A300269 - OEIS

%I #40 Mar 19 2022 06:36:47

%S 1,0,2,4,4,0,20,48,80,0,94,344,424,0,1096,4864,3856,0,16444,52432,

%T 65248,0,182362,720928,671104,0,4152320,11156656,9256396,0,34636834,

%U 135397376,130150588,0,533834992,2773200896,1857304312,0,7065319328,27541477824,26817356776

%N Number of solutions to 1 +- 8 +- 27 +- ... +- n^3 == 0 (mod n).

%H Alois P. Heinz, <a href="/A300269/b300269.txt">Table of n, a(n) for n = 1..1000</a>

%e Solutions for n = 7:

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

%e 1 +8 +27 +64 +125 +216 +343 =  784.

%e 1 +8 +27 +64 +125 +216 -343 =   98.

%e 1 +8 +27 -64 +125 -216 +343 =  224.

%e 1 +8 +27 -64 +125 -216 -343 = -462.

%e 1 +8 +27 -64 -125 +216 +343 =  406.

%e 1 +8 +27 -64 -125 +216 -343 = -280.

%e 1 +8 -27 -64 +125 +216 +343 =  602.

%e 1 +8 -27 -64 +125 +216 -343 =  -84.

%e 1 -8 +27 +64 +125 -216 +343 =  336.

%e 1 -8 +27 +64 +125 -216 -343 = -350.

%e 1 -8 +27 +64 -125 +216 +343 =  518.

%e 1 -8 +27 +64 -125 +216 -343 = -168.

%e 1 -8 +27 -64 -125 -216 +343 =  -42.

%e 1 -8 +27 -64 -125 -216 -343 = -728.

%e 1 -8 -27 +64 +125 +216 +343 =  714.

%e 1 -8 -27 +64 +125 +216 -343 =   28.

%e 1 -8 -27 -64 +125 -216 +343 =  154.

%e 1 -8 -27 -64 +125 -216 -343 = -532.

%e 1 -8 -27 -64 -125 +216 +343 =  336.

%e 1 -8 -27 -64 -125 +216 -343 = -350.

%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^3, m-i^3]))

%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^3, m - i^3}}]];

%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     i3 = 2 * i * i * i

%o     a = ary.clone

%o     (0..n - 1).each{|j| a[(j + i3) % n] += ary[j]}

%o     ary = a

%o   }

%o   ary[((n * (n + 1)) ** 2 / 4) % n] / 2

%o end

%o def A300269(n)

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

%o end

%o p A300269(100)

%o (PARI) a(n) = my (v=vector(n,k,k==1)); for (i=2, n, v = vector(n, k, v[1 + (k-i^3)%n] + v[1 + (k+i^3)%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), A300268 (k=2), this sequence (k=3).

%Y Cf. A113263, A195938.

%K nonn

%O 1,3

%A _Seiichi Manyama_, Mar 01 2018

%E More terms from _Alois P. Heinz_, Mar 01 2018