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A063698 Positions of negative coefficients in cyclotomic polynomial Phi_n(x), converted from binary to decimal. (The constant term in the least significant bit (bit-0), the term x in the next bit (bit-1) and so on).

%I A063698 #20 Aug 06 2018 22:26:20
%S A063698 0,1,0,0,0,0,2,0,0,0,10,0,4,0,42,146,0,0,8,0,68,2322,682,0,16,0,2730,
%T A063698 0,1092,0,56,0,0,599186,43690,8726850,64,0,174762,9585810,4112,0,792,
%U A063698 0,279620,2101256,2796202,0,256,0,32800,2454267026,4473924,0,512
%N A063698 Positions of negative coefficients in cyclotomic polynomial Phi_n(x), converted from binary to decimal. (The constant term in the least significant bit (bit-0), the term x in the next bit (bit-1) and so on).
%C A063698 Maple procedures Phi_pos_terms and Phi_neg_terms are modeled after the formula given in Lam and Leung paper and they compute correct results for all integers x > 1 and for all n with at most two distinct odd prime factors (that is, up to n=104). Other procedures as in A063696 and A063694.
%H A063698 Antti Karttunen, <a href="/A063698/b063698.txt">Table of n, a(n) for n = 0..3003</a>
%H A063698 D. M. Bloom, <a href="http://www.jstor.org/stable/2313417">On the Coefficients of the Cyclotomic Polynomials</a>, Amer.Math.Monthly 75, 372-377, 1968.
%H A063698 T.Y. Lam and K. H. Leung, <a href="http://www.jstor.org/stable/2974668">On the Cyclotomic Polynomial Phi_pq(X)</a>, Amer.Math.Monthly 103, 562-564, August-September 1996.
%H A063698 H. Lenstra, <a href="http://hdl.handle.net/1887/3800">Vanishing sums of roots of unity</a>, in Proc. Bicentennial Congress Wiskundig Genootschap (Vrije Univ. Amsterdam, 1978), Part II, pp. 249-268.
%H A063698 <a href="/index/Cy#CyclotomicPolynomialsCoefficients">Index entries for cyclotomic polynomials, positions of coefficients</a>
%p A063698 with(numtheory); [seq(Phi_neg_terms(j,2),j=0..104)];
%p A063698 Phi_neg_terms := proc(n,x) local a,m,p,q,e,f,r,s; if(n < 2) then RETURN(n); fi; a := op(2, ifactors(n)); m := nops(a); p := a[1][1]; e := a[1][2]; if(1 = m) then RETURN(0); fi; if(2 = m) then q := a[2][1]; f := a[2][2]; r := inv_p_mod_q(p,q)-1; s := inv_p_mod_q(q,p)-1;
%p A063698 RETURN( x^((s+1)*(q^f)*(p^(e-1))) * x^((r+1)*(p^e)*(q^(f-1))) * x^(-((p^e) * (q^f))) * (`if`((p-2)=s,1,(((x^((p-s-1)*((q^f)*(p^(e-1)))))-1)/((x^((q^f)*(p^(e-1))))-1)))) * (`if`((q-2)=r,1,(((x^((q-r-1)*((p^e)*(q^(f-1)))))-1)/((x^((p^e)*(q^(f-1))))-1)))) ); fi;
%p A063698 if((3 = m) and (2 = p)) then if(1 = e) then RETURN(every_other_pos(Phi_neg_terms(n/2,x),x,0)+every_other_pos(Phi_pos_terms(n/2,x),x,1)); else RETURN(dilate(Phi_neg_terms((n/(2^(e-1))),x),x,2^(e-1))); fi; else printf(`Cannot handle argument %a with >=3 distinct odd prime factors!\n`,n); RETURN(0); fi; end;
%t A063698 a[n_] := 2^(Flatten[Position[CoefficientList[Cyclotomic[n, x], x], _?Negative]] - 1) // Total; a[0] = 0; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Mar 05 2016 *)
%o A063698 (PARI) a(n)=my(p); if(n<1, 0, p=polcyclo(n); sum(i=0, n, 2^i*(polcoeff(p, i)<0))) \\ _Michel Marcus_, Mar 05 2016
%Y A063698 A013594, A063696 gives the positions of the positive and A063697 the nonzero terms. This sequence in binary: A063699. A019320[n] = A063696[n]-A063698[n] for up to n=104
%K A063698 nonn
%O A063698 0,7
%A A063698 _Antti Karttunen_, Aug 03 2001