Help:=proc() : print( `IsConst(L),Di(L),IsPol(L) , GuessPol(L,x) `): end:


#Di(L): the sequence of differences of consectutive entries of L
#For example,Di([1,4,9]); should be [3,5].
Di:=proc(L) local i:

[seq(L[i]-L[i-1],i=2..nops(L))]:

end:

#IsConst(L): is the sequence of numbers L a constant sequence:
IsConst:=proc(L) 
   evalb( nops(convert(L,set))<=1 ):
end:


#IsPol(L): inputs a list of length L and returns
#a conjectured degree if it (conj.) a polynomial
#and FAIL otherwise
IsPol:=proc(L) local L1, k:

L1:=L:

for k from 1 to nops(L)-4 do
 L1:=Di(L1):

 if L1=[0$nops(L1)] then
    RETURN(k-1):
 fi:

od:

FAIL:

end:


#GuessPol(L,x): guesses a polynomial expression P(x)
#such that P(i)=L[i] for all i between 1 and nops(L)
GuessPol:=proc(L,x) local d,P,var, eq,a,i,c:

d:=IsPol(L):


if d=FAIL then
 RETURN(d):
fi:

P:=add( a[i]*x^i, i=0..d):

var:={ seq( a[i], i=0..d)}:

eq:={ seq( subs(x=c,P)=L[c], c=1..nops(L))}:


var:=solve(eq,var):


if var=NULL then
 RETURN(FAIL):
fi:

subs( var, P):


end: