A300269 Number of solutions to 1 +- 8 +- 27 +- ... +- n^3 == 0 (mod n).

1, 0, 2, 4, 4, 0, 20, 48, 80, 0, 94, 344, 424, 0, 1096, 4864, 3856, 0, 16444, 52432, 65248, 0, 182362, 720928, 671104, 0, 4152320, 11156656, 9256396, 0, 34636834, 135397376, 130150588, 0, 533834992, 2773200896, 1857304312, 0, 7065319328, 27541477824, 26817356776

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Example

Solutions for n = 7:
-----------------------------------
1 +8 +27 +64 +125 +216 +343 =  784.
1 +8 +27 +64 +125 +216 -343 =   98.
1 +8 +27 -64 +125 -216 +343 =  224.
1 +8 +27 -64 +125 -216 -343 = -462.
1 +8 +27 -64 -125 +216 +343 =  406.
1 +8 +27 -64 -125 +216 -343 = -280.
1 +8 -27 -64 +125 +216 +343 =  602.
1 +8 -27 -64 +125 +216 -343 =  -84.
1 -8 +27 +64 +125 -216 +343 =  336.
1 -8 +27 +64 +125 -216 -343 = -350.
1 -8 +27 +64 -125 +216 +343 =  518.
1 -8 +27 +64 -125 +216 -343 = -168.
1 -8 +27 -64 -125 -216 +343 =  -42.
1 -8 +27 -64 -125 -216 -343 = -728.
1 -8 -27 +64 +125 +216 +343 =  714.
1 -8 -27 +64 +125 +216 -343 =   28.
1 -8 -27 -64 +125 -216 +343 =  154.
1 -8 -27 -64 +125 -216 -343 = -532.
1 -8 -27 -64 -125 +216 +343 =  336.
1 -8 -27 -64 -125 +216 -343 = -350.

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^3, m-i^3]))
    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^3, m - i^3}}]];
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|
    i3 = 2 * i * i * i
    a = ary.clone
    (0..n - 1).each{|j| a[(j + i3) % n] += ary[j]}
    ary = a
  }
  ary[((n * (n + 1)) ** 2 / 4) % n] / 2
end
def A300269(n)
  (1..n).map{|i| A(i)}
end
p A300269(100)
(PARI) a(n) = my (v=vector(n, k, k==1)); for (i=2, n, v = vector(n, k, v[1 + (k-i^3)%n] + v[1 + (k+i^3)%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), A300268 (k=2), this sequence (k=3).

Cf. A113263, A195938.

Sequence in context: A300190 A099211 A261761 * A094225 A358641 A057277

Adjacent sequences: A300266 A300267 A300268 * A300270 A300271 A300272

Keyword

nonn

Author

Seiichi Manyama, Mar 01 2018

Extensions

More terms from Alois P. Heinz, Mar 01 2018

Status

approved