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A051258 Fibocyclotomic numbers: numbers formed from cyclotomic polynomials and Fibonacci numbers (A000045). 10
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

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

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

Sequence in context: A199458 A287480 A287755 * A063704 A224918 A224508

Adjacent sequences:  A051255 A051256 A051257 * A051259 A051260 A051261

KEYWORD

nonn,nice

AUTHOR

Antti Karttunen, Oct 24 1999

STATUS

approved

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Last modified December 11 21:15 EST 2017. Contains 295919 sequences.