0,4

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 zero-th cyclotomic polynomial is defined.

T. D. Noe, Table of n, a(n) for n=0..500

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.

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 22th 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).

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;

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

Sequence in context: A100404 A103114 A004561 this_sequence A063704 A116891 A079620

Adjacent sequences: A051255 A051256 A051257 this_sequence A051259 A051260 A051261

nonn,nice

Antti Karttunen Oct 24 1999