# [seqfan] Re: Integer Sequence Analysis in Mathematica 7

Alexander Povolotsky apovolot at gmail.com
Sat Nov 22 02:45:03 CET 2008

```Dear Maximilian,

I was not talking ggf, I meant closed form analytical formula

For example PURRS (The Parma University's Recurrence Relation Solver)
http://www.cs.unipr.it/purrs/  yields the following:

Exact solution for x(n) = x(-1+n)+x(-3+n)
x(n) = 1/2*(I*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-I*sqrt(3)*(29/54-1/18*sqrt(93))^(1
/3))^(-1)*(-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)+3/2*(29/54+1/18
*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)+3/2*(29/54-1/18
*sqrt(93))^(1/3))^(-1)*(29/54+1/18*sqrt(93))^(1/3)*(1/3+(1/2*I)*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54
-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(1)+1/2*(I*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-I*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(-(1/2*I)
*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)+3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)
*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)+3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(29
/54-1/18*sqrt(93))^(1/3)*(1/3+(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1/2
*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2*(29
/54-1/18*sqrt(93))^(1/3))^n*x(1)+(I*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-I
*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)
*sqrt(3)+3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18
*sqrt(93))^(1/3)+3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(29/54+1/18*sqrt(93))^(1
/3)*(29/54-1/18*sqrt(93))^(1/3)*(1/3+(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)
-1/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2
*(29/54-1/18*sqrt(93))^(1/3))^n*x(0)+(-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)
*sqrt(3)-3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18
*sqrt(93))^(1/3)-3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(29/54-1/18*sqrt(93))^(2
/3)*x(0)*(1/3+(29/54+1/18*sqrt(93))^(1/3)+(29/54-1/18*sqrt(93))^(1/3))^n*((1/2
*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)
*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)+(29
/54+1/18*sqrt(93))^(2/3)*(-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-3/2*(29
/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54
-1/18*sqrt(93))^(1/3))^(-1)*x(0)*(1/3+(29/54+1/18*sqrt(93))^(1/3)+(29/54-1/18
*sqrt(93))^(1/3))^n*((1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18
*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18
*sqrt(93))^(1/3))^(-1)+(-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54
+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54-1
/18*sqrt(93))^(1/3))^(-1)*(29/54-1/18*sqrt(93))^(1/3)*x(1)*(1/3+(29/54+1/18
*sqrt(93))^(1/3)+(29/54-1/18*sqrt(93))^(1/3))^n*((1/2*I)*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54
-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)-1/6*(I*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-I*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(-(1/2*I)
*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)+3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)
*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)+3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(29
/54-1/18*sqrt(93))^(1/3)*(1/3+(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1/2
*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2*(29
/54-1/18*sqrt(93))^(1/3))^n*x(0)-1/6*(I*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-I
*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)
*sqrt(3)+3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18
*sqrt(93))^(1/3)+3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(29/54+1/18*sqrt(93))^(1
/3)*(1/3+(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18
*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18
*sqrt(93))^(1/3))^n*x(0)+(-3/2*(29/54-1/18*sqrt(93))^(2/3)-(3/2*I)*sqrt(3)*(29
/54-1/18*sqrt(93))^(2/3)+3*(29/54+1/18*sqrt(93))^(1/3)*(29/54-1/18*sqrt(93))^(1
/3)-3/2*(29/54+1/18*sqrt(93))^(2/3)+(3/2*I)*(29/54+1/18*sqrt(93))^(2/3)
*sqrt(3))^(-1)*(1/3-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18
*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18
*sqrt(93))^(1/3))^n*x(2)-(1/2*I)*sqrt(3)*(-3/2*(29/54-1/18*sqrt(93))^(2/3)-(3/2
*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(2/3)+3*(29/54+1/18*sqrt(93))^(1/3)*(29/54-1
/18*sqrt(93))^(1/3)-3/2*(29/54+1/18*sqrt(93))^(2/3)+(3/2*I)*(29/54+1/18
*sqrt(93))^(2/3)*sqrt(3))^(-1)*(29/54-1/18*sqrt(93))^(2/3)*(1/3-(1/2*I)*(29/54+1
/18*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29
/54-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(0)+1/9*(-(1/2*I)
*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)
*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*x(0)
*(1/3+(29/54+1/18*sqrt(93))^(1/3)+(29/54-1/18*sqrt(93))^(1/3))^n*((1/2*I)*(29/54
+1/18*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)
*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)-1/3*(29/54+1
/18*sqrt(93))^(1/3)*(-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1
/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18
*sqrt(93))^(1/3))^(-1)*x(0)*(1/3+(29/54+1/18*sqrt(93))^(1/3)+(29/54-1/18
*sqrt(93))^(1/3))^n*((1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18
*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18
*sqrt(93))^(1/3))^(-1)-1/2*(29/54+1/18*sqrt(93))^(1/3)*(-3/2*(29/54-1/18
*sqrt(93))^(2/3)-(3/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(2/3)+3*(29/54+1/18
*sqrt(93))^(1/3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54+1/18*sqrt(93))^(2/3)+(3
/2*I)*(29/54+1/18*sqrt(93))^(2/3)*sqrt(3))^(-1)*(1/3-(1/2*I)*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54
-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(1)+(1/2*I)*sqrt(3)*(
-3/2*(29/54-1/18*sqrt(93))^(2/3)-(3/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(2/3)+3
*(29/54+1/18*sqrt(93))^(1/3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54+1/18
*sqrt(93))^(2/3)+(3/2*I)*(29/54+1/18*sqrt(93))^(2/3)*sqrt(3))^(-1)*(29/54-1/18
*sqrt(93))^(1/3)*(1/3-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1
/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18
*sqrt(93))^(1/3))^n*x(1)+1/2*(I*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-I*sqrt(3)
*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)
+3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)+3/2
*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(29/54+1/18*sqrt(93))^(2/3)*(1/3+(1/2*I)*(29
/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)
*(29/54-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(0)+1/2*(I*(29
/54+1/18*sqrt(93))^(1/3)*sqrt(3)-I*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(
-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)+3/2*(29/54+1/18*sqrt(93))^(1/3)+(1
/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)+3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)
*(29/54-1/18*sqrt(93))^(2/3)*(1/3+(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1
/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2
*(29/54-1/18*sqrt(93))^(1/3))^n*x(0)-(I*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-I
*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)
*sqrt(3)+3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18
*sqrt(93))^(1/3)+3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(1/3+(1/2*I)*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54
-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(2)-(1/6*I)*sqrt(3)*(
-3/2*(29/54-1/18*sqrt(93))^(2/3)-(3/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(2/3)+3
*(29/54+1/18*sqrt(93))^(1/3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54+1/18
*sqrt(93))^(2/3)+(3/2*I)*(29/54+1/18*sqrt(93))^(2/3)*sqrt(3))^(-1)*(29/54-1/18
*sqrt(93))^(1/3)*(1/3-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1
/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18
*sqrt(93))^(1/3))^n*x(0)+1/6*(29/54+1/18*sqrt(93))^(1/3)*(-3/2*(29/54-1/18
*sqrt(93))^(2/3)-(3/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(2/3)+3*(29/54+1/18
*sqrt(93))^(1/3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54+1/18*sqrt(93))^(2/3)+(3
/2*I)*(29/54+1/18*sqrt(93))^(2/3)*sqrt(3))^(-1)*(1/3-(1/2*I)*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54
-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(0)+(1/2*I)*(I*(29/54
+1/18*sqrt(93))^(1/3)*sqrt(3)-I*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(-(1/2
*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)+3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)
*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)+3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(29
/54+1/18*sqrt(93))^(2/3)*sqrt(3)*(1/3+(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)
*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54-1/18
*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(0)+(1/2*I)*(29/54+1/18
*sqrt(93))^(2/3)*sqrt(3)*(-3/2*(29/54-1/18*sqrt(93))^(2/3)-(3/2*I)*sqrt(3)*(29
/54-1/18*sqrt(93))^(2/3)+3*(29/54+1/18*sqrt(93))^(1/3)*(29/54-1/18*sqrt(93))^(1
/3)-3/2*(29/54+1/18*sqrt(93))^(2/3)+(3/2*I)*(29/54+1/18*sqrt(93))^(2/3)
*sqrt(3))^(-1)*(1/3-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18
*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18
*sqrt(93))^(1/3))^n*x(0)-2/3*(-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-3/2
*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29
/54-1/18*sqrt(93))^(1/3))^(-1)*x(1)*(1/3+(29/54+1/18*sqrt(93))^(1/3)+(29/54-1/18
*sqrt(93))^(1/3))^n*((1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18
*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18
*sqrt(93))^(1/3))^(-1)+(29/54+1/18*sqrt(93))^(1/3)*(-(1/2*I)*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54
-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*x(1)*(1/3+(29/54+1
/18*sqrt(93))^(1/3)+(29/54-1/18*sqrt(93))^(1/3))^n*((1/2*I)*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54
-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)+2/3*(I*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-I*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(-(1/2*I)
*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)+3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)
*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)+3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(1/3
+(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)-(1
/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n
*x(1)-1/2*(-3/2*(29/54-1/18*sqrt(93))^(2/3)-(3/2*I)*sqrt(3)*(29/54-1/18
*sqrt(93))^(2/3)+3*(29/54+1/18*sqrt(93))^(1/3)*(29/54-1/18*sqrt(93))^(1/3)-3/2
*(29/54+1/18*sqrt(93))^(2/3)+(3/2*I)*(29/54+1/18*sqrt(93))^(2/3)*sqrt(3))^(-1)
*(29/54-1/18*sqrt(93))^(1/3)*(1/3-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1
/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2
*(29/54-1/18*sqrt(93))^(1/3))^n*x(1)-(29/54+1/18*sqrt(93))^(1/3)*(-(1/2*I)*(29
/54+1/18*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)
*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(29/54-1/18
*sqrt(93))^(1/3)*x(0)*(1/3+(29/54+1/18*sqrt(93))^(1/3)+(29/54-1/18*sqrt(93))^(1
/3))^n*((1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18*sqrt(93))^(1
/3)-(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18*sqrt(93))^(1
/3))^(-1)+(1/6*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)*(-3/2*(29/54-1/18
*sqrt(93))^(2/3)-(3/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(2/3)+3*(29/54+1/18
*sqrt(93))^(1/3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54+1/18*sqrt(93))^(2/3)+(3
/2*I)*(29/54+1/18*sqrt(93))^(2/3)*sqrt(3))^(-1)*(1/3-(1/2*I)*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54
-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(0)-1/3*(-(1/2*I)*(29
/54+1/18*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)
*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(29/54-1/18
*sqrt(93))^(1/3)*x(0)*(1/3+(29/54+1/18*sqrt(93))^(1/3)+(29/54-1/18*sqrt(93))^(1
/3))^n*((1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18*sqrt(93))^(1
/3)-(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18*sqrt(93))^(1
/3))^(-1)-(1/2*I)*(I*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-I*sqrt(3)*(29/54-1/18
*sqrt(93))^(1/3))^(-1)*(-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)+3/2*(29/54
+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)+3/2*(29/54-1
/18*sqrt(93))^(1/3))^(-1)*sqrt(3)*(29/54-1/18*sqrt(93))^(2/3)*(1/3+(1/2*I)*(29
/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)
*(29/54-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(0)-(1/2*I)*(29
/54+1/18*sqrt(93))^(1/3)*sqrt(3)*(-3/2*(29/54-1/18*sqrt(93))^(2/3)-(3/2*I)
*sqrt(3)*(29/54-1/18*sqrt(93))^(2/3)+3*(29/54+1/18*sqrt(93))^(1/3)*(29/54-1/18
*sqrt(93))^(1/3)-3/2*(29/54+1/18*sqrt(93))^(2/3)+(3/2*I)*(29/54+1/18
*sqrt(93))^(2/3)*sqrt(3))^(-1)*(1/3-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)
-1/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2
*(29/54-1/18*sqrt(93))^(1/3))^n*x(1)-1/9*(I*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)
-I*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(-(1/2*I)*(29/54+1/18*sqrt(93))^(1
/3)*sqrt(3)+3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18
*sqrt(93))^(1/3)+3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(1/3+(1/2*I)*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54
-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(0)+1/6*(-3/2*(29/54-1
/18*sqrt(93))^(2/3)-(3/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(2/3)+3*(29/54+1/18
*sqrt(93))^(1/3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54+1/18*sqrt(93))^(2/3)+(3
/2*I)*(29/54+1/18*sqrt(93))^(2/3)*sqrt(3))^(-1)*(29/54-1/18*sqrt(93))^(1/3)*(1/3
-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)+(1
/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n
*x(0)-(29/54+1/18*sqrt(93))^(1/3)*(-3/2*(29/54-1/18*sqrt(93))^(2/3)-(3/2*I)
*sqrt(3)*(29/54-1/18*sqrt(93))^(2/3)+3*(29/54+1/18*sqrt(93))^(1/3)*(29/54-1/18
*sqrt(93))^(1/3)-3/2*(29/54+1/18*sqrt(93))^(2/3)+(3/2*I)*(29/54+1/18
*sqrt(93))^(2/3)*sqrt(3))^(-1)*(29/54-1/18*sqrt(93))^(1/3)*(1/3-(1/2*I)*(29/54+1
/18*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29
/54-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(0)+(1/2*I)*(I*(29
/54+1/18*sqrt(93))^(1/3)*sqrt(3)-I*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(
-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)+3/2*(29/54+1/18*sqrt(93))^(1/3)+(1
/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)+3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)
*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)*(1/3+(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)
*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54-1/18
*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(1)-2/3*(-3/2*(29/54-1/18
*sqrt(93))^(2/3)-(3/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(2/3)+3*(29/54+1/18
*sqrt(93))^(1/3)*(29/54-1/18*sqrt(93))^(1/3)-3/2*(29/54+1/18*sqrt(93))^(2/3)+(3
/2*I)*(29/54+1/18*sqrt(93))^(2/3)*sqrt(3))^(-1)*(1/3-(1/2*I)*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54
-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(1)-(1/2*I)*(I*(29/54
+1/18*sqrt(93))^(1/3)*sqrt(3)-I*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(-(1/2
*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)+3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)
*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)+3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(29
/54+1/18*sqrt(93))^(1/3)*sqrt(3)*(1/3+(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)
*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54-1/18
*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(1)-(1/6*I)*(I*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-I*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(-(1/2*I)
*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)+3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)
*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)+3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)
*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)*(1/3+(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)
*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54-1/18
*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(0)+(-(1/2*I)*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54
-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(1/3+(29/54+1/18
*sqrt(93))^(1/3)+(29/54-1/18*sqrt(93))^(1/3))^n*x(2)*((1/2*I)*(29/54+1/18
*sqrt(93))^(1/3)*sqrt(3)-3/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54
-1/18*sqrt(93))^(1/3)-3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)+(1/6*I)*(I*(29/54+1
/18*sqrt(93))^(1/3)*sqrt(3)-I*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(-(1/2
*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)+3/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)
*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)+3/2*(29/54-1/18*sqrt(93))^(1/3))^(-1)*(29
/54+1/18*sqrt(93))^(1/3)*sqrt(3)*(1/3+(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)
*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1/3)-(1/2*I)*sqrt(3)*(29/54-1/18
*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1/3))^n*x(0)-1/2*(29/54+1/18
*sqrt(93))^(2/3)*(-3/2*(29/54-1/18*sqrt(93))^(2/3)-(3/2*I)*sqrt(3)*(29/54-1/18
*sqrt(93))^(2/3)+3*(29/54+1/18*sqrt(93))^(1/3)*(29/54-1/18*sqrt(93))^(1/3)-3/2
*(29/54+1/18*sqrt(93))^(2/3)+(3/2*I)*(29/54+1/18*sqrt(93))^(2/3)*sqrt(3))^(-1)
*(1/3-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1
/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1
/3))^n*x(0)+1/9*(-3/2*(29/54-1/18*sqrt(93))^(2/3)-(3/2*I)*sqrt(3)*(29/54-1/18
*sqrt(93))^(2/3)+3*(29/54+1/18*sqrt(93))^(1/3)*(29/54-1/18*sqrt(93))^(1/3)-3/2
*(29/54+1/18*sqrt(93))^(2/3)+(3/2*I)*(29/54+1/18*sqrt(93))^(2/3)*sqrt(3))^(-1)
*(1/3-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1/2*(29/54+1/18*sqrt(93))^(1
/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2*(29/54-1/18*sqrt(93))^(1
/3))^n*x(0)-1/2*(-3/2*(29/54-1/18*sqrt(93))^(2/3)-(3/2*I)*sqrt(3)*(29/54-1/18
*sqrt(93))^(2/3)+3*(29/54+1/18*sqrt(93))^(1/3)*(29/54-1/18*sqrt(93))^(1/3)-3/2
*(29/54+1/18*sqrt(93))^(2/3)+(3/2*I)*(29/54+1/18*sqrt(93))^(2/3)*sqrt(3))^(-1)
*(29/54-1/18*sqrt(93))^(2/3)*(1/3-(1/2*I)*(29/54+1/18*sqrt(93))^(1/3)*sqrt(3)-1
/2*(29/54+1/18*sqrt(93))^(1/3)+(1/2*I)*sqrt(3)*(29/54-1/18*sqrt(93))^(1/3)-1/2
*(29/54-1/18*sqrt(93))^(1/3))^n*x(0)
for each n >= 0

But when trying to add initial conditions for exact solution PURRS demo gives
PURRS Demo Error
memory limit exceeded

ARP

On Fri, Nov 21, 2008 at 8:28 PM, Maximilian Hasler
<maximilian.hasler at gmail.com> wrote:
> On Fri, Nov 21, 2008 at 8:09 PM, Alexander Povolotsky
> <apovolot at gmail.com> wrote:
>> Could you if it is allowed  (just to get a real "by example" feel of
>> those new exciting  Mathematica 7 capabilities) show us what close
>> formula is suggested by Mathematica 7 for
>> A001609                  a(n) = a(n-1) + a(n-3).
>>        1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144, 211, 309, 453, 664,
>> 973, 1426, 2090, 3063, 4489, 6579, 9642, 14131, 20710, 30352, 44483,
>> 65193, 95545, 140028, 205221, 300766, 440794, 646015, 946781, 1387575,
>> 2033590, 2980371, 4367946, 6401536, 9381907
>
>
> I don't think you need M-- (I like the minuses ;-P) for this:
>
> ggf(v)={    local( p,q,B=#v\2 );    B<4 & return;
>    if( !#q = qflll( matrix(B, B, x, y, v[x-y+B+1]), 4)[1], return);
>    if ( polcoeff( p = Ser( q[,1] ), 0 )<0, p=-p );
>    q = Pol( Ser( v )*p );    if( Ser(v) == q /= Pol(p), q)
> }
>
> ggf( [       1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144, 211, 309,
> 453, 664])
> %1 = (3*x^2 + 1)/(-x^3 - x + 1)
>
> Vec(%+O(x^90))
> %2 = [1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144, 211, 309, 453,
> 664, 973, 1426, 2090, 3063, 4489, 6579, 9642, 14131, 20710, 30352,
> 44483, 65193, 95545, 140028, 205221, 300766, 440794, 646015, 946781,
> 1387575, 2033590, 2980371, 4367946, 6401536, 9381907, 13749853,
> 20151389, 29533296, 43283149, 63434538, 92967834, 136250983,
> 199685521, 292653355, 428904338, 628589859, 921243214, 1350147552,
> 1978737411, 2899980625, 4250128177, 6228865588, 9128846213,
> 13378974390, 19607839978, 28736686191, 42115660581, 61723500559,
> 90460186750, 132575847331, 194299347890, 284759534640, 417335381971,
> 611634729861, 896394264501, 1313729646472, 1925364376333,
> 2821758640834, 4135488287306, 6060852663639, 8882611304473,
> 13018099591779, 19078952255418, 27961563559891, 40979663151670,
> 60058615407088, 88020178966979, 128999842118649, 189058457525737,
> 277078636492716, 406078478611365, 595136936137102, 872215572629818]
>
> Maximilian
>

```