> #ok to post
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> #Yifan Zhang, hw7, 09/29/30
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> 
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> 
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> #Q1. 
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> GFseq:=proc(f,x,N) local  f1,i:
> f1:=taylor(f,x=0,N+1):
> [seq(coeff(f1,x,i),i=0..N)]:
> end:
> GFseq(1/(1-x-x^2-x^3),x,30);
[1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 
19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770,
8646064, 15902591, 29249425, 53798080]

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> #OEIS: A73 		Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 3 with a(0) = a(1) = 0 and a(2) = 1.
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> 
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> GFseq(1/(1-x+x^2-x^3),x,30);
[1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1,
0, 0, 1, 1, 0]

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> GFseq(exp(-x)/(1-x),x,30);
[1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760, 16687/45360, 16481/
44800, 1468457/3991680, 16019531/43545600, 63633137/172972800, 2467007773/
6706022400, 34361893981/93405312000, 15549624751/42268262400, 8178130767479/
22230464256000, 138547156531409/376610217984000, 92079694567171/250298560512000
, 4282366656425369/11640679464960000, 72289643288657479/196503623737344000, 
6563440628747948887/17841281393295360000, 39299278806015611311/
106826515449937920000, 9923922230666898717143/26976017466662584320000, 
79253545592131482810517/215433472824041472000000, 5934505493938805432851513/
16131658445064225423360000, 14006262966463963871240459/
38072970106357874688000000, 461572649528573755888451011/
1254684545727217532928000000, 116167945043852116348068366947/
315777214062132212662272000000, 3364864615063302680426807870189/
9146650338351415815045120000000]

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> 
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> 
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> 
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> #Q2.
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> Rec:=proc(P,n)  local L, INI,k,i:
> option remember:
> 
> #We first unpack P
> L:=P[1]:
> INI:=P[2]:
> 
> k:=nops(L):
> 
> if nops(INI)<>k then
>  print(INI, `should have`, k, `entries `):
>  RETURN(FAIL):
> fi:
> 
> 
> if n<k then
> #For this case we use the initial condition, since list in Maple start at index 1, we need to access INI[n+1]
>  RETURN(INI[n+1]):
> else
> #We use the definig recurrence
> add(L[i]*Rec(P,n-i),i=1..k):
> fi:
> 
> end:
> SeqRec:=proc(P,N) local n:
> [seq(Rec(P,n),n=0..N)]:
> end:
> evalb(SeqRec([[1,1],[0,1]],30)=GFseq(x/(1-x-x^2),x,30));
true

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> #Both of them are: f(n)=f(n-1)+f(n-2), for f(0)=0 and f(1) = 1.
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> 
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> 
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> 
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> #Q3.
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> #1) a(n) = 3*a(n-1)-5*a(n-2), a(0)=1, a(1)=0
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> #This is a second-order linear recurrence with constant coefficient [[3,-5], [1,0]];
> 
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> #2)a(n) = c1*a(n-1)+c2*a(n-2)+...+ck*a(n-k), c1=1, c2=c3=...=ck=0
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> #This is a k-th order linear recurrence with constant coefficient [[c1,c2,c3,...,ck], [1,0,0,0,.....,0]].
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> 
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> 
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> 
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