"Recurrence equations:"

"I(N+1) = I(N)+J(N)     +L(N)"

"J(N+1) = I(N)"

"K(N+1) =      J(N)+K(N)"

"L(N+1) =           K(N)"

"M(N+1) =                L(N)+2M(N)"

""

"Initial conditions:"

"I(0) = 1"

"J(0) = 0"

"K(0) = 0"

"L(0) = 0"

"M(0) = 0"

""

"Values of the equations for the first few N's:"

"  N  = 0 1 2 3  4  5  6   7   8 ... to infinity"

"I(N) = 1 1 2 3  6 11 21  39  73 ..."

"J(N) = 0 1 1 2  3  6 11  21  39 ..."

"K(N) = 0 0 1 2  4  7 13  24  45 ..."

"L(N) = 0 0 0 1  2  4  7  13  24 ..."

"M(N) = 0 0 0 0  1  4 12  31  75 ..."

""

"This can be solved using a simple iterates function."

"Here is an example of the of the iterates function."

"NEWTON(u, x, x0, n) :=    ITERATES(x - u/DIF(u, x), x, x0, n)"

"First a vector v is defined so v=[n,i,j,k,l,m]."

"initial value of v is v0:=[0,1,0,0,0,0]."

v0:=[0,1,0,0,0,0]

"The recurrence equations:"

FRED_I(v):=v SUB 2+v SUB 3+v SUB 5

FRED_J(v):=v SUB 2

FRED_K(v):=v SUB 3+v SUB 4

FRED_L(v):=v SUB 4

FRED_M(v):=v SUB 5+2*v SUB 6

"This can be one equation."

FRED_AUX(v):=[v SUB 1+1,FRED_I(v),FRED_J(v),FRED_K(v),FRED_L(v),FRED_M(v)]

"This is a solution:"

FRED(n):=ITERATES(FRED_AUX(v_),v_,v0,n)

"Test the it with n=8."

FRED(8)

;Simp(#40)
[[0,1,0,0,0,0],[1,1,1,0,0,0],[2,2,1,1,0,0],[3,3,2,2,1,0],[4,6,3,4,2,1],[5,11,~
6,7,4,4],[6,21,11,13,7,12],[7,39,21,24,13,31],[8,73,39,45,24,75]]

"There are simplier solutions. This is one example."

FRED(16)` SUB 5

;Simp(User)
[0,0,0,1,2,4,7,13,24,45,84,157,293,547,1021,1906,3558]

"LOOKUP 1 2 4 7 13 24 45 84 157 293 547 1021 1906 3558"

FRED(22)` SUB 5

;Simp(#46)
[0,0,0,1,2,4,7,13,24,45,84,157,293,547,1021,1906,3558,6642,12399,23146,43208,~
80659,150571]

[6642,12399,23146,43208,80659,150571]

"1+(x^4-x^3+2*x-1)*F(x) = 0"

;Simp(#36')
FRED_AUX(v):=[v SUB 1+1,v SUB 5+v SUB 3+v SUB 2,v SUB 2,v SUB 4+v SUB 3,v SUB~
 4,2*v SUB 6+v SUB 5]

FRED1_AUX(v):=[v SUB 4+v SUB 2+v SUB 1,v SUB 1,v SUB 3+v SUB 2,v SUB 3]

FRED1(n):=ITERATES(FRED1_AUX(v_),v_,[1,0,0,0],n)

FRED1(22)` SUB 4

;Simp(#53)
[0,0,0,1,2,4,7,13,24,45,84,157,293,547,1021,1906,3558,6642,12399,23146,43208,~
80659,150571]