A299754 Number of distinct sums of n complex n-th roots of 1.

1, 3, 10, 25, 126, 127, 1716, 2241, 18469, 15231, 352716, 36973, 5200300, 1799995, 30333601, 24331777, 1166803110, 12247363, 17672631900, 723276561

Offset

1,2

Comments

a(n) == 1 (mod n).

Also, a(n) equals the size of the set { f(x) mod Phi_n(x) }, where f(x) runs over the polynomials of degree at most n-1 with nonnegative integer coefficients such that f(1)=n (i.e. the coefficients sum to n), Phi_n(x) is the n-th cyclotomic polynomial. In particular, for prime n, Phi_n(x)=1+x+...+x^(n-1) and thus all f(x) mod Phi_n(x) are distinct, implying that a(n)=binomial(2*n-1,n). - Max Alekseyev, Feb 20 2018

Links

Table of n, a(n) for n=1..20.

Formula

For prime p, a(p) = binomial(2*p-1,p). - Conjectured by Robert Israel, Feb 18 2018; proved by Max Alekseyev, Feb 20 2018

a(n) = A299807(n,n). - Max Alekseyev, Feb 25 2018

Example

From M. F. Hasler, Feb 18 2018: (Start)
For n=2, the n-th roots of unity are U[2] = {-1, 1}, and taking x, y in this set, we can get x + y = -2, 0 or 2.
For n=3, the n-th roots of unity are U[3] = {1, w, w^2} where w = exp(2i*Pi/3) = -1/2 + i sqrt(3)/2, and taking x, y, z in this set, we can get x + y + z to be any of the 10 distinct values { 3, 2 + w, 2 + w^2, 1 + 2w, 1 + 2w^2, 0, w - 1, w^2 - 1, 3w, 3w^2 }. (End)

Maple

nexti:= proc(i, N) local ip, j, k;
  ip:= i;
  for k from N to 1 by -1 while i[k]=N-1 do od;
  if k=0 then return NULL fi;
  ip[k]:= ip[k]+1;
  for j from k+1 to N do ip[j]:= ip[k] od;
  ip
end proc:
f:= proc(N) local S, i, P, z;
  S:= {}:
  i:= <(0$N)>:
  P:= numtheory:-cyclotomic(N, z);
  while i <> NULL do
    S:= S union {rem(add(z^i[k], k=1..N), P, z)};
    i:= nexti(i, N);
  od;
  nops(S);
end proc:
seq(f(N), N=1..10); # Robert Israel, Feb 18 2018

Mathematica

a[n_] := (t = Table[Exp[2 k Pi I/n], {k, 0, n - 1}]; b[0] = 1; iter = Table[{b[j], b[j - 1], n}, {j, 1, n}]; msets = Table[Array[b, n], Evaluate[Sequence @@ iter]]; tot = Total /@ (t[[#]] & /@ Flatten[msets, n - 1]) // N; u = Union[tot, SameTest -> (Chop[Abs[#1 - #2]] == 0 &)]; u // Length); Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 10}] (* Jean-François Alcover, Feb 19 2018 *)

Prog

(PARI) a(n, U=vector(n, k, bestappr(exp(2*Pi/n*k*I), 5*2^n)), S=[])={forvec(i=vector(n, k, [1, n]), S=setunion(S, [vecsum(vecextract(U, i))]); #S} \\ Not very efficient for n > 8. - M. F. Hasler, Feb 18 2018

Crossrefs

Cf. A103314, A107754, A107861, A108380, A107848, A107753, A108381.

Sequence in context: A098702 A100001 A227937 * A190529 A343584 A335970

Adjacent sequences: A299751 A299752 A299753 * A299755 A299756 A299757

Keyword

nonn

Author

David W. Wilson, Feb 18 2018

Extensions

a(1) through a(11) from Robert Israel, Feb 18 2018

a(12)-a(13) from Chai Wah Wu, Feb 20 2018

a(14)-a(20) from Max Alekseyev, Feb 22 2018

Status

approved