A300307 Number of solutions to 1 +- 3 +- 6 +- ... +- n*(n+1)/2 == 0 mod n.

1, 2, 0, 4, 4, 16, 12, 32, 20, 112, 88, 384, 308, 1264, 1056, 4096, 3852, 15120, 13820, 52608, 49824, 190848, 182356, 704512, 671540, 2582128, 2475220, 9615744, 9256428, 35868672, 34636840, 134217728, 130021392, 505292976, 491156304, 1909836416, 1857282536

Offset: 1

Author

Seiichi Manyama, Mar 02 2018

Keywords

• nonn

Examples

			Solutions for n = 7:
------------------------------
1 +3 +6 +10 +15 +21 +28 =  84.
1 +3 +6 +10 +15 +21 -28 =  28.
1 +3 +6 +10 +15 -21 +28 =  42.
1 +3 +6 +10 +15 -21 -28 = -14.
1 +3 -6 +10 -15 +21 +28 =  42.
1 +3 -6 +10 -15 +21 -28 = -14.
1 +3 -6 +10 -15 -21 +28 =   0.
1 +3 -6 [phone] = -56.
1 -3 +6 -10 -15 +21 +28 =  28.
1 -3 +6 -10 -15 +21 -28 = -28.
1 -3 +6 -10 -15 -21 +28 = -14.
1 -3 [phone] = -70.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..2^11

Crossrefs

Cf. A000079, A000217, A000325, A058498, A300190, A300218.

Programs

Ruby

def A(n)
  ary = [1] + Array.new(n - 1, 0)
  (1..n).each{|i|
    it = i * (i + 1)
    a = ary.clone
    (0..n - 1).each{|j| a[(j + it) % n] += ary[j]}
    ary = a
  }
  ary[(n * (n + 1) * (n + 2) / 6) % n] / 2
end
def A300307(n)
  (1..n).map{|i| A(i)}
end
p A300307(100)

Formula

a(2^n) = 2^A000325(n) for n>0 (conjectured).