A063697 -id:A063697 - cp's OEIS frontend

A063672 Sequence A019320 in binary.

10, 1, 11, 111, 101, 11111, 11, 1111111, 10001, 1001001, 1011, 11111111111, 1101, 1111111111111, 101011, 10010111, 100000001, 11111111111111111, 111001, 1111111111111111111, 11001101, 100100110111, 1010101011

Offset: 0

Antti Karttunen, Aug 03 2001

Paolo Xausa, Table of n, a(n) for n = 0..900

Cf. A063697, A063699, A063671.

map(convert, A019320,binary); or up to n=104 with: map(convert,[seq(Phi_pos_terms(j,2)-Phi_neg_terms(j,2),j=0..104)],binary);

A063672[n_] := If[n == 0, 10, FromDigits[IntegerDigits[Cyclotomic[n, 2], 2]]];
Array[A063672, 30, 0] (* Paolo Xausa, Feb 26 2024 *)

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).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 03 2001

nonn

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.

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

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

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;

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 *)

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

A063671 Positions of nonzero coefficients in cyclotomic polynomial Phi_n(x), A063670 in binary.

Original entry on oeis.org

10, 11, 11, 111, 101, 11111, 111, 1111111, 10001, 1001001, 11111, 11111111111, 10101, 1111111111111, 1111111, 110111011, 100000001, 11111111111111111, 1001001, 1111111111111111111, 101010101, 1101101011011, 11111111111

Offset: 0

Antti Karttunen, Aug 03 2001

a(0) could also be 1. - T. D. Noe, Oct 29 2007

			Phi_15(x) = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1, thus the 1-bits of a(15) are at positions 0,1,3,4,5,7 and 8, thus we get a(15) = 110111011.

Cf. A063672, A063697, A063699.

Cf. A013595.

Maple
```
map(convert, A063670,binary);
```

a[n_] := FromDigits[Abs[CoefficientList[Cyclotomic[n, x], x]]]; a[0]=10;Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Nov 02 2016 *)

a(n) = A063697(n) (the positive terms) + A063699(n) (the negative terms) (computed in any base, up to n=104).

A063696 Positions of positive coefficients in cyclotomic polynomial Phi_n(x), converted from binary to decimal.

Original entry on oeis.org

0, 2, 3, 7, 5, 31, 5, 127, 17, 73, 21, 2047, 17, 8191, 85, 297, 257, 131071, 65, 524287, 273, 4681, 1365, 8388607, 257, 1082401, 5461, 262657, 4369, 536870911, 387, 2147483647, 65537, 1198665, 87381, 17454241, 4097, 137438953471, 349525

Offset: 0

Antti Karttunen, Aug 03 2001

nonn

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 A063698 and A063694.

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. W. 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

Cf. A013594, A063697 (binary version), A063698 (negative terms), A063670 (nonzero terms).

A019320(n) = a(n) - A063698(n) for up to n=104.

with(numtheory); [seq(Phi_pos_terms(j,2),j=0..104)];
inv_p_mod_q := (p,q) -> op(2,op(1,msolve(p*x=1,q))); # Find's p's inverse modulo q.
dilate := proc(nn,x,e) local n,i,s; n := nn; i := 0; s := 0; while(n > 0) do s := s + (((x^e)^i)*(n mod x)); n := floor(n/x); i := i+1; od; RETURN(s); end;
Phi_pos_terms := proc(n,x) local a,m,p,q,e,f,r,s; if(n < 2) then RETURN(x); fi; a := op(2, ifactors(n)); m := nops(a); p := a[1][1]; e := a[1][2]; if(1 = m) then RETURN(((x^(p^e))-1)/((x^(p^(e-1)))-1)); 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( (`if`(0=s,1,(((x^((s+1)*((q^f)*(p^(e-1)))))-1)/((x^((q^f)*(p^(e-1))))-1)))) * (`if`(0=r,1,(((x^((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_pos_terms(n/2,x),x,0)+every_other_pos(Phi_neg_terms(n/2,x),x,1)); else RETURN(dilate(Phi_pos_terms((n/(2^(e-1))),x),x,2^(e-1))); fi; else printf(`Cannot handle argument %a with three or more distinct odd prime factors!\n`,n); RETURN(0); fi; end;

a[n_] := 2^(Flatten[Position[CoefficientList[Cyclotomic[n, x], x], ?Positive]] - 1) // Total; a[0] = 0; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover, Mar 05 2016 *)

a(n)=local(p); if(n<1,0,p=polcyclo(n); sum(i=0,n,2^i*(polcoeff(p,i)>0)))

A063699 Positions of negative coefficients in cyclotomic polynomial Phi_n(x), A063698 in binary.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 10, 0, 0, 0, 1010, 0, 100, 0, 101010, 10010010, 0, 0, 1000, 0, 1000100, 100100010010, 1010101010, 0, 10000, 0, 101010101010, 0, 10001000100, 0, 111000, 0, 0, 10010010010010010010, 1010101010101010, 100001010010100101000010

Offset: 0

Antti Karttunen, Aug 03 2001

			E.g. Phi_15(x) = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1, thus the 1-bits of a(15) are at positions 1,4 and 7, thus we get a(15) = 10010010

Index entries for cyclotomic polynomials, positions of coefficients

Cf. A063697, A063671, A063672.

Maple
```
map(convert, A063698,binary);
```

A118887 Number of ways to place n objects with weights 1,2,...,n evenly spaced around the circumference of a circular disk so that the disk will exactly balance on the center point.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 0, 24, 0, 732, 0, 720, 48, 0, 0

Offset: 1

Hugo Pfoertner, May 03 2006

The position of weight 1 is kept fixed at position 1. Mirror configurations are counted only once. Proposed in the seqfan mailing list by Brendan McKay, Sep 12 2005
Also number of permutations p1,p2,...,pn such that the polynomial p1 + p2*x + ... + pn*x^(n-1) has exp(2*pi*i/n) as a zero. Also number of equiangular polygons whose sides are some permutation of 1,2,3,...,n. - T. D. Noe, Sep 13 2005
No solutions exist if n is a prime power. Proved by W. Edwin Clark, Sep 14 2005
Murray Klamkin proved that solutions do exist if n is not a prime power. - Jonathan Sondow, Oct 17 2013

			The smallest n for which a solution exists is n=6 with 4 solutions:
...........Weight
......1..2..3..4..5..6
.Count...at.position
..1...1..4..5..2..3..6
..2...1..5..3..4..2..6
..3...1..6..2..4..3..5
..4...1..6..3..2..5..4
Configurations 1 is the mirror image of configuration 4, ditto for configurations 2 and 3. Therefore a(6)=2.

Andrew Bernoff, Bernoff's Puzzler, MuddMath Newsletter Volume 4, No. 1, Page 10, Spring 2005
Marius Munteanu and Laura Munteanu, Rational equiangular polygons, Applied Math., 4 (2013), 1460-1465.
Hugo Pfoertner, Balanced weights on circle (Tables of configurations)
G. J. Woeginger, Nothing new about equiangular polygons, Amer. Math. Monthly, 120 (2013), 849-850.

Cf. A118888 (configurations with minimum imbalance), A063697 (positions of positive coefficients in cyclotomic polynomial in binary), A063699 (positions of negative coefficients in cyclotomic polynomial in binary), A326921.

Needs["DiscreteMath`Combinatorica`"]; Table[eLst=E^(2.*Pi*I*Range[n]/n); Count[(Permutations[Range[n]]), q_List/;Chop[q.eLst]===0]/(2n), {n,10}] (* very slow for n>10 *) (* T. D. Noe, May 05 2006 *)

a(A000961(n)) = 0, a(A024619(n)) > 0. - Jonathan Sondow, Oct 17 2013

cp's OEIS Frontend

A063672 Sequence A019320 in binary.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A063671 Positions of nonzero coefficients in cyclotomic polynomial Phi_n(x), A063670 in binary.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A063696 Positions of positive coefficients in cyclotomic polynomial Phi_n(x), converted from binary to decimal.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A063699 Positions of negative coefficients in cyclotomic polynomial Phi_n(x), A063698 in binary.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A118887 Number of ways to place n objects with weights 1,2,...,n evenly spaced around the circumference of a circular disk so that the disk will exactly balance on the center point.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula