 6349, 16599, 43429, 113666, 297544, 778938, 2039239, 5338745, 13976959, 36592092, 95799274, 250805684, 656617729, 1719047451, 4500524569, 11782526198, 30847053964, 80758635630, 211428852859, 553527922877, 1449154915699, 3793936824144, 9932655556654, 26004029845736, 68079433980469, 178234272095583, 466623382306189, 1221635874822890, 3198284242162384, 8373216851664162, 21921366312829999, 57390882086825729, 150251279947647079, 393362957756115396, 1029837593320698994, 2696149822205981468, 7058611873297245289, 18479685797685754275, 48380445519760017409, 126661650761594297822, 331604506765022875924, 868151869533474329814, 2272851101835400113379, 5950401435972726010181, 15578353206082777917019, 40784658182275607740728, 106775621340744045305014, 279542205839956528174160, 731850996179125539217309, 1916010782697420089477607, 5016181351913134729215349, 13132533273041984098168274, 34381418467212817565289304, 90011722128596468597699466, 235653747918576588227808919, 616949521627133296085727113, 1615194816962823300029372239, 4228634929261336604002389420, 11070709970821186511977795834, 28983494983202222931930997892, 75879774978785482283815197649, 198655829953154223919514594859, 520087714880677189474728586729, 1361607314688877344504671165126, 3564734229185954844039284908444, 9332595372868987187613183559998, 24433051889421006718800265771339, 63966560295394032968787613753805, 167466628996761092187562575489859, 438433326694889243593900112715552, 1147833351087906638594137762656574, 3005066726568830672188513175253944, 7867366828618585377971401763105029, 20597033759286925461725692114060911, 53923734449242191007205674579077469, 141174169588439647559891331623171258, 369598774316076751672468320290436064, 967622153359790607457513629248136690, 2533267685763295070700072567453973759, 6632180903930094604642704073113784337, 17363275026026988743228039651887378999, 45457644174150871625041414882548352404, 119009657496425626131896204995757677954, 311571328315126006770647200104724681196, 815704327448952394180045395318416365369, 2135541654031731175769488985850524414643, 5590920634646241133128421562233156878289, 14637220249906992223615775700848946219950, 38320740115074735537718905540313681781284, 100325000095317214389540940920092099123622, 262654260170876907630903917219962615589299
Generating function in Maple syntax:
-(2*x^6+3*x^5-x^4+x^3-5*x^2+4*x-1)/(x^2-3*x+1)/(x-1)^2
Generating function in latex syntax:
-\frac{2 x^{6}+3 x^{5}-x^{4}+x^{3}-5 x^{2}+4 x -1}{\left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}
Generating function in sympy syntax:
(-2*x**6 - 3*x**5 + x**4 - x**3 + 5*x**2 - 4*x + 1)/((x - 1)**2*(x**2 - 3*x + 1))
Implicit equation for the generating function in Maple syntax:
(x^2-3*x+1)*(x-1)^2*F(x)+2*x^6+3*x^5-x^4+x^3-5*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+2 x^{6}+3 x^{5}-x^{4}+x^{3}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(Or(n = 0,n = 1),1,n = 2,2,1/10*(315-139*5^(1/2))*(3/2-1/2*5^(1/2))^(-n)+1/10*(139*5^(1/2)+315)*(3/2+1/2*5^(1/2))^(-n)+3*n-14)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(315-139 \sqrt{5}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(139 \sqrt{5}+315\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+3 n -14 & \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) = 139
a(n+2) = -a(n)+3*a(n+1)-3*n+11, 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) = 19
a \! \left(5\right) = 53
a \! \left(6\right) = 139
a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)-3 n +11, \quad n \geq 7
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/16780/
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[14,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[13,x]
F[10,x] = F[1,x]+F[11,x]
F[11,x] = F[12,x]
F[12,x] = F[10,x]*F[4,x]
F[13,x] = F[7,x]
F[14,x] = F[15,x]+F[35,x]
F[15,x] = F[16,x]
F[16,x] = F[17,x]*F[4,x]
F[17,x] = F[10,x]+F[18,x]
F[18,x] = F[15,x]+F[19,x]
F[19,x] = F[20,x]+F[21,x]+F[34,x]
F[20,x] = 0
F[21,x] = F[22,x]*F[4,x]
F[22,x] = F[23,x]+F[27,x]
F[23,x] = F[11,x]+F[24,x]
F[24,x] = F[25,x]
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[11,x]+F[24,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[29,x]
F[29,x] = F[30,x]*F[4,x]
F[30,x] = F[15,x]+F[28,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[19,x]+F[31,x]
F[34,x] = F[18,x]*F[4,x]
F[35,x] = F[20,x]+F[36,x]+F[58,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[38,x]+F[46,x]
F[38,x] = F[39,x]+F[7,x]
F[39,x] = F[20,x]+F[40,x]+F[42,x]
F[40,x] = F[4,x]*F[41,x]
F[41,x] = F[11,x]
F[42,x] = F[4,x]*F[43,x]
F[43,x] = F[44,x]+F[45,x]
F[44,x] = F[11,x]
F[45,x] = F[39,x]
F[46,x] = F[47,x]+F[51,x]
F[47,x] = F[48,x]
F[48,x] = F[4,x]*F[49,x]
F[49,x] = F[30,x]+F[50,x]
F[50,x] = F[47,x]
F[51,x] = 2*F[20,x]+F[52,x]+F[54,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[28,x]
F[54,x] = F[4,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[19,x]
F[57,x] = F[51,x]
F[58,x] = F[4,x]*F[59,x]
F[59,x] = F[60,x]+F[72,x]
F[60,x] = F[15,x]+F[61,x]
F[61,x] = F[20,x]+F[34,x]+F[62,x]
F[62,x] = F[4,x]*F[63,x]
F[63,x] = F[64,x]+F[68,x]
F[64,x] = F[11,x]+F[65,x]
F[65,x] = F[20,x]+F[40,x]+F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[11,x]+F[65,x]
F[68,x] = F[28,x]+F[69,x]
F[69,x] = 2*F[20,x]+F[52,x]+F[70,x]
F[70,x] = F[4,x]*F[71,x]
F[71,x] = F[19,x]+F[69,x]
F[72,x] = F[73,x]
F[73,x] = F[20,x]+F[21,x]+F[74,x]
F[74,x] = F[4,x]*F[75,x]
F[75,x] = F[18,x]+F[76,x]
F[76,x] = F[73,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_{14}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)
F_{13}\! \left(x \right) = F_{7}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)+F_{35}\! \left(x \right)
F_{15}\! \left(x \right) = F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{17}\! \left(x \right) = F_{10}\! \left(x \right)+F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)+F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{34}\! \left(x \right)
F_{20}\! \left(x \right) = 0
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{4}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)+F_{27}\! \left(x \right)
F_{23}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \left(x \right)
F_{24}\! \left(x \right) = F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{11}\! \left(x \right)+F_{24}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{31}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{15}\! \left(x \right)+F_{28}\! \left(x \right)
F_{31}\! \left(x \right) = F_{32}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{19}\! \left(x \right)+F_{31}\! \left(x \right)
F_{34}\! \left(x \right) = F_{18}\! \left(x \right) F_{4}\! \left(x \right)
F_{35}\! \left(x \right) = F_{20}\! \left(x \right)+F_{36}\! \left(x \right)+F_{58}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{38}\! \left(x \right)+F_{46}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{7}\! \left(x \right)
F_{39}\! \left(x \right) = F_{20}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)
F_{40}\! \left(x \right) = F_{4}\! \left(x \right) F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{11}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{44}\! \left(x \right)+F_{45}\! \left(x \right)
F_{44}\! \left(x \right) = F_{11}\! \left(x \right)
F_{45}\! \left(x \right) = F_{39}\! \left(x \right)
F_{46}\! \left(x \right) = F_{47}\! \left(x \right)+F_{51}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)
F_{48}\! \left(x \right) = F_{4}\! \left(x \right) F_{49}\! \left(x \right)
F_{49}\! \left(x \right) = F_{30}\! \left(x \right)+F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = F_{47}\! \left(x \right)
F_{51}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)
F_{52}\! \left(x \right) = F_{4}\! \left(x \right) F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{28}\! \left(x \right)
F_{54}\! \left(x \right) = F_{4}\! \left(x \right) F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)+F_{57}\! \left(x \right)
F_{56}\! \left(x \right) = F_{19}\! \left(x \right)
F_{57}\! \left(x \right) = F_{51}\! \left(x \right)
F_{58}\! \left(x \right) = F_{4}\! \left(x \right) F_{59}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)+F_{72}\! \left(x \right)
F_{60}\! \left(x \right) = F_{15}\! \left(x \right)+F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{20}\! \left(x \right)+F_{34}\! \left(x \right)+F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{4}\! \left(x \right) F_{63}\! \left(x \right)
F_{63}\! \left(x \right) = F_{64}\! \left(x \right)+F_{68}\! \left(x \right)
F_{64}\! \left(x \right) = F_{11}\! \left(x \right)+F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = F_{20}\! \left(x \right)+F_{40}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{4}\! \left(x \right) F_{67}\! \left(x \right)
F_{67}\! \left(x \right) = F_{11}\! \left(x \right)+F_{65}\! \left(x \right)
F_{68}\! \left(x \right) = F_{28}\! \left(x \right)+F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = 2 F_{20}\! \left(x \right)+F_{52}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{4}\! \left(x \right) F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{19}\! \left(x \right)+F_{69}\! \left(x \right)
F_{72}\! \left(x \right) = F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{4}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{18}\! \left(x \right)+F_{76}\! \left(x \right)
F_{76}\! \left(x \right) = F_{73}\! \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_14(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_13(x))
Eq(F_10(x), F_1(x) + F_11(x))
Eq(F_11(x), F_12(x))
Eq(F_12(x), F_10(x)*F_4(x))
Eq(F_13(x), F_7(x))
Eq(F_14(x), F_15(x) + F_35(x))
Eq(F_15(x), F_16(x))
Eq(F_16(x), F_17(x)*F_4(x))
Eq(F_17(x), F_10(x) + F_18(x))
Eq(F_18(x), F_15(x) + F_19(x))
Eq(F_19(x), F_20(x) + F_21(x) + F_34(x))
Eq(F_20(x), 0)
Eq(F_21(x), F_22(x)*F_4(x))
Eq(F_22(x), F_23(x) + F_27(x))
Eq(F_23(x), F_11(x) + F_24(x))
Eq(F_24(x), F_25(x))
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_11(x) + F_24(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_29(x))
Eq(F_29(x), F_30(x)*F_4(x))
Eq(F_30(x), F_15(x) + F_28(x))
Eq(F_31(x), F_32(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_19(x) + F_31(x))
Eq(F_34(x), F_18(x)*F_4(x))
Eq(F_35(x), F_20(x) + F_36(x) + F_58(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_38(x) + F_46(x))
Eq(F_38(x), F_39(x) + F_7(x))
Eq(F_39(x), F_20(x) + F_40(x) + F_42(x))
Eq(F_40(x), F_4(x)*F_41(x))
Eq(F_41(x), F_11(x))
Eq(F_42(x), F_4(x)*F_43(x))
Eq(F_43(x), F_44(x) + F_45(x))
Eq(F_44(x), F_11(x))
Eq(F_45(x), F_39(x))
Eq(F_46(x), F_47(x) + F_51(x))
Eq(F_47(x), F_48(x))
Eq(F_48(x), F_4(x)*F_49(x))
Eq(F_49(x), F_30(x) + F_50(x))
Eq(F_50(x), F_47(x))
Eq(F_51(x), 2*F_20(x) + F_52(x) + F_54(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_28(x))
Eq(F_54(x), F_4(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_19(x))
Eq(F_57(x), F_51(x))
Eq(F_58(x), F_4(x)*F_59(x))
Eq(F_59(x), F_60(x) + F_72(x))
Eq(F_60(x), F_15(x) + F_61(x))
Eq(F_61(x), F_20(x) + F_34(x) + F_62(x))
Eq(F_62(x), F_4(x)*F_63(x))
Eq(F_63(x), F_64(x) + F_68(x))
Eq(F_64(x), F_11(x) + F_65(x))
Eq(F_65(x), F_20(x) + F_40(x) + F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_11(x) + F_65(x))
Eq(F_68(x), F_28(x) + F_69(x))
Eq(F_69(x), 2*F_20(x) + F_52(x) + F_70(x))
Eq(F_70(x), F_4(x)*F_71(x))
Eq(F_71(x), F_19(x) + F_69(x))
Eq(F_72(x), F_73(x))
Eq(F_73(x), F_20(x) + F_21(x) + F_74(x))
Eq(F_74(x), F_4(x)*F_75(x))
Eq(F_75(x), F_18(x) + F_76(x))
Eq(F_76(x), F_73(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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