0132_0231_1023_1032_2031
Counting sequence:
1, 1, 2, 6, 19, 59, 180, 545, 1647, 4977, 15043, 45473, 137464, 415553, 1256212, 3797508, 11479796, 34703205, 104907130, 317132270, 958684865, 2898086256, 8760860065, 26483914652, 80060374207, 242021000371, 731624916824, 2211688316710, 6685892043586, 20211325475112, 61098455493741, 184699477939605, 558343036259833, 1687860461856762, 5102370323776270, 15424369199521011, 46627577009552335, 140954285368717024, 426102144654341469, 1288098742114050735, 3893898189087405097, 11771180742008411151, 35584057243546862741, 107569933523591049440, 325181879038491058773, 983018683671617109176, 2971646929726662244704, 8983232589202585475944, 27156142590309067059441, 82092506574034926933826, 248164098166611826900274, 750195385535062094263721, 2267826493178935728950816, 6855596691648668965204193, 20724339423631493100493488, 62649286978782089427598719, 189387612252397439997886315, 572515177815383865105434476, 1730702578329968621200996622, 5231881233380839489964408838, 15815878235193725305386052320, 47811101436037936800303550137, 144532057375131235989682638601, 436917681911896212834994928257, 1320793907138496397433444635314, 3992735055950310851319517622630, 12069962725337500795806251162771, 36487269540693934547993925133843, 110300327252918249353290900747772, 333435807755689839796621752574937, 1007970154419898554456639796963727, 3047074754927657496535085745214785, 9211249481350867112086783841279195, 27845433352258214199313137466446745, 84176219539473730752043719180187416, 254463123138395936015172365619636857, 769237219152872389614978771143189372, 2325389596858086799731968336462102876, 7029608867770063455789400052428341900, 21250374948180015531102631590768488941, 64239482442425956374491971941840498378, 194194743129658713804583681111809854406, 587047043739693923177064819338881784769, 1774632134781411616932579956672936452320, 5364679453518019842286093712201543958097, 16217324748570113553170023571602811600468, 49024666670086867290362817080207426003903, 148200642175895025994835877762945517350259, 448007744532998338613541315994356164501376, 1354318957156247235324335300427061774888822, 4094080649486824161359278690852058713473674, 12376328542057533083802355446595663393720552, 37413407623063702785562106738931156708881253, 113100025198330275019547176372568058228792477, 341899241810240259151102015305431566863424025, 1033554955849319240008019625070196533981261098, 3124417126825852476695228272745308735867537422, 9445054012034463155196619047924219357554070323, 28552220036278340140043647563606275996337400327, 86312822347159245334638926483949628237246841736, 260922033105182509449144120053201739044792147157
Generating function in Maple syntax:
-(x-1)*(x^3-2*x^2+3*x-1)/(x^5-3*x^4+7*x^3-8*x^2+5*x-1)
Generating function in latex syntax:
-\frac{\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)}{x^{5}-3 x^{4}+7 x^{3}-8 x^{2}+5 x -1}
Generating function in sympy syntax:
(1 - x)*(x**3 - 2*x**2 + 3*x - 1)/(x**5 - 3*x**4 + 7*x**3 - 8*x**2 + 5*x - 1)
Implicit equation for the generating function in Maple syntax:
(x^5-3*x^4+7*x^3-8*x^2+5*x-1)*F(x)+(x-1)*(x^3-2*x^2+3*x-1) = 0
Implicit equation for the generating function in latex syntax:
\left(x^{5}-3 x^{4}+7 x^{3}-8 x^{2}+5 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) = 0
Explicit closed form in Maple syntax:
-122/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+3)-122/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+3)-122/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+3)-122/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+3)-122/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+3)+432/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+2)+432/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+2)+432/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+2)+432/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+2)+432/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+2)-959/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n+1)-959/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n+1)-959/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n+1)-959/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n+1)-959/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n+1)-171/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n-1)-171/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n-1)-171/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n-1)-171/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n-1)-171/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n-1)+1409/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 1)^(-n)+1409/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 2)^(-n)+1409/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 3)^(-n)+1409/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 4)^(-n)+1409/2617*RootOf(_Z^5-3*_Z^4+7*_Z^3-8*_Z^2+5*_Z-1,index = 5)^(-n)
Explicit closed form in latex syntax:
-\frac{122 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +3}}{2617}-\frac{122 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +3}}{2617}-\frac{122 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +3}}{2617}-\frac{122 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +3}}{2617}-\frac{122 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +3}}{2617}+\frac{432 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +2}}{2617}+\frac{432 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +2}}{2617}+\frac{432 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +2}}{2617}+\frac{432 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +2}}{2617}+\frac{432 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +2}}{2617}-\frac{959 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n +1}}{2617}-\frac{959 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n +1}}{2617}-\frac{959 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n +1}}{2617}-\frac{959 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n +1}}{2617}-\frac{959 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n +1}}{2617}-\frac{171 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n -1}}{2617}-\frac{171 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n -1}}{2617}-\frac{171 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n -1}}{2617}-\frac{171 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n -1}}{2617}-\frac{171 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n -1}}{2617}+\frac{1409 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =1\right)^{-n}}{2617}+\frac{1409 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =2\right)^{-n}}{2617}+\frac{1409 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =3\right)^{-n}}{2617}+\frac{1409 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =4\right)^{-n}}{2617}+\frac{1409 \mathit{RootOf}\! \left(\textit{\_Z}^{5}-3 \textit{\_Z}^{4}+7 \textit{\_Z}^{3}-8 \textit{\_Z}^{2}+5 \textit{\_Z} -1, \mathit{index} =5\right)^{-n}}{2617}
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 19
a(n+5) = a(n)-3*a(n+1)+7*a(n+2)-8*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) = 19
a \! \left(n +5\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+7 a \! \left(n +2\right)-8 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/18752/
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[22,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[20,x]
F[17,x] = F[18,x]+F[4,x]
F[18,x] = F[19,x]
F[19,x] = F[17,x]*F[4,x]
F[20,x] = F[21,x]
F[21,x] = F[15,x]
F[22,x] = F[2,x]+F[23,x]
F[23,x] = F[24,x]+F[25,x]+F[38,x]
F[24,x] = 0
F[25,x] = F[26,x]*F[4,x]
F[26,x] = F[27,x]+F[28,x]
F[27,x] = F[21,x]+F[4,x]
F[28,x] = F[29,x]+F[31,x]
F[29,x] = F[24,x]+F[25,x]+F[30,x]
F[30,x] = F[2,x]*F[4,x]
F[31,x] = F[32,x]
F[32,x] = F[33,x]*F[4,x]
F[33,x] = F[34,x]+F[37,x]
F[34,x] = F[29,x]+F[35,x]
F[35,x] = F[36,x]
F[36,x] = F[34,x]*F[4,x]
F[37,x] = F[31,x]
F[38,x] = F[39,x]*F[4,x]
F[39,x] = F[40,x]+F[43,x]
F[40,x] = F[2,x]+F[41,x]
F[41,x] = F[24,x]+F[25,x]+F[42,x]
F[42,x] = F[4,x]*F[40,x]
F[43,x] = F[23,x]+F[44,x]
F[44,x] = F[32,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_{22}\! \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_{20}\! \left(x \right)
F_{17}\! \left(x \right) = F_{18}\! \left(x \right)+F_{4}\! \left(x \right)
F_{18}\! \left(x \right) = F_{19}\! \left(x \right)
F_{19}\! \left(x \right) = F_{17}\! \left(x \right) F_{4}\! \left(x \right)
F_{20}\! \left(x \right) = F_{21}\! \left(x \right)
F_{21}\! \left(x \right) = F_{15}\! \left(x \right)
F_{22}\! \left(x \right) = F_{2}\! \left(x \right)+F_{23}\! \left(x \right)
F_{23}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{38}\! \left(x \right)
F_{24}\! \left(x \right) = 0
F_{25}\! \left(x \right) = F_{26}\! \left(x \right) F_{4}\! \left(x \right)
F_{26}\! \left(x \right) = F_{27}\! \left(x \right)+F_{28}\! \left(x \right)
F_{27}\! \left(x \right) = F_{21}\! \left(x \right)+F_{4}\! \left(x \right)
F_{28}\! \left(x \right) = F_{29}\! \left(x \right)+F_{31}\! \left(x \right)
F_{29}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{30}\! \left(x \right)
F_{30}\! \left(x \right) = F_{2}\! \left(x \right) F_{4}\! \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_{34}\! \left(x \right)+F_{37}\! \left(x \right)
F_{34}\! \left(x \right) = F_{29}\! \left(x \right)+F_{35}\! \left(x \right)
F_{35}\! \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_{31}\! \left(x \right)
F_{38}\! \left(x \right) = F_{39}\! \left(x \right) F_{4}\! \left(x \right)
F_{39}\! \left(x \right) = F_{40}\! \left(x \right)+F_{43}\! \left(x \right)
F_{40}\! \left(x \right) = F_{2}\! \left(x \right)+F_{41}\! \left(x \right)
F_{41}\! \left(x \right) = F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{42}\! \left(x \right)
F_{42}\! \left(x \right) = F_{4}\! \left(x \right) F_{40}\! \left(x \right)
F_{43}\! \left(x \right) = F_{23}\! \left(x \right)+F_{44}\! \left(x \right)
F_{44}\! \left(x \right) = F_{32}\! \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_22(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_20(x))
Eq(F_17(x), F_18(x) + F_4(x))
Eq(F_18(x), F_19(x))
Eq(F_19(x), F_17(x)*F_4(x))
Eq(F_20(x), F_21(x))
Eq(F_21(x), F_15(x))
Eq(F_22(x), F_2(x) + F_23(x))
Eq(F_23(x), F_24(x) + F_25(x) + F_38(x))
Eq(F_24(x), 0)
Eq(F_25(x), F_26(x)*F_4(x))
Eq(F_26(x), F_27(x) + F_28(x))
Eq(F_27(x), F_21(x) + F_4(x))
Eq(F_28(x), F_29(x) + F_31(x))
Eq(F_29(x), F_24(x) + F_25(x) + F_30(x))
Eq(F_30(x), F_2(x)*F_4(x))
Eq(F_31(x), F_32(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_29(x) + F_35(x))
Eq(F_35(x), F_36(x))
Eq(F_36(x), F_34(x)*F_4(x))
Eq(F_37(x), F_31(x))
Eq(F_38(x), F_39(x)*F_4(x))
Eq(F_39(x), F_40(x) + F_43(x))
Eq(F_40(x), F_2(x) + F_41(x))
Eq(F_41(x), F_24(x) + F_25(x) + F_42(x))
Eq(F_42(x), F_4(x)*F_40(x))
Eq(F_43(x), F_23(x) + F_44(x))
Eq(F_44(x), F_32(x))
Pack JSON:
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Specification JSON:
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