A300268 Number of solutions to 1 +- 4 +- 9 +- ... +- n^2 == 0 (mod n).

1, 0, 2, 4, 6, 0, 10, 48, 32, 0, 94, 344, 370, 0, 1268, 4608, 3856, 0, 13798, 55960, 50090, 0, 182362, 721952, 690496, 0, 2485592, 9586984, 9256746, 0, 34636834, 135335936, 130150588, 0, 493452348, 1908875264, 1857293524, 0, 7049188508, 27603824928

Offset

1,3

Links

Seiichi Manyama, Table of n, a(n) for n = 1..3334 (terms 1..1000 from Alois P. Heinz)

Example

Solutions for n = 7:
-------------------------------
1 +4 +9 +16 +25 +36 +49 =  140.
1 +4 +9 +16 +25 +36 -49 =   42.
1 +4 +9 -16 -25 -36 +49 =  -14.
1 +4 +9 -16 -25 -36 -49 = -112.
1 +4 -9 +16 -25 -36 +49 =    0.
1 +4 -9 +16 -25 -36 -49 =  -98.
1 -4 +9 -16 +25 -36 +49 =   28.
1 -4 +9 -16 +25 -36 -49 =  -70.
1 -4 -9 +16 +25 -36 +49 =   42.
1 -4 -9 +16 +25 -36 -49 =  -56.

Maple

b:= proc(n, i, m) option remember; `if`(i=0, `if`(n=0, 1, 0),
      add(b(irem(n+j, m), i-1, m), j=[i^2, m-i^2]))
    end:
a:= n-> b(0, n-1, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 01 2018

Mathematica

b[n_, i_, m_] := b[n, i, m] = If[i == 0, If[n == 0, 1, 0],
     Sum[b[Mod[n + j, m], i - 1, m], {j, {i^2, m - i^2}}]];
a[n_] := b[0, n - 1, n];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)

Prog

(Ruby)
def A(n)
  ary = [1] + Array.new(n - 1, 0)
  (1..n).each{|i|
    i2 = 2 * i * i
    a = ary.clone
    (0..n - 1).each{|j| a[(j + i2) % n] += ary[j]}
    ary = a
  }
  ary[(n * (n + 1) * (2 * n + 1) / 6) % n] / 2
end
def A300268(n)
  (1..n).map{|i| A(i)}
end
p A300268(100)
(PARI) a(n) = my (v=vector(n, k, k==1)); for (i=2, n, v = vector(n, k, v[1 + (k-i^2)%n] + v[1 + (k+i^2)%n])); v[1] \\ Rémy Sigrist, Mar 01 2018

Crossrefs

Number of solutions to 1 +- 2^k +- 3^k +- ... +- n^k == 0 (mod n): A300190 (k=1), this sequence (k=2), A300269 (k=3).

Cf. A083527, A215573.

Sequence in context: A013670 A121206 A062004 * A009285 A013082 A352639

Adjacent sequences: A300265 A300266 A300267 * A300269 A300270 A300271

Keyword

nonn

Author

Seiichi Manyama, Mar 01 2018

Status

approved