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A051258
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Fibocyclotomic numbers: numbers formed from cyclotomic polynomials and Fibonacci numbers (A000045).
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9
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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; internal format)
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OFFSET
| 0,4
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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.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..500
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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.
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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 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).
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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]] (* From Harvey P. Dale, Oct 02 2011 *)
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CROSSREFS
| Cf. A019320, A054433, A063704, A063706, A063708.
Sequence in context: A103114 A004561 A199458 * A063704 A116891 A079620
Adjacent sequences: A051255 A051256 A051257 * A051259 A051260 A051261
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KEYWORD
| nonn,nice
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AUTHOR
| Antti Karttunen Oct 24 1999
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