0132_0231_1032_2013
Counting sequence:
1, 1, 2, 6, 20, 66, 215, 697, 2258, 7317, 23716, 76875, 249192, 807758, 2618345, 8487346, 27511664, 89178844, 289072543, 937026464, 3037364213, 9845593169, 31914415909, 103450338135, 335333489737, 1086980974332, 3523440618781, 11421206154755, 37021753491238, 120005734332624, 388997681487923, 1260933046612173, 4087304947312383, 13248968117069782, 42946430087766220, 139210528773716408, 451249877623197838, 1462723070220796613, 4741405784807160269, 15369231041668966594, 49819246344427378969, 161488710762280368538, 523464436281672419939, 1696806016706970842130, 5500183887915027157604, 17828804532170882562435, 57791935237796498128868, 187332121595871719759902, 607235657314674698462648, 1968349796997713301914229, 6380391000874956114938650, 20681989241007561476775273, 67040511922622770765665477, 217311313069243786554106655, 704412980055767026131000449, 2283349354724752313200754191, 7401459688191982738592084636, 23991775679256336620867586817, 77769159664823336426573252191, 252088143696759242028231245522, 817141813880525432064005139986, 2648759018175670165710975492221, 8585932352487276772025472128665, 27831234874760730334008210751053, 90214737649279161118742424717958, 292430390737314945077403507199314, 947910902974975125498167902611520, 3072646032833061109985967400156622, 9959958909064257882537009925266631, 32285131580477797928734610518164237, 104652010182509089086278617109993462, 339228700615315977444886651121429277, 1099607270997158652107972420434959108, 3564368664080031314816967613357662055, 11553874104483341210266121734821142852, 37451795648277928344402333096353352122, 121399712736708221184079769560762938697, 393516251956611845013355998261548502986, 1275579958659616700051280090889278833336, 4134782802092428057033363756220138609836, 13402867224759701057700959574386344814347, 43445292883009879313305648170648654700516, 140827588756804206245792902950047144571845, 456491565345423491623610899518571378989501, 1479713961384194804651394365443287707951165, 4796481630188476585130471830982889775002867, 15547758978508507586519529449665118925185513, 50397943303348600340667739889787041267129812, 163364552583976126251251664091135649288323601, 529544963379278740998141336491369035339503075, 1716515999370275253596005050996726025897952062, 5564073647857169678752190766348711401608477488, 18035902706491664108492067744933299576135929095, 58463242405734536420325913236612655559063290781, 189508158710650708569791718748430383359357309179, 614289265187565338765801657780948511563597752234, 1991214013644844373951509787590765084713268211648, 6454505186453110204987938053972586732041513391004, 20922229813806817934491794193363189841269374500818, 67819249924918610363545453678403420549113592056709, 219835586422215046131422169384174937357998062801249
Generating function in Maple syntax:
-(x^4+x^3-4*x^2+4*x-1)/(x^5-x^4-4*x^3+7*x^2-5*x+1)
Generating function in latex syntax:
-\frac{x^{4}+x^{3}-4 x^{2}+4 x -1}{x^{5}-x^{4}-4 x^{3}+7 x^{2}-5 x +1}
Generating function in sympy syntax:
(-x**4 - x**3 + 4*x**2 - 4*x + 1)/(x**5 - x**4 - 4*x**3 + 7*x**2 - 5*x + 1)
Implicit equation for the generating function in Maple syntax:
(x^5-x^4-4*x^3+7*x^2-5*x+1)*F(x)+x^4+x^3-4*x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}-x^{4}-4 x^{3}+7 x^{2}-5 x +1\right) F \! \left(x \right)+x^{4}+x^{3}-4 x^{2}+4 x -1 = 0
Explicit closed form in Maple syntax:
1798/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n+3)+1798/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n+3)+1798/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n+3)+1798/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n+3)+1798/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n+3)+978/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n+2)+978/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n+2)+978/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n+2)+978/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n+2)+978/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n+2)-4931/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n+1)-4931/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n+1)-4931/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n+1)-4931/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n+1)-4931/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n+1)-503/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n-1)-503/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n-1)-503/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n-1)-503/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n-1)-503/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n-1)+8007/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 1)^(-n)+8007/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 2)^(-n)+8007/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 3)^(-n)+8007/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 4)^(-n)+8007/27007*RootOf(_Z^5-_Z^4-4*_Z^3+7*_Z^2-5*_Z+1,index = 5)^(-n)
Explicit closed form in latex syntax:
\frac{1798 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +3}}{27007}+\frac{1798 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +3}}{27007}+\frac{1798 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +3}}{27007}+\frac{1798 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +3}}{27007}+\frac{1798 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +3}}{27007}+\frac{978 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +2}}{27007}+\frac{978 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +2}}{27007}+\frac{978 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +2}}{27007}+\frac{978 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +2}}{27007}+\frac{978 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +2}}{27007}-\frac{4931 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n +1}}{27007}-\frac{4931 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n +1}}{27007}-\frac{4931 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n +1}}{27007}-\frac{4931 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n +1}}{27007}-\frac{4931 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n +1}}{27007}-\frac{503 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n -1}}{27007}-\frac{503 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n -1}}{27007}-\frac{503 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n -1}}{27007}-\frac{503 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n -1}}{27007}-\frac{503 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n -1}}{27007}+\frac{8007 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =1\right)^{-n}}{27007}+\frac{8007 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =2\right)^{-n}}{27007}+\frac{8007 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =3\right)^{-n}}{27007}+\frac{8007 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =4\right)^{-n}}{27007}+\frac{8007 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-\textit{\_Z}^{4}-4 \textit{\_Z}^{3}+7 \textit{\_Z}^{2}-5 \textit{\_Z} +1, \mathit{index} =5\right)^{-n}}{27007}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 20
a(n+5) = -a(n)+a(n+1)+4*a(n+2)-7*a(n+3)+5*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) = 20
a \! \left(n +5\right) = -a \! \left(n \right)+a \! \left(n +1\right)+4 a \! \left(n +2\right)-7 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 5
Specification 1
Strategy pack name: point_placements
Tree: http://permpal.com/tree/15333/
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[11,x]
F[18,x] = F[14,x]
F[19,x] = F[2,x]+F[20,x]
F[20,x] = F[21,x]+F[22,x]+F[35,x]
F[21,x] = 0
F[22,x] = F[23,x]*F[4,x]
F[23,x] = F[24,x]+F[28,x]
F[24,x] = F[11,x]+F[25,x]
F[25,x] = F[26,x]
F[26,x] = F[27,x]*F[4,x]
F[27,x] = F[7,x]
F[28,x] = F[29,x]+F[32,x]
F[29,x] = F[21,x]+F[22,x]+F[30,x]
F[30,x] = F[31,x]*F[4,x]
F[31,x] = F[2,x]+F[29,x]
F[32,x] = F[33,x]
F[33,x] = F[34,x]*F[4,x]
F[34,x] = F[20,x]
F[35,x] = F[36,x]*F[4,x]
F[36,x] = F[31,x]+F[37,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[29,x]
F[42,x] = F[38,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_{11}\! \left(x \right)
F_{18}\! \left(x \right) = F_{14}\! \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_{35}\! \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_{28}\! \left(x \right)
F_{24}\! \left(x \right) = F_{11}\! \left(x \right)+F_{25}\! \left(x \right)
F_{25}\! \left(x \right) = F_{26}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right) F_{4}\! \left(x \right)
F_{27}\! \left(x \right) = F_{7}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{32}\! \left(x \right)
F_{29}\! \left(x \right) = F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{31}\! \left(x \right) F_{4}\! \left(x \right)
F_{31}\! \left(x \right) = F_{2}\! \left(x \right)+F_{29}\! \left(x \right)
F_{32}\! \left(x \right) = F_{33}\! \left(x \right)
F_{33}\! \left(x \right) = F_{34}\! \left(x \right) F_{4}\! \left(x \right)
F_{34}\! \left(x \right) = F_{20}\! \left(x \right)
F_{35}\! \left(x \right) = F_{36}\! \left(x \right) F_{4}\! \left(x \right)
F_{36}\! \left(x \right) = F_{31}\! \left(x \right)+F_{37}\! \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_{29}\! \left(x \right)
F_{42}\! \left(x \right) = F_{38}\! \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_11(x))
Eq(F_18(x), F_14(x))
Eq(F_19(x), F_2(x) + F_20(x))
Eq(F_20(x), F_21(x) + F_22(x) + F_35(x))
Eq(F_21(x), 0)
Eq(F_22(x), F_23(x)*F_4(x))
Eq(F_23(x), F_24(x) + F_28(x))
Eq(F_24(x), F_11(x) + F_25(x))
Eq(F_25(x), F_26(x))
Eq(F_26(x), F_27(x)*F_4(x))
Eq(F_27(x), F_7(x))
Eq(F_28(x), F_29(x) + F_32(x))
Eq(F_29(x), F_21(x) + F_22(x) + F_30(x))
Eq(F_30(x), F_31(x)*F_4(x))
Eq(F_31(x), F_2(x) + F_29(x))
Eq(F_32(x), F_33(x))
Eq(F_33(x), F_34(x)*F_4(x))
Eq(F_34(x), F_20(x))
Eq(F_35(x), F_36(x)*F_4(x))
Eq(F_36(x), F_31(x) + F_37(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_29(x))
Eq(F_42(x), F_38(x))
Pack JSON:
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Specification JSON:
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