A026119 -id:A026119 - cp's OEIS frontend

A300190 Number of solutions to 1 +- 2 +- 3 +- ... +- n == 0 (mod n).

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

Offset: 1

Seiichi Manyama, Feb 28 2018

nonn

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

			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.

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

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).

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

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-François Alcover, Apr 29 2020, after Alois P. Heinz *)

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)

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.

A000116 Number of even sequences with period 2n (bisection of A000013).

Original entry on oeis.org

1, 2, 4, 8, 20, 56, 180, 596, 2068, 7316, 26272, 95420, 349716, 1290872, 4794088, 17896832, 67110932, 252648992, 954444608, 3616828364, 13743921632, 52357746896, 199911300472, 764877836468, 2932031358484, 11258999739560, 43303843861744, 166799988689300

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Cf. A000013, A026119.

a000116 n = a000116_list !! n
a000116_list = bis a000013_list where bis (x:_:xs) = x : bis xs
-- Reinhard Zumkeller, Jul 08 2013

with(numtheory):
a:= n-> `if`(n=0, 1, add(phi(2*d)*2^(2*n/d), d=divisors(2*n))/(4*n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 25 2012

a[n_] := Sum[ EulerPhi[2d]*2^(2n/d), {d, Divisors[2n]}]/(4n); a[0]=1; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Sep 13 2012, after Alois P. Heinz *)

a(2*n) + a(n) = 2 * A000208(2*n); a(2*n+1) = 2 * A000208(2*n+1). - Reinhard Zumkeller, Jul 08 2013

a(n) ~ 4^(n-1) / n. - Cedric Lorand, Apr 18 2022

More terms from David W. Wilson, Jan 13 2000

A100447 Bisection of A000031.

Original entry on oeis.org

2, 4, 8, 20, 60, 188, 632, 2192, 7712, 27596, 99880, 364724, 1342184, 4971068, 18512792, 69273668, 260301176, 981706832, 3714566312, 14096303344, 53634713552, 204560302844, 781874936816, 2994414645860, 11488774559636

Offset: 0

N. J. A. Sloane, Nov 21 2004

Cf. A000031, A100446. Equals 2*A026119.

with(numtheory):seq((1/(2*n+1))*add(phi(d)*2^((2*n+1)/d),d=divisors(2*n+1)),n=0..30); (C. Ronaldo)

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005

cp's OEIS Frontend

A300190 Number of solutions to 1 +- 2 +- 3 +- ... +- n == 0 (mod n).

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A000116 Number of even sequences with period 2n (bisection of A000013).

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A100447 Bisection of A000031.

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