0132_0231_0312_0321_1023
Counting sequence:
1, 1, 2, 6, 19, 58, 178, 547, 1680, 5160, 15849, 48680, 149520, 459249, 1410578, 4332574, 13307451, 40873682, 125543042, 385604003, 1184378240, 3637804080, 11173473201, 34319193840, 105411007360, 323768691201, 994451794242, 3054447196246, 9381699272579, 28815780920778, 88507338164338, 271849266569347, 834981881356240, 2564637201312280, 7877253532341849, 24194893211812440, 74314334955393360, 228255621197209649, 701084503271099538, 2153372951556879534, 6614060140912454891, 20315009304811254562, 62397316362719543682, 191652636277514206403, 588658857996441418880, 1808058881047232545760, 5553431963056492300001, 17057302110888404112480, 52391306355716003464320, 160919292149157270147201, 494261746595303722254082, 1518119244030738766114086, 4662885717885657565025139, 14321999608103119147182298, 43989856321744792485406898, 135114335439089296255131747, 415002120207485350347552400, 1274674217336097724380881800, 3915147131125637676912370249, 12025329178145309084829531400, 36935659631564539628521923600, 113447451808483158838097259249, 348452537472469147922208938898, 1070267942870826674310319208894, 3287315620789398642140532962331, 10096951948032181915505710566642, 31012671249494903272905241969922, 95255061426403281786545939663203, 292574820606452628633002996593920, 898640181120820607256408521185040, 2760162933539370200282866261248401, 8477808559798159947695055840675920, 26039491039908847821708573046618880, 79979995872145093058152176879920001, 245657633242694068921868720219187522, 754534582208192894639619677863505526, 2317544251456814929155302637423817699, 7118310391740993763329624455610720618, 21863808124188480115706624324476479858, 67154434041814062381635555932989751747, 206264068265726365979080989617156158160, 633537702529626584289619968720334233720, 1945903733506505335134788837769515973049, 5976820834743428309001811946992714325560, 18357736138494185145315974820081321052240, 56385574446459434186793093329293144896049, 173187640440615225534246303097102633040658, 531943836625596433439053072164396505848654, 1633859348184754549358636335002401341389771, 5018380110548419439423362214775013512313922, 15413896527830239526174893413392334409749762, 47343605095049876694444263503015008064790403, 145415336047513294973397516388817636365295360, 446641524559821705006944080365070229993321920, 1371854282246684752189101611184833363635006401, 4213634577692517674846258086440302719205524160, 12942129921589861434572320823297160751244599040, 39751602522456447350899120000711245387375481601, 122096587862815174975354516477389205517696962562, 375018258932342868685789612169024474585129962566, 1151864249397895191973695635684487508391520770259
Generating function in Maple syntax:
(x^4-x^3-x^2-2*x+1)/(x^4-x^3-3*x+1)
Generating function in latex syntax:
\frac{x^{4}-x^{3}-x^{2}-2 x +1}{x^{4}-x^{3}-3 x +1}
Generating function in sympy syntax:
(x**4 - x**3 - x**2 - 2*x + 1)/(x**4 - x**3 - 3*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^4-x^3-3*x+1)*F(x)-x^4+x^3+x^2+2*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{4}-x^{3}-3 x +1\right) F \! \left(x \right)-x^{4}+x^{3}+x^{2}+2 x -1 = 0
Explicit closed form in Maple syntax:
-13/337*Sum(_alpha^(-n+2),_alpha = RootOf(_Z^4-_Z^3-3*_Z+1))+91/674*Sum(_alpha^(-n+1),_alpha = RootOf(_Z^4-_Z^3-3*_Z+1))-59/674*Sum(_alpha^(-n),_alpha = RootOf(_Z^4-_Z^3-3*_Z+1))+57/674*Sum(_alpha^(-n-1),_alpha = RootOf(_Z^4-_Z^3-3*_Z+1))+piecewise(n = 0,1,0)
Explicit closed form in latex syntax:
-\frac{13 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-\textit{\_Z}^{3}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +2}\right)}{337}+\frac{91 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-\textit{\_Z}^{3}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n +1}\right)}{674}-\frac{59 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-\textit{\_Z}^{3}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n}\right)}{674}+\frac{57 \left(\munderset{\underline{\hspace{1.25 ex}}\alpha =\mathit{RootOf}\! \left(\textit{\_Z}^{4}-\textit{\_Z}^{3}-3 \textit{\_Z} +1\right)}{\textcolor{gray}{\sum}}\! \underline{\hspace{1.25 ex}}\alpha^{-n -1}\right)}{674}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \mathit{\text{otherwise}} \end{array}\right.\right)
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+1) = a(n)-3*a(n+3)+a(n+4), n >= 5
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(n +1\right) = a \! \left(n \right)-3 a \! \left(n +3\right)+a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18577/
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[16,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[14,x]+F[4,x]
F[14,x] = F[15,x]
F[15,x] = x^2
F[16,x] = F[17,x]+F[2,x]
F[17,x] = F[18,x]+F[19,x]+F[68,x]
F[18,x] = 0
F[19,x] = F[20,x]*F[4,x]
F[20,x] = F[21,x]+F[34,x]
F[21,x] = F[22,x]+F[25,x]
F[22,x] = F[23,x]
F[23,x] = F[24,x]*F[4,x]
F[24,x] = F[1,x]+F[4,x]
F[25,x] = F[18,x]+F[26,x]+F[32,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[28,x]+F[31,x]
F[28,x] = F[29,x]+F[4,x]
F[29,x] = F[30,x]
F[30,x] = F[28,x]*F[4,x]
F[31,x] = F[15,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[4,x]
F[34,x] = F[35,x]+F[60,x]
F[35,x] = F[18,x]+F[19,x]+F[36,x]
F[36,x] = F[37,x]*F[4,x]
F[37,x] = F[2,x]+F[38,x]
F[38,x] = F[18,x]+F[39,x]+F[59,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[44,x]
F[41,x] = F[4,x]+F[42,x]
F[42,x] = F[18,x]+F[26,x]+F[43,x]
F[43,x] = x^2
F[44,x] = F[38,x]+F[45,x]
F[45,x] = F[18,x]+F[46,x]+F[51,x]+F[52,x]
F[46,x] = F[4,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[43,x]
F[49,x] = F[50,x]
F[50,x] = 2*F[18,x]+F[46,x]+F[51,x]
F[51,x] = F[38,x]*F[4,x]
F[52,x] = F[4,x]*F[53,x]
F[53,x] = F[54,x]+F[57,x]
F[54,x] = F[38,x]+F[55,x]
F[55,x] = F[56,x]
F[56,x] = F[4,x]*F[54,x]
F[57,x] = F[58,x]
F[58,x] = F[38,x]*F[4,x]
F[59,x] = F[2,x]*F[4,x]
F[60,x] = F[18,x]+F[52,x]+F[61,x]+F[66,x]
F[61,x] = F[4,x]*F[62,x]
F[62,x] = F[63,x]+F[64,x]
F[63,x] = F[32,x]
F[64,x] = F[65,x]
F[65,x] = 2*F[18,x]+F[61,x]+F[66,x]
F[66,x] = F[4,x]*F[67,x]
F[67,x] = F[38,x]
F[68,x] = F[4,x]*F[69,x]
F[69,x] = F[70,x]+F[73,x]
F[70,x] = F[2,x]+F[71,x]
F[71,x] = F[18,x]+F[39,x]+F[72,x]
F[72,x] = F[4,x]*F[70,x]
F[73,x] = F[38,x]+F[74,x]
F[74,x] = F[58,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_{16}\! \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_{14}\! \left(x \right)+F_{4}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = x^{2}
F_{16}\! \left(x \right) = F_{17}\! \left(x \right)+F_{2}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{68}\! \left(x \right)
F_{18}\! \left(x \right) = 0
F_{19}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{34}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)+F_{25}\! \left(x \right)
F_{22}\! \left(x \right) = F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right) F_{4}\! \left(x \right)
F_{24}\! \left(x \right) = F_{1}\! \left(x \right)+F_{4}\! \left(x \right)
F_{25}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)+F_{32}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \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_{4}\! \left(x \right)
F_{29}\! \left(x \right) = F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{28}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{15}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{60}\! \left(x \right)
F_{35}\! \left(x \right) = F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{2}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{59}\! \left(x \right)
F_{39}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)+F_{44}\! \left(x \right)
F_{41}\! \left(x \right) = F_{4}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{18}\! \left(x \right)+F_{26}\! \left(x \right)+F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = x^{2}
F_{44}\! \left(x \right) = F_{38}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{51}\! \left(x \right)+F_{52}\! \left(x \right)
F_{46}\! \left(x \right) = F_{4}\! \left(x \right) F_{47}\! \left(x \right)
F_{47}\! \left(x \right) = F_{48}\! \left(x \right)+F_{49}\! \left(x \right)
F_{48}\! \left(x \right) = F_{43}\! \left(x \right)
F_{49}\! \left(x \right) = F_{50}\! \left(x \right)
F_{50}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{38}\! \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_{57}\! \left(x \right)
F_{54}\! \left(x \right) = F_{38}\! \left(x \right)+F_{55}\! \left(x \right)
F_{55}\! \left(x \right) = F_{56}\! \left(x \right)
F_{56}\! \left(x \right) = F_{4}\! \left(x \right) F_{54}\! \left(x \right)
F_{57}\! \left(x \right) = F_{58}\! \left(x \right)
F_{58}\! \left(x \right) = F_{38}\! \left(x \right) F_{4}\! \left(x \right)
F_{59}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{60}\! \left(x \right) = F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{61}\! \left(x \right)+F_{66}\! \left(x \right)
F_{61}\! \left(x \right) = F_{4}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{63}\! \left(x \right)+F_{64}\! \left(x \right)
F_{63}\! \left(x \right) = F_{32}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)
F_{65}\! \left(x \right) = 2 F_{18}\! \left(x \right)+F_{61}\! \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_{38}\! \left(x \right)
F_{68}\! \left(x \right) = F_{4}\! \left(x \right) F_{69}\! \left(x \right)
F_{69}\! \left(x \right) = F_{70}\! \left(x \right)+F_{73}\! \left(x \right)
F_{70}\! \left(x \right) = F_{2}\! \left(x \right)+F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{72}\! \left(x \right)
F_{72}\! \left(x \right) = F_{4}\! \left(x \right) F_{70}\! \left(x \right)
F_{73}\! \left(x \right) = F_{38}\! \left(x \right)+F_{74}\! \left(x \right)
F_{74}\! \left(x \right) = F_{58}\! \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_16(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_14(x) + F_4(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), x**2)
Eq(F_16(x), F_17(x) + F_2(x))
Eq(F_17(x), F_18(x) + F_19(x) + F_68(x))
Eq(F_18(x), 0)
Eq(F_19(x), F_20(x)*F_4(x))
Eq(F_20(x), F_21(x) + F_34(x))
Eq(F_21(x), F_22(x) + F_25(x))
Eq(F_22(x), F_23(x))
Eq(F_23(x), F_24(x)*F_4(x))
Eq(F_24(x), F_1(x) + F_4(x))
Eq(F_25(x), F_18(x) + F_26(x) + F_32(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_28(x) + F_31(x))
Eq(F_28(x), F_29(x) + F_4(x))
Eq(F_29(x), F_30(x))
Eq(F_30(x), F_28(x)*F_4(x))
Eq(F_31(x), F_15(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_4(x))
Eq(F_34(x), F_35(x) + F_60(x))
Eq(F_35(x), F_18(x) + F_19(x) + F_36(x))
Eq(F_36(x), F_37(x)*F_4(x))
Eq(F_37(x), F_2(x) + F_38(x))
Eq(F_38(x), F_18(x) + F_39(x) + F_59(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_44(x))
Eq(F_41(x), F_4(x) + F_42(x))
Eq(F_42(x), F_18(x) + F_26(x) + F_43(x))
Eq(F_43(x), x**2)
Eq(F_44(x), F_38(x) + F_45(x))
Eq(F_45(x), F_18(x) + F_46(x) + F_51(x) + F_52(x))
Eq(F_46(x), F_4(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_43(x))
Eq(F_49(x), F_50(x))
Eq(F_50(x), 2*F_18(x) + F_46(x) + F_51(x))
Eq(F_51(x), F_38(x)*F_4(x))
Eq(F_52(x), F_4(x)*F_53(x))
Eq(F_53(x), F_54(x) + F_57(x))
Eq(F_54(x), F_38(x) + F_55(x))
Eq(F_55(x), F_56(x))
Eq(F_56(x), F_4(x)*F_54(x))
Eq(F_57(x), F_58(x))
Eq(F_58(x), F_38(x)*F_4(x))
Eq(F_59(x), F_2(x)*F_4(x))
Eq(F_60(x), F_18(x) + F_52(x) + F_61(x) + F_66(x))
Eq(F_61(x), F_4(x)*F_62(x))
Eq(F_62(x), F_63(x) + F_64(x))
Eq(F_63(x), F_32(x))
Eq(F_64(x), F_65(x))
Eq(F_65(x), 2*F_18(x) + F_61(x) + F_66(x))
Eq(F_66(x), F_4(x)*F_67(x))
Eq(F_67(x), F_38(x))
Eq(F_68(x), F_4(x)*F_69(x))
Eq(F_69(x), F_70(x) + F_73(x))
Eq(F_70(x), F_2(x) + F_71(x))
Eq(F_71(x), F_18(x) + F_39(x) + F_72(x))
Eq(F_72(x), F_4(x)*F_70(x))
Eq(F_73(x), F_38(x) + F_74(x))
Eq(F_74(x), F_58(x))
Pack JSON:
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Specification JSON:
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