A051258 Fibocyclotomic numbers: numbers formed from cyclotomic polynomials and Fibonacci numbers (A000045).

1, 1, 1, 2, 1, 7, 0, 20, 3, 10, 1, 143, 2, 376, 4, 11, 21, 2583, 6, 6764, 15, 74, 33, 46367, 18, 7435, 88, 2618, 104, 832039, 25, 2178308, 987, 3399, 609, 20160, 136, 39088168, 1596, 23228, 861, 267914295, 182, 701408732, 4895, 35920, 10945, 4807526975

Offset: 0

Author

Antti Karttunen, Oct 24 1999

Keywords

• nonn
• nice

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.

Examples

			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.)
		

Links

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

Crossrefs

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

Programs

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 *)

PARI

a(n)=my(P=polcyclo(n));sum(i=1,poldegree(P),polcoeff(P,i)*fibonacci(i)) \\ Charles R Greathouse IV, Jan 05 2013

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.