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

%I A300190 #65 Apr 29 2020 07:44:25
%S A300190 1,0,2,4,4,0,10,32,30,0,94,344,316,0,1096,4096,3856,0,13798,52432,
%T A300190 49940,0,182362,699072,671092,0,2485534,9586984,9256396,0,34636834,
%U A300190 134217728,130150588,0,490853416,1908874584,1857283156,0,7048151672,27487790720
%N A300190 Number of solutions to 1 +- 2 +- 3 +- ... +- n == 0 (mod n).
%C A300190 Apparently a(2*n + 1) = A053656(2*n + 1) for n >= 0. - _Georg Fischer_, Mar 26 2019
%H A300190 Seiichi Manyama, <a href="/A300190/b300190.txt">Table of n, a(n) for n = 1..3334</a> (terms 1..1000 from Alois P. Heinz)
%F A300190 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.
%F A300190 a(2^n) = 2^A000325(n) for n > 1.
%e A300190 Solutions for n = 7:
%e A300190 --------------------------
%e A300190 1 +2 +3 +4 +5 +6 +7 =  28.
%e A300190 1 +2 +3 +4 +5 +6 -7 =  14.
%e A300190 1 +2 -3 +4 -5 -6 +7 =   0.
%e A300190 1 +2 -3 +4 -5 -6 -7 = -14.
%e A300190 1 +2 -3 -4 +5 +6 +7 =  14.
%e A300190 1 +2 -3 -4 +5 +6 -7 =   0.
%e A300190 1 -2 +3 +4 -5 +6 +7 =  14.
%e A300190 1 -2 +3 +4 -5 +6 -7 =   0.
%e A300190 1 -2 -3 -4 -5 +6 +7 =   0.
%e A300190 1 -2 -3 -4 -5 +6 -7 = -14.
%p A300190 b:= proc(n, i, m) option remember; `if`(i=0, `if`(n=0, 1, 0),
%p A300190       add(b(irem(n+j, m), i-1, m), j=[i, m-i]))
%p A300190     end:
%p A300190 a:= n-> b(0, n-1, n):
%p A300190 seq(a(n), n=1..60);  # _Alois P. Heinz_, Mar 01 2018
%t A300190 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}}]];
%t A300190 a[n_] := b[0, n - 1, n];
%t A300190 Array[a, 60] (* _Jean-François Alcover_, Apr 29 2020, after _Alois P. Heinz_ *)
%o A300190 (Ruby)
%o A300190 def A(n)
%o A300190   ary = [1] + Array.new(n - 1, 0)
%o A300190   (1..n).each{|i|
%o A300190     i1 = 2 * i
%o A300190     a = ary.clone
%o A300190     (0..n - 1).each{|j| a[(j + i1) % n] += ary[j]}
%o A300190     ary = a
%o A300190   }
%o A300190   ary[(n * (n + 1) / 2) % n] / 2
%o A300190 end
%o A300190 def A300190(n)
%o A300190   (1..n).map{|i| A(i)}
%o A300190 end
%o A300190 p A300190(100)
%Y A300190 Number of solutions to 1 +- 2^k +- 3^k +- ... +- n^k == 0 (mod n): this sequence (k=1), A300268 (k=2), A300269 (k=3).
%Y A300190 Cf. A000016, A026119, A053656, A058377, A063776, A300307, A300328.
%Y A300190 Cf. A016825 (4n+2).
%K A300190 nonn
%O A300190 1,3
%A A300190 _Seiichi Manyama_, Feb 28 2018