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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). 6
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, 0, 1092, 0, 56, 0, 0, 599186, 43690, 8726850, 64, 0, 174762, 9585810, 4112, 0, 792, 0, 279620, 2101256, 2796202, 0, 256, 0, 32800, 2454267026, 4473924, 0, 512 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

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.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..3003

D. M. Bloom, On the Coefficients of the Cyclotomic Polynomials, Amer.Math.Monthly 75, 372-377, 1968.

T.Y. Lam and K. H. Leung, On the Cyclotomic Polynomial Phi_pq(X), Amer.Math.Monthly 103, 562-564, August-September 1996.

H. Lenstra, Vanishing sums of roots of unity, in Proc. Bicentennial Congress Wiskundig Genootschap (Vrije Univ. Amsterdam, 1978), Part II, pp. 249-268.

Index entries for cyclotomic polynomials, positions of coefficients

MAPLE

with(numtheory); [seq(Phi_neg_terms(j, 2), j=0..104)];

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;

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;

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;

MATHEMATICA

a[n_] := 2^(Flatten[Position[CoefficientList[Cyclotomic[n, x], x], _?Negative]] - 1) // Total; a[0] = 0; Table[a[n], {n, 0, 60}] (* Jean-Fran├žois Alcover, Mar 05 2016 *)

PROG

(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

CROSSREFS

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

Sequence in context: A329039 A181502 A223154 * A287466 A288014 A179677

Adjacent sequences:  A063695 A063696 A063697 * A063699 A063700 A063701

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 03 2001

STATUS

approved

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Last modified April 5 23:12 EDT 2020. Contains 333260 sequences. (Running on oeis4.)