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
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
Programs
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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;
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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.
Comments