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A051258 Fibocyclotomic numbers: numbers formed from cyclotomic polynomials and Fibonacci numbers (A000045). 11
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A199458 A287480 A287755 * A063704 A224918 A224508
KEYWORD
nonn,nice
AUTHOR
Antti Karttunen, Oct 24 1999
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)