A300268 - OEIS

%I #42 Mar 19 2022 06:37:01

%S 1,0,2,4,6,0,10,48,32,0,94,344,370,0,1268,4608,3856,0,13798,55960,

%T 50090,0,182362,721952,690496,0,2485592,9586984,9256746,0,34636834,

%U 135335936,130150588,0,493452348,1908875264,1857293524,0,7049188508,27603824928

%N Number of solutions to 1 +- 4 +- 9 +- ... +- n^2 == 0 (mod n).

%H Seiichi Manyama, <a href="/A300268/b300268.txt">Table of n, a(n) for n = 1..3334</a> (terms 1..1000 from Alois P. Heinz)

%e Solutions for n = 7:

%e -------------------------------

%e 1 +4 +9 +16 +25 +36 +49 =  140.

%e 1 +4 +9 +16 +25 +36 -49 =   42.

%e 1 +4 +9 -16 -25 -36 +49 =  -14.

%e 1 +4 +9 -16 -25 -36 -49 = -112.

%e 1 +4 -9 +16 -25 -36 +49 =    0.

%e 1 +4 -9 +16 -25 -36 -49 =  -98.

%e 1 -4 +9 -16 +25 -36 +49 =   28.

%e 1 -4 +9 -16 +25 -36 -49 =  -70.

%e 1 -4 -9 +16 +25 -36 +49 =   42.

%e 1 -4 -9 +16 +25 -36 -49 =  -56.

%p b:= proc(n, i, m) option remember; `if`(i=0, `if`(n=0, 1, 0),

%p       add(b(irem(n+j, m), i-1, m), j=[i^2, m-i^2]))

%p     end:

%p a:= n-> b(0, n-1, n):

%p seq(a(n), n=1..60);  # _Alois P. Heinz_, Mar 01 2018

%t b[n_, i_, m_] := b[n, i, m] = If[i == 0, If[n == 0, 1, 0],

%t      Sum[b[Mod[n + j, m], i - 1, m], {j, {i^2, m - i^2}}]];

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

%t Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Mar 19 2022, after _Alois P. Heinz_ *)

%o (Ruby)

%o def A(n)

%o   ary = [1] + Array.new(n - 1, 0)

%o   (1..n).each{|i|

%o     i2 = 2 * i * i

%o     a = ary.clone

%o     (0..n - 1).each{|j| a[(j + i2) % n] += ary[j]}

%o     ary = a

%o   }

%o   ary[(n * (n + 1) * (2 * n + 1) / 6) % n] / 2

%o end

%o def A300268(n)

%o   (1..n).map{|i| A(i)}

%o end

%o p A300268(100)

%o (PARI) a(n) = my (v=vector(n,k,k==1)); for (i=2, n, v = vector(n, k, v[1 + (k-i^2)%n] + v[1 + (k+i^2)%n])); v[1] \\ _Rémy Sigrist_, Mar 01 2018

%Y Number of solutions to 1 +- 2^k +- 3^k +- ... +- n^k == 0 (mod n): A300190 (k=1), this sequence (k=2), A300269 (k=3).

%Y Cf. A083527, A215573.

%K nonn

%O 1,3

%A _Seiichi Manyama_, Mar 01 2018