0132_0231_1023_1032_2013
Counting sequence:
1, 1, 2, 6, 19, 58, 175, 529, 1603, 4860, 14732, 44651, 135330, 410169, 1243181, 3767957, 11420290, 34613714, 104910579, 317973106, 963743583, 2921007069, 8853270127, 26833345508, 81329093208, 246500064667, 747116185418, 2264431838009, 6863258552065, 20801826383937, 63048182962266, 191092517622102, 579181644505211, 1755439624251106, 5320555828428215, 16126054084883905, 48876401025459187, 148139313227331668, 448994927266318748, 1360857157488781571, 4124617207512421578, 12501287894095388753, 37890109832837200493, 114841001607726974517, 348071190831868902362, 1054967757081662280034, 3197498091761096245979, 9691285802988232344890, 29373284305377673163591, 89027384851090845390413, 269832789926496819907903, 817835260928984053659644, 2478774037065610911996904, 7512907574871711065299571, 22770845339086758748707090, 69016075639059510703372529, 209180582700642767872897729, 634004697801399405077524737, 1921602625084420128612033138, 5824178687533983949711392230, 17652482850264036756799535779, 53502848641274228251393300618, 162161597153925738551043744447, 491495018663847715290099741745, 1489670536126240644517492034595, 4515037227102359124057294169580, 13684610568408668574329158209004, 41476638395113881711378988258779, 125711398505590322849598595000690, 381018238838107888755405703729641, 1154826849856718218207637263731565, 3500160667417917499494771855059541, 10608624746869597232552975083242738, 32153643707709391152761531812656434, 97454366456631538508770774042835331, 295374099053858580402983028761771778, 895248325591488020768191817541898287, 2713405024481252170907356033978425725, 8224049815469554489121615194261603983, 24926243873324900377055824348477344340, 75548865531402769504824113507578365112, 228980792777609904128577166435625422475, 694017085395791541944393943917903440730, 2103493961124790205665673659606062908713, 6375472502906902236398618651218615160129, 19323397350562029883388714851131966385217, 58567217566613062842773222555953871644746, 177511175248653837210455166488631625986486, 538018001321631299151626865275719468774027, 1630676881839404163939996559016126337309490, 4942412868033104003744228782132344257360839, 14979941906421734031039491432224707840869217, 45402653625145710062942211976204921188749523, 137610744359512989003631938216058293032139204, 417083924642928017695681611479203828816217724, 1264138211047485874984320591631289270286071027, 3831472090415494372197354455800496858323711130, 11612795382135185320879870017354295210442603265, 35197180980304636448375073694372206755117860141, 106679012950328845483861517055625222195142014197, 323333047905870400601798417361244154979462066122
Generating function in Maple syntax:
-(x^2-x+1)*(x^2+2*x-1)/(x^5-x^4-3*x^3+4*x^2-4*x+1)
Generating function in latex syntax:
-\frac{\left(x^{2}-x +1\right) \left(x^{2}+2 x -1\right)}{x^{5}-x^{4}-3 x^{3}+4 x^{2}-4 x +1}
Generating function in sympy syntax:
(-x**2 + x - 1)*(x**2 + 2*x - 1)/(x**5 - x**4 - 3*x**3 + 4*x**2 - 4*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^5-x^4-3*x^3+4*x^2-4*x+1)*F(x)+(x^2-x+1)*(x^2+2*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}-x^{4}-3 x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)+\left(x^{2}-x +1\right) \left(x^{2}+2 x -1\right) = 0
Explicit closed form in Maple syntax:
570/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+3)+570/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+3)+570/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+3)+570/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+3)+570/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+3)+859/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+2)+859/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+2)+859/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+2)+859/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+2)+859/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+2)+12/1801*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n+1)+12/1801*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n+1)+12/1801*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n+1)+12/1801*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n+1)+12/1801*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n+1)+641/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n-1)+641/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n-1)+641/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n-1)+641/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n-1)+641/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n-1)+3164/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 1)^(-n)+3164/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 2)^(-n)+3164/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 3)^(-n)+3164/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 4)^(-n)+3164/23413*RootOf(_Z^5-_Z^4-3*_Z^3+4*_Z^2-4*_Z+1,index = 5)^(-n)
Explicit closed form in latex syntax:
\frac{570 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{23413}+\frac{570 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{23413}+\frac{570 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{23413}+\frac{570 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{23413}+\frac{570 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{23413}+\frac{859 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{23413}+\frac{859 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{23413}+\frac{859 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{23413}+\frac{859 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{23413}+\frac{859 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{23413}+\frac{12 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{1801}+\frac{12 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{1801}+\frac{12 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{1801}+\frac{12 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{1801}+\frac{12 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{1801}+\frac{641 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{23413}+\frac{641 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{23413}+\frac{641 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{23413}+\frac{641 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{23413}+\frac{641 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{23413}+\frac{3164 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{23413}+\frac{3164 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{23413}+\frac{3164 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{23413}+\frac{3164 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{23413}+\frac{3164 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-3 \textit{\_Z}^{3}+4 \textit{\_Z}^{2}-4 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{23413}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+5) = -a(n)+a(n+1)+3*a(n+2)-4*a(n+3)+4*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 +5\right) = -a \! \left(n \right)+a \! \left(n +1\right)+3 a \! \left(n +2\right)-4 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/18751/
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[19,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[7,x]
F[14,x] = F[15,x]
F[15,x] = F[16,x]*F[4,x]
F[16,x] = F[17,x]+F[18,x]
F[17,x] = F[4,x]
F[18,x] = F[15,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[32,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[27,x]
F[24,x] = F[25,x]+F[4,x]
F[25,x] = F[26,x]
F[26,x] = F[4,x]*F[7,x]
F[27,x] = F[28,x]+F[30,x]
F[28,x] = F[21,x]+F[22,x]+F[29,x]
F[29,x] = F[2,x]*F[4,x]
F[30,x] = F[31,x]
F[31,x] = F[20,x]*F[4,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[2,x]+F[35,x]
F[35,x] = F[21,x]+F[22,x]+F[36,x]
F[36,x] = F[34,x]*F[4,x]
F[37,x] = F[20,x]+F[38,x]
F[38,x] = F[39,x]
F[39,x] = F[4,x]*F[40,x]
F[40,x] = F[41,x]+F[42,x]
F[41,x] = F[28,x]
F[42,x] = F[39,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_{19}\! \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_{7}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)
F_{17}\! \left(x \right) = F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{15}\! \left(x \right)
F_{19}\! \left(x \right) = F_{2}\! \left(x \right)+F_{20}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{32}\! \left(x \right)
F_{21}\! \left(x \right) = 0
F_{22}\! \left(x \right) = F_{23}\! \left(x \right) F_{4}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{27}\! \left(x \right)
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_{26}\! \left(x \right) = F_{4}\! \left(x \right) F_{7}\! \left(x \right)
F_{27}\! \left(x \right) = F_{28}\! \left(x \right)+F_{30}\! \left(x \right)
F_{28}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{29}\! \left(x \right)
F_{29}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right)
F_{31}\! \left(x \right) = F_{20}\! \left(x \right) F_{4}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right) F_{4}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{2}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{36}\! \left(x \right)
F_{36}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{37}\! \left(x \right) = F_{20}\! \left(x \right)+F_{38}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \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_{42}\! \left(x \right)
F_{41}\! \left(x \right) = F_{28}\! \left(x \right)
F_{42}\! \left(x \right) = F_{39}\! \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_19(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_7(x))
Eq(F_14(x), F_15(x))
Eq(F_15(x), F_16(x)*F_4(x))
Eq(F_16(x), F_17(x) + F_18(x))
Eq(F_17(x), F_4(x))
Eq(F_18(x), F_15(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_32(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_27(x))
Eq(F_24(x), F_25(x) + F_4(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_4(x)*F_7(x))
Eq(F_27(x), F_28(x) + F_30(x))
Eq(F_28(x), F_21(x) + F_22(x) + F_29(x))
Eq(F_29(x), F_2(x)*F_4(x))
Eq(F_30(x), F_31(x))
Eq(F_31(x), F_20(x)*F_4(x))
Eq(F_32(x), F_33(x)*F_4(x))
Eq(F_33(x), F_34(x) + F_37(x))
Eq(F_34(x), F_2(x) + F_35(x))
Eq(F_35(x), F_21(x) + F_22(x) + F_36(x))
Eq(F_36(x), F_34(x)*F_4(x))
Eq(F_37(x), F_20(x) + F_38(x))
Eq(F_38(x), F_39(x))
Eq(F_39(x), F_4(x)*F_40(x))
Eq(F_40(x), F_41(x) + F_42(x))
Eq(F_41(x), F_28(x))
Eq(F_42(x), F_39(x))
Pack JSON:
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Specification JSON:
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