#C6.txt, Feb. 8, 2016, Dr. Z.'s ExpMath class RU (Spring 2016)
#c=introducing Dr. Z.'s second-most favorite "ansatz"
Help:=proc() print(` RecToSeq(R,N), RecToSeqW(R,N,w)`): end:

#RecToSeq(R,N): inputs a C-finite sequence [ini,rec] where ini  is a list
#of numbers of length k say, and rec is a list of coefficients of the
#C-finite sequence defined UNIQELY by
#x(n)=ini[n+1], for n=0..k-1, 
#and rec is a list of numbers [c1,c2,...,ck]
#and for n>=k defined recursively by
#x(n)=c1*x(n-1)+c2*x(n-2)+...+ck*x(n-k).
#For example the sequence 2^n is coded as
#[[1],[2]], and the sequence of Fib. numbers are given by
#[[0,1],[1,1]]
#It outputs the list consisting of the first N terms of that sequence starting at n=0
RecToSeq:=proc(R,N) local ini,rec,Se,k,n,nt:

ini:=R[1]:
rec:=R[2]:
k:=nops(ini):

if nops(rec)<>k then
 RETURN(FAIL):
fi:

Se:=ini:

for n from k+1 to N do

#nt:=c1*Se[n]+c2*Se[n-1]+...+ ck*se[n-k+1]

nt:=add(rec[i]*Se[n-i],i=1..k):

Se:=[op(Se),nt]:

od:


Se:

end:



#RecToSeqW(R,N,w): inputs a C-finite sequence [ini,rec] where ini  is a list
#of numbers of length k say, and rec is a list of coefficients of the
#C-finite sequence defined UNIQELY by
#x(n)=ini[n+1], for n=0..k-1, 
#and rec is a list of numbers [c1,c2,...,ck]
#and for n>=k defined recursively by
#x(n)=c1*x(n-1)+c2*x(n-2)+...+ck*x(n-k).
#For example the sequence 2^n is coded as
#[[1],[2]], and the sequence of Fib. numbers are given by
#[[0,1],[1,1]]
#It outputs the list consisting of the terms N-w+1 through w of that sequence starting at n=N-w+1
RecToSeqW:=proc(R,N,w) local ini,rec,Se,k,n,nt:

ini:=R[1]:
rec:=R[2]:
k:=nops(ini):



if nops(rec)<>k then
 RETURN(FAIL):
fi:

if w<k then
  RETURN(FAIL):
fi:

Se:=ini:

for n from k+1 to N do

nt:=add(rec[i]*Se[n-i],i=1..k):

Se:=[op(Se),nt]:

od:


for n from k+1 to w do

nt:=add(rec[i]*Se[n-i],i=1..k):

Se:=[op(Se),nt]:

od:


for n from w+1 to N do

nt:=add(rec[i]*Se[w-i+1],i=1..k):

Se:=[op(2..w,Se),nt]:

od:



Se:

end: