A107848 -id:A107848 - cp's OEIS frontend

A107847 Related to sums of the n-th roots of unity: sums in a circular wedge (excluding the origin).

1, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 333, 630, 1161, 2182, 4080, 7710, 14508, 27594, 52371, 99858, 190557, 364722, 698634, 1342176, 2580795, 4971008, 9586377, 18512790, 35786499, 69273666, 134215680, 260300986, 505286415, 981706806

Offset: 1

T. D. Noe, May 25 2005

nonn

Consider the 2^n sums formed from all the subsets of the n-th roots of unity. The number A103314(n) tells how many of these sums are zero. The remaining sums fall into n wedges centered at the origin. The number a(n) gives the number of sums that fall into each wedge. Interestingly, a(n) coincides with A059966(n) when n is either p^k or pq for primes p and q.

Max Alekseyev and M. F. Hasler, Table of n, a(n) for n = 1..164
T. D. Noe, Sums of Roots of Unity Plots

Cf. A103314 (number of subsets of the n-th roots of unity summing to zero), A107848 (number of subsets of the n-th roots of unity summing to a real number).

Cf. also A110981.

a(n) = (2^n - A103314(n))/n.

a(n) = A001037(n) - A110981(n). - Max Alekseyev, Jan 14 2008

A299807 Rectangular array read by antidiagonals: T(n,k) is the number of distinct sums of k complex n-th roots of 1, n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 9, 10, 5, 1, 1, 6, 15, 16, 15, 6, 1, 1, 7, 19, 35, 25, 21, 7, 1, 1, 8, 28, 37, 70, 36, 28, 8, 1, 1, 9, 33, 84, 61, 126, 49, 36, 9, 1, 1, 10, 45, 96, 210, 91, 210, 64, 45, 10, 1, 1, 11, 51, 163, 225, 462, 127, 330, 81, 55, 11, 1, 1, 12, 66, 180, 477, 456, 924, 169, 495, 100, 66

Offset: 1

Max Alekseyev, Feb 24 2018

			Array starts:
  n=1:  1,  1,  1,   1,   1,    1,    1,    1,     1,     1,     1, ...
  n=2:  1,  2,  3,   4,   5,    6,    7,    8,     9,    10,    11, ...
  n=3:  1,  3,  6,  10,  15,   21,   28,   36,    45,    55,    66, ...
  n=4:  1,  4,  9,  16,  25,   36,   49,   64,    81,   100,   121, ...
  n=5:  1,  5, 15,  35,  70,  126,  210,  330,   495,   715,  1001, ...
  n=6:  1,  6, 19,  37,  61,   91,  127,  169,   217,   271,   331, ...
  n=7:  1,  7, 28,  84, 210,  462,  924, 1716,  3003,  5005,  8008, ...
  n=8:  1,  8, 33,  96, 225,  456,  833, 1408,  2241,  3400,  4961, ...
  n=9:  1,  9, 45, 163, 477, 1197, 2674, 5454, 10341, 18469, 31383, ...
  n=10: 1, 10, 51, 180, 501, 1131, 2221, 3951,  6531, 10201, 15231, ...
  ...

Max Alekseyev, Table of n, a(n) for n = 1..351

Rows: A000012 (n=1), A000027 (n=2), A000217 (n=3), A000290 (n=4), A000332 (n=5), A354343 (n=6), A000579 (n=7), A014820 (n=8).
Columns: A000012 (k=0), A000027 (k=1), A031940 (k=2).
Diagonal: A299754 (n=k).
Cf. A103314, A107754, A107861, A108380, A107848, A107753, A108381, A143008.

From Chai Wah Wu, May 28 2018: (Start)
The following are all conjectures.
For m >= 0, the 2^(m+1)-th row are the figurate numbers based on the 2^m-dimensional regular convex polytope with g.f.: (1+x)^(2^m-1)/(1-x)^(2^m+1).
For p prime, the n=p row corresponds to binomial(k+p-1,p-1) for k = 0,1,2,3,..., which is the maximum possible (i.e., the number of combinations with repetitions of k choices from p categories) with g.f.: 1/(1-x)^p.
(End)

Row 6 corrected by Max Alekseyev, Aug 14 2022

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

Original entry on oeis.org

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

Offset: 1

David W. Wilson, Feb 18 2018

nonn

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

			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)

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

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

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 *)

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

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

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

A354343 Number of distinct sums of n complex 6th power roots of unity.

Original entry on oeis.org

1, 6, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, 3367, 3571, 3781, 3997, 4219, 4447, 4681, 4921, 5167, 5419, 5677, 5941, 6211, 6487, 6769, 7057, 7351, 7651, 7957

Offset: 0

Max Alekseyev, Aug 15 2022

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Same as A003215 except for a(1) = 6.
Row 6 of A299807.
Cf. A000012, A000027, A000217, A000290, A000332, A000579, A014820, A103314, A107754, A107861, A108380, A107848, A107753, A108381, A143008, A299754.

LinearRecurrence[{3,-3,1},{1,6,19,37,61},60] (* Harvey P. Dale, Nov 03 2024 *)

a(n)=if(n==1, 6, 3*n*(n+1)+1) \\ Charles R Greathouse IV, Aug 15 2022

For n >= 2, a(n) = 3*n^2 + 3*n + 1 = A003215(n).
For n >= 5, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f. (1 + 3*x + 4*x^2 - 3*x^3 + x^4) / (1 - x)^3.

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A107847 Related to sums of the n-th roots of unity: sums in a circular wedge (excluding the origin).

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A299807 Rectangular array read by antidiagonals: T(n,k) is the number of distinct sums of k complex n-th roots of 1, n >= 1, k >= 0.

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

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A354343 Number of distinct sums of n complex 6th power roots of unity.

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