OFFSET
0,4
COMMENTS
For all primes p, a(p) = fib(p+1)-1 and for all n of the form 2^i*p^j (where p is an odd prime and i >= 0 and j >= 2) fib(p)|a(2^i*p^j).
a(0) depends on how the zeroth cyclotomic polynomial is defined.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..500
FORMULA
a(n) = Sum (coefficient_of_term_i_of_cp_n times Fib(exponent_of_term_i_of_cp_n)), i=1..degree of cp_n, where cp_n is the n-th cyclotomic polynomial.
EXAMPLE
a(22) = fib(10)-fib(9)+fib(8)-fib(7)+fib(6)-fib(5)+fib(4)-fib(3)+fib(2)-fib(1) = 33 as the 22nd cyclotomic polynomial is x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1 (The constant term does not affect the result, as fib(0)=0.)
MAPLE
get_coefficient := proc(e); if(1 = nops(e)) then if(`integer` = op(0, e)) then RETURN(e); else RETURN(1); fi; else if(2 = nops(e)) then if(`*` = op(0, e)) then RETURN(op(1, e)); else RETURN(1); fi; else RETURN(`Cannot find coefficient!`); fi; fi; end;
get_exponent := proc(e); if((1 = e) or (-1 = e)) then RETURN(0); else if(1 = nops(e)) then RETURN(1); else if(2 = nops(e)) then if(`^` = op(0, e)) then RETURN(op(2, e)); else RETURN(get_exponent(op(2, e))); fi; else RETURN(`Cannot find exponent!`); fi; fi; fi; end;
fibo_cyclotomic := proc(j) local i, p; p := sort(cyclotomic(j, x)); RETURN(add((get_coefficient(op(i, p))*fibonacci(get_exponent(op(i, p)))), i=1..nops(p))); end;
MATHEMATICA
f[n_]:=Module[{cy=CoefficientList[Cyclotomic[n, x], x]}, Total[ Times@@@ Thread[ {Fibonacci[ Range[0, Length[cy]- 1]], cy}]]]; Join[{1}, Array[f, 50]] (* Harvey P. Dale, Oct 02 2011 *)
PROG
(PARI) a(n)=my(P=polcyclo(n)); sum(i=1, poldegree(P), polcoeff(P, i)*fibonacci(i)) \\ Charles R Greathouse IV, Jan 05 2013
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Antti Karttunen, Oct 24 1999
STATUS
approved