0132_0231_0312_1203_1320
Counting sequence:
1, 1, 2, 6, 19, 56, 156, 421, 1118, 2946, 7735, 20276, 53112, 139081, 364154, 953406, 2496091, 6534896, 17108628, 44791021, 117264470, 307002426, 803742847, 2104226156, 5508935664, 14422580881, 37758807026, 98853840246, 258802713763, 677554301096, 1773860189580, 4644026267701, 12158218613582, 31830629573106, 83333670105799, 218170380744356, 571177472127336, 1495362035637721, 3914908634785898, 10249363868720046, 26833182971374315, 70250185045402976, 183917372164834692, 481501931449101181, 1260588422182468934, 3300263335098305706, 8640201583112448271, 22620341414239039196, 59220822659604669408, 155042126564574969121, 405905557034120238050, 1062674544537785745126, 2782118076579236997427, 7283679685199925247256, 19068920979020538744444, 49923083251861690986181, 130700328776564534214206, 342177903077831911656546, 895833380456931200755543, 2345322238292961690610196, 6140133334421953871075160, 16075077764972899922615401, 42085099960496745896771162, 110180222116517337767698206, 288455566389055267406323579, 755186477050648464451272656, 1977103864762890125947494516, 5176125117238021913391211021, 13551271486951175614226138678, 35477689343615504929287205146, 92881796543895339173635476895, 243167700288070512591619225676, 636621304320316198601222200272, 1666696212672878083212047375281, 4363467333698318051034919925714, 11423705788422076069892712402006, 29907650031567910158643217280451, 78299244306281654406036939439496, 204990082887277053059467601038188, 536671004355549504772365863675221, 1405022930179371461257629989987630, 3678397786182564879000524106287826, 9630170428368323175743942328876007, 25212113498922404648231302880340356, 66006170068398890768949966312145224, 172806396706274267658618596056095481, 452413020050423912206905821856141386, 1184432663444997468962098869512328846, 3100884970284568494679390786680845323, 8118222247408708015076073490530207296, 21253781771941555550548829684909776740, 55643123068415958636570415564199123101, 145675587433306320359162417007687592742, 381383639231503002440916835458863655306, 998475330261202686963588089368903373359, 2614042351552105058449847432647846464956, 6843651724395112488385954208574636021696, 17916912821633232406708015193076061600321, 46907086740504584731738091370653548779458, 122804347399880521788506258918884584738246, 321505955459136980633780685386000205435475
Generating function in Maple syntax:
(x^4-x^3+5*x^2-4*x+1)/(x^2-3*x+1)/(x-1)^2
Generating function in latex syntax:
\frac{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:
(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)-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)-x^{4}+x^{3}-5 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
piecewise(n = 0,1,1/10*(9*5^(1/2)-15)*(3/2-1/2*5^(1/2))^(-n)+1/10*(-9*5^(1/2)-15)*(3/2+1/2*5^(1/2))^(-n)-2*n+3)
Explicit closed form in latex syntax:
\left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(9 \sqrt{5}-15\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(-9 \sqrt{5}-15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}-2 n +3 & \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(n+2) = -a(n)+3*a(n+1)+2*n-1, 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 +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)+2 n -1, \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18596/
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[12,x]*F[4,x]
F[4,x] = F[20,x]+F[5,x]
F[5,x] = F[1,x]+F[6,x]
F[6,x] = F[7,x]
F[7,x] = F[12,x]*F[8,x]
F[8,x] = F[13,x]+F[9,x]
F[9,x] = F[1,x]+F[10,x]
F[10,x] = F[11,x]
F[11,x] = F[12,x]*F[9,x]
F[12,x] = x
F[13,x] = F[14,x]+F[17,x]
F[14,x] = F[15,x]
F[15,x] = F[12,x]*F[16,x]
F[16,x] = F[1,x]+F[14,x]
F[17,x] = F[18,x]
F[18,x] = F[12,x]*F[19,x]
F[19,x] = F[10,x]+F[17,x]
F[20,x] = F[21,x]+F[32,x]
F[21,x] = F[22,x]
F[22,x] = F[12,x]*F[23,x]
F[23,x] = F[24,x]+F[9,x]
F[24,x] = F[21,x]+F[25,x]
F[25,x] = F[26,x]+F[27,x]+F[31,x]
F[26,x] = 0
F[27,x] = F[12,x]*F[28,x]
F[28,x] = F[29,x]+F[30,x]
F[29,x] = F[10,x]
F[30,x] = F[25,x]
F[31,x] = F[12,x]*F[24,x]
F[32,x] = F[26,x]+F[33,x]+F[63,x]
F[33,x] = F[12,x]*F[34,x]
F[34,x] = F[35,x]+F[44,x]
F[35,x] = F[36,x]+F[6,x]
F[36,x] = F[37,x]
F[37,x] = F[12,x]*F[38,x]
F[38,x] = F[39,x]+F[40,x]
F[39,x] = F[12,x]
F[40,x] = F[41,x]
F[41,x] = F[42,x]
F[42,x] = F[12,x]*F[43,x]
F[43,x] = F[12,x]+F[41,x]
F[44,x] = F[32,x]+F[45,x]
F[45,x] = 2*F[26,x]+F[46,x]+F[50,x]
F[46,x] = F[12,x]*F[47,x]
F[47,x] = F[48,x]+F[49,x]
F[48,x] = F[36,x]
F[49,x] = F[45,x]
F[50,x] = F[12,x]*F[51,x]
F[51,x] = F[52,x]+F[59,x]
F[52,x] = F[53,x]
F[53,x] = F[26,x]+F[54,x]+F[58,x]
F[54,x] = F[12,x]*F[55,x]
F[55,x] = F[56,x]+F[57,x]
F[56,x] = F[12,x]
F[57,x] = F[53,x]
F[58,x] = F[12,x]*F[21,x]
F[59,x] = F[60,x]
F[60,x] = F[61,x]
F[61,x] = F[12,x]*F[62,x]
F[62,x] = F[53,x]+F[60,x]
F[63,x] = F[12,x]*F[64,x]
F[64,x] = F[65,x]+F[80,x]
F[65,x] = F[21,x]+F[66,x]
F[66,x] = F[26,x]+F[67,x]+F[79,x]
F[67,x] = F[12,x]*F[68,x]
F[68,x] = F[69,x]+F[72,x]
F[69,x] = F[10,x]+F[70,x]
F[70,x] = F[71,x]
F[71,x] = F[12,x]*F[39,x]
F[72,x] = F[66,x]+F[73,x]
F[73,x] = 2*F[26,x]+F[74,x]+F[78,x]
F[74,x] = F[12,x]*F[75,x]
F[75,x] = F[76,x]+F[77,x]
F[76,x] = F[70,x]
F[77,x] = F[73,x]
F[78,x] = F[12,x]*F[52,x]
F[79,x] = F[12,x]*F[65,x]
F[80,x] = F[81,x]+F[84,x]
F[81,x] = F[26,x]+F[54,x]+F[82,x]
F[82,x] = F[12,x]*F[83,x]
F[83,x] = F[21,x]+F[81,x]
F[84,x] = F[85,x]
F[85,x] = F[12,x]*F[86,x]
F[86,x] = F[66,x]+F[84,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_{12}\! \left(x \right) F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{20}\! \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_{12}\! \left(x \right) F_{8}\! \left(x \right)
F_{8}\! \left(x \right) = F_{13}\! \left(x \right)+F_{9}\! \left(x \right)
F_{9}\! \left(x \right) = F_{1}\! \left(x \right)+F_{10}\! \left(x \right)
F_{10}\! \left(x \right) = F_{11}\! \left(x \right)
F_{11}\! \left(x \right) = F_{12}\! \left(x \right) F_{9}\! \left(x \right)
F_{12}\! \left(x \right) = x
F_{13}\! \left(x \right) = F_{14}\! \left(x \right)+F_{17}\! \left(x \right)
F_{14}\! \left(x \right) = F_{15}\! \left(x \right)
F_{15}\! \left(x \right) = F_{12}\! \left(x \right) F_{16}\! \left(x \right)
F_{16}\! \left(x \right) = F_{1}\! \left(x \right)+F_{14}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)
F_{18}\! \left(x \right) = F_{12}\! \left(x \right) F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{10}\! \left(x \right)+F_{17}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{32}\! \left(x \right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right)
F_{22}\! \left(x \right) = F_{12}\! \left(x \right) F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{9}\! \left(x \right)
F_{24}\! \left(x \right) = F_{21}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)
F_{26}\! \left(x \right) = 0
F_{27}\! \left(x \right) = F_{12}\! \left(x \right) F_{28}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{30}\! \left(x \right)
F_{29}\! \left(x \right) = F_{10}\! \left(x \right)
F_{30}\! \left(x \right) = F_{25}\! \left(x \right)
F_{31}\! \left(x \right) = F_{12}\! \left(x \right) F_{24}\! \left(x \right)
F_{32}\! \left(x \right) = F_{26}\! \left(x \right)+F_{33}\! \left(x \right)+F_{63}\! \left(x \right)
F_{33}\! \left(x \right) = F_{12}\! \left(x \right) F_{34}\! \left(x \right)
F_{34}\! \left(x \right) = F_{35}\! \left(x \right)+F_{44}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right)+F_{6}\! \left(x \right)
F_{36}\! \left(x \right) = F_{37}\! \left(x \right)
F_{37}\! \left(x \right) = F_{12}\! \left(x \right) F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right)+F_{40}\! \left(x \right)
F_{39}\! \left(x \right) = F_{12}\! \left(x \right)
F_{40}\! \left(x \right) = F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{12}\! \left(x \right) F_{43}\! \left(x \right)
F_{43}\! \left(x \right) = F_{12}\! \left(x \right)+F_{41}\! \left(x \right)
F_{44}\! \left(x \right) = F_{32}\! \left(x \right)+F_{45}\! \left(x \right)
F_{45}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)
F_{46}\! \left(x \right) = F_{12}\! \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_{36}\! \left(x \right)
F_{49}\! \left(x \right) = F_{45}\! \left(x \right)
F_{50}\! \left(x \right) = F_{12}\! \left(x \right) F_{51}\! \left(x \right)
F_{51}\! \left(x \right) = F_{52}\! \left(x \right)+F_{59}\! \left(x \right)
F_{52}\! \left(x \right) = F_{53}\! \left(x \right)
F_{53}\! \left(x \right) = F_{26}\! \left(x \right)+F_{54}\! \left(x \right)+F_{58}\! \left(x \right)
F_{54}\! \left(x \right) = F_{12}\! \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_{12}\! \left(x \right)
F_{57}\! \left(x \right) = F_{53}\! \left(x \right)
F_{58}\! \left(x \right) = F_{12}\! \left(x \right) F_{21}\! \left(x \right)
F_{59}\! \left(x \right) = F_{60}\! \left(x \right)
F_{60}\! \left(x \right) = F_{61}\! \left(x \right)
F_{61}\! \left(x \right) = F_{12}\! \left(x \right) F_{62}\! \left(x \right)
F_{62}\! \left(x \right) = F_{53}\! \left(x \right)+F_{60}\! \left(x \right)
F_{63}\! \left(x \right) = F_{12}\! \left(x \right) F_{64}\! \left(x \right)
F_{64}\! \left(x \right) = F_{65}\! \left(x \right)+F_{80}\! \left(x \right)
F_{65}\! \left(x \right) = F_{21}\! \left(x \right)+F_{66}\! \left(x \right)
F_{66}\! \left(x \right) = F_{26}\! \left(x \right)+F_{67}\! \left(x \right)+F_{79}\! \left(x \right)
F_{67}\! \left(x \right) = F_{12}\! \left(x \right) F_{68}\! \left(x \right)
F_{68}\! \left(x \right) = F_{69}\! \left(x \right)+F_{72}\! \left(x \right)
F_{69}\! \left(x \right) = F_{10}\! \left(x \right)+F_{70}\! \left(x \right)
F_{70}\! \left(x \right) = F_{71}\! \left(x \right)
F_{71}\! \left(x \right) = F_{12}\! \left(x \right) F_{39}\! \left(x \right)
F_{72}\! \left(x \right) = F_{66}\! \left(x \right)+F_{73}\! \left(x \right)
F_{73}\! \left(x \right) = 2 F_{26}\! \left(x \right)+F_{74}\! \left(x \right)+F_{78}\! \left(x \right)
F_{74}\! \left(x \right) = F_{12}\! \left(x \right) F_{75}\! \left(x \right)
F_{75}\! \left(x \right) = F_{76}\! \left(x \right)+F_{77}\! \left(x \right)
F_{76}\! \left(x \right) = F_{70}\! \left(x \right)
F_{77}\! \left(x \right) = F_{73}\! \left(x \right)
F_{78}\! \left(x \right) = F_{12}\! \left(x \right) F_{52}\! \left(x \right)
F_{79}\! \left(x \right) = F_{12}\! \left(x \right) F_{65}\! \left(x \right)
F_{80}\! \left(x \right) = F_{81}\! \left(x \right)+F_{84}\! \left(x \right)
F_{81}\! \left(x \right) = F_{26}\! \left(x \right)+F_{54}\! \left(x \right)+F_{82}\! \left(x \right)
F_{82}\! \left(x \right) = F_{12}\! \left(x \right) F_{83}\! \left(x \right)
F_{83}\! \left(x \right) = F_{21}\! \left(x \right)+F_{81}\! \left(x \right)
F_{84}\! \left(x \right) = F_{85}\! \left(x \right)
F_{85}\! \left(x \right) = F_{12}\! \left(x \right) F_{86}\! \left(x \right)
F_{86}\! \left(x \right) = F_{66}\! \left(x \right)+F_{84}\! \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_12(x)*F_4(x))
Eq(F_4(x), F_20(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_12(x)*F_8(x))
Eq(F_8(x), F_13(x) + F_9(x))
Eq(F_9(x), F_1(x) + F_10(x))
Eq(F_10(x), F_11(x))
Eq(F_11(x), F_12(x)*F_9(x))
Eq(F_12(x), x)
Eq(F_13(x), F_14(x) + F_17(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_12(x)*F_16(x))
Eq(F_16(x), F_1(x) + F_14(x))
Eq(F_17(x), F_18(x))
Eq(F_18(x), F_12(x)*F_19(x))
Eq(F_19(x), F_10(x) + F_17(x))
Eq(F_20(x), F_21(x) + F_32(x))
Eq(F_21(x), F_22(x))
Eq(F_22(x), F_12(x)*F_23(x))
Eq(F_23(x), F_24(x) + F_9(x))
Eq(F_24(x), F_21(x) + F_25(x))
Eq(F_25(x), F_26(x) + F_27(x) + F_31(x))
Eq(F_26(x), 0)
Eq(F_27(x), F_12(x)*F_28(x))
Eq(F_28(x), F_29(x) + F_30(x))
Eq(F_29(x), F_10(x))
Eq(F_30(x), F_25(x))
Eq(F_31(x), F_12(x)*F_24(x))
Eq(F_32(x), F_26(x) + F_33(x) + F_63(x))
Eq(F_33(x), F_12(x)*F_34(x))
Eq(F_34(x), F_35(x) + F_44(x))
Eq(F_35(x), F_36(x) + F_6(x))
Eq(F_36(x), F_37(x))
Eq(F_37(x), F_12(x)*F_38(x))
Eq(F_38(x), F_39(x) + F_40(x))
Eq(F_39(x), F_12(x))
Eq(F_40(x), F_41(x))
Eq(F_41(x), F_42(x))
Eq(F_42(x), F_12(x)*F_43(x))
Eq(F_43(x), F_12(x) + F_41(x))
Eq(F_44(x), F_32(x) + F_45(x))
Eq(F_45(x), 2*F_26(x) + F_46(x) + F_50(x))
Eq(F_46(x), F_12(x)*F_47(x))
Eq(F_47(x), F_48(x) + F_49(x))
Eq(F_48(x), F_36(x))
Eq(F_49(x), F_45(x))
Eq(F_50(x), F_12(x)*F_51(x))
Eq(F_51(x), F_52(x) + F_59(x))
Eq(F_52(x), F_53(x))
Eq(F_53(x), F_26(x) + F_54(x) + F_58(x))
Eq(F_54(x), F_12(x)*F_55(x))
Eq(F_55(x), F_56(x) + F_57(x))
Eq(F_56(x), F_12(x))
Eq(F_57(x), F_53(x))
Eq(F_58(x), F_12(x)*F_21(x))
Eq(F_59(x), F_60(x))
Eq(F_60(x), F_61(x))
Eq(F_61(x), F_12(x)*F_62(x))
Eq(F_62(x), F_53(x) + F_60(x))
Eq(F_63(x), F_12(x)*F_64(x))
Eq(F_64(x), F_65(x) + F_80(x))
Eq(F_65(x), F_21(x) + F_66(x))
Eq(F_66(x), F_26(x) + F_67(x) + F_79(x))
Eq(F_67(x), F_12(x)*F_68(x))
Eq(F_68(x), F_69(x) + F_72(x))
Eq(F_69(x), F_10(x) + F_70(x))
Eq(F_70(x), F_71(x))
Eq(F_71(x), F_12(x)*F_39(x))
Eq(F_72(x), F_66(x) + F_73(x))
Eq(F_73(x), 2*F_26(x) + F_74(x) + F_78(x))
Eq(F_74(x), F_12(x)*F_75(x))
Eq(F_75(x), F_76(x) + F_77(x))
Eq(F_76(x), F_70(x))
Eq(F_77(x), F_73(x))
Eq(F_78(x), F_12(x)*F_52(x))
Eq(F_79(x), F_12(x)*F_65(x))
Eq(F_80(x), F_81(x) + F_84(x))
Eq(F_81(x), F_26(x) + F_54(x) + F_82(x))
Eq(F_82(x), F_12(x)*F_83(x))
Eq(F_83(x), F_21(x) + F_81(x))
Eq(F_84(x), F_85(x))
Eq(F_85(x), F_12(x)*F_86(x))
Eq(F_86(x), F_66(x) + F_84(x))
Pack JSON:
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Specification JSON:
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