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A300190 Number of solutions to 1 +- 2 +- 3 +- ... +- n == 0 (mod n). 8
1, 0, 2, 4, 4, 0, 10, 32, 30, 0, 94, 344, 316, 0, 1096, 4096, 3856, 0, 13798, 52432, 49940, 0, 182362, 699072, 671092, 0, 2485534, 9586984, 9256396, 0, 34636834, 134217728, 130150588, 0, 490853416, 1908874584, 1857283156, 0, 7048151672, 27487790720 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Apparently a(2*n + 1) = A053656(2*n + 1) for n >= 0. - Georg Fischer, Mar 26 2019

LINKS

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

FORMULA

a(4*n+1) = A000016(n), a(4*n+2) = 0, a(4*n+3) = A000016(n), a(4*n+4) = 2 * A000016(n) for n > 0.

a(2^n) = 2^A000325(n) for n > 1.

EXAMPLE

Solutions for n = 7:

--------------------------

1 +2 +3 +4 +5 +6 +7 =  28.

1 +2 +3 +4 +5 +6 -7 =  14.

1 +2 -3 +4 -5 -6 +7 =   0.

1 +2 -3 +4 -5 -6 -7 = -14.

1 +2 -3 -4 +5 +6 +7 =  14.

1 +2 -3 -4 +5 +6 -7 =   0.

1 -2 +3 +4 -5 +6 +7 =  14.

1 -2 +3 +4 -5 +6 -7 =   0.

1 -2 -3 -4 -5 +6 +7 =   0.

1 -2 -3 -4 -5 +6 -7 = -14.

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, m-i]))

    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, m - i}}]];

a[n_] := b[0, n - 1, n];

Array[a, 60] (* Jean-Fran├žois Alcover, Apr 29 2020, after Alois P. Heinz *)

PROG

(Ruby)

def A(n)

  ary = [1] + Array.new(n - 1, 0)

  (1..n).each{|i|

    i1 = 2 * i

    a = ary.clone

    (0..n - 1).each{|j| a[(j + i1) % n] += ary[j]}

    ary = a

  }

  ary[(n * (n + 1) / 2) % n] / 2

end

def A300190(n)

  (1..n).map{|i| A(i)}

end

p A300190(100)

CROSSREFS

Number of solutions to 1 +- 2^k +- 3^k +- ... +- n^k == 0 (mod n): this sequence (k=1), A300268 (k=2), A300269 (k=3).

Cf. A000016, A026119, A053656, A058377, A063776, A300307, A300328.

Cf. A016825 (4n+2).

Sequence in context: A118434 A090132 A199051 * A099211 A261761 A300269

Adjacent sequences:  A300187 A300188 A300189 * A300191 A300192 A300193

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Feb 28 2018

STATUS

approved

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Last modified September 26 06:05 EDT 2020. Contains 337346 sequences. (Running on oeis4.)