cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

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

Views

Author

Max Alekseyev, Feb 24 2018

Keywords

Examples

			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, ...
  ...

Links

Crossrefs

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.

Formula

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)

Extensions

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

Views

Author

David W. Wilson, Feb 18 2018

Keywords

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

Examples

			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)

Crossrefs

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

Programs

  • 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 *)
  • 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

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

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

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

Views

Author

Max Alekseyev, Aug 15 2022

Keywords

Links

Crossrefs

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.

Programs

Formula

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.
Showing 1-3 of 3 results.

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