A051258 - OEIS

%I #20 Apr 15 2014 01:38:40

%S 1,1,1,2,1,7,0,20,3,10,1,143,2,376,4,11,21,2583,6,6764,15,74,33,46367,

%T 18,7435,88,2618,104,832039,25,2178308,987,3399,609,20160,136,

%U 39088168,1596,23228,861,267914295,182,701408732,4895,35920,10945,4807526975

%N Fibocyclotomic numbers: numbers formed from cyclotomic polynomials and Fibonacci numbers (A000045).

%C 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).

%C a(0) depends on how the zeroth cyclotomic polynomial is defined.

%H T. D. Noe, <a href="/A051258/b051258.txt">Table of n, a(n) for n = 0..500</a>

%F 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.

%e 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.)

%p 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;

%p 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;

%p 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;

%t 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 *)

%o (PARI) a(n)=my(P=polcyclo(n));sum(i=1,poldegree(P),polcoeff(P,i)*fibonacci(i)) \\ _Charles R Greathouse IV_, Jan 05 2013

%Y Cf. A019320, A054433, A063704, A063706, A063708.

%K nonn,nice

%O 0,4

%A _Antti Karttunen_, Oct 24 1999