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A300269 Number of solutions to 1 +- 8 +- 27 +- ... +- n^3 == 0 (mod n).

%I A300269 #40 Mar 19 2022 06:36:47
%S A300269 1,0,2,4,4,0,20,48,80,0,94,344,424,0,1096,4864,3856,0,16444,52432,
%T A300269 65248,0,182362,720928,671104,0,4152320,11156656,9256396,0,34636834,
%U A300269 135397376,130150588,0,533834992,2773200896,1857304312,0,7065319328,27541477824,26817356776
%N A300269 Number of solutions to 1 +- 8 +- 27 +- ... +- n^3 == 0 (mod n).
%H A300269 Alois P. Heinz, <a href="/A300269/b300269.txt">Table of n, a(n) for n = 1..1000</a>
%e A300269 Solutions for n = 7:
%e A300269 -----------------------------------
%e A300269 1 +8 +27 +64 +125 +216 +343 =  784.
%e A300269 1 +8 +27 +64 +125 +216 -343 =   98.
%e A300269 1 +8 +27 -64 +125 -216 +343 =  224.
%e A300269 1 +8 +27 -64 +125 -216 -343 = -462.
%e A300269 1 +8 +27 -64 -125 +216 +343 =  406.
%e A300269 1 +8 +27 -64 -125 +216 -343 = -280.
%e A300269 1 +8 -27 -64 +125 +216 +343 =  602.
%e A300269 1 +8 -27 -64 +125 +216 -343 =  -84.
%e A300269 1 -8 +27 +64 +125 -216 +343 =  336.
%e A300269 1 -8 +27 +64 +125 -216 -343 = -350.
%e A300269 1 -8 +27 +64 -125 +216 +343 =  518.
%e A300269 1 -8 +27 +64 -125 +216 -343 = -168.
%e A300269 1 -8 +27 -64 -125 -216 +343 =  -42.
%e A300269 1 -8 +27 -64 -125 -216 -343 = -728.
%e A300269 1 -8 -27 +64 +125 +216 +343 =  714.
%e A300269 1 -8 -27 +64 +125 +216 -343 =   28.
%e A300269 1 -8 -27 -64 +125 -216 +343 =  154.
%e A300269 1 -8 -27 -64 +125 -216 -343 = -532.
%e A300269 1 -8 -27 -64 -125 +216 +343 =  336.
%e A300269 1 -8 -27 -64 -125 +216 -343 = -350.
%p A300269 b:= proc(n, i, m) option remember; `if`(i=0, `if`(n=0, 1, 0),
%p A300269       add(b(irem(n+j, m), i-1, m), j=[i^3, m-i^3]))
%p A300269     end:
%p A300269 a:= n-> b(0, n-1, n):
%p A300269 seq(a(n), n=1..60);  # _Alois P. Heinz_, Mar 01 2018
%t A300269 b[n_, i_, m_] := b[n, i, m] = If[i == 0, If[n == 0, 1, 0],
%t A300269      Sum[b[Mod[n + j, m], i - 1, m], {j, {i^3, m - i^3}}]];
%t A300269 a[n_] := b[0, n - 1, n];
%t A300269 Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Mar 19 2022, after _Alois P. Heinz_ *)
%o A300269 (Ruby)
%o A300269 def A(n)
%o A300269   ary = [1] + Array.new(n - 1, 0)
%o A300269   (1..n).each{|i|
%o A300269     i3 = 2 * i * i * i
%o A300269     a = ary.clone
%o A300269     (0..n - 1).each{|j| a[(j + i3) % n] += ary[j]}
%o A300269     ary = a
%o A300269   }
%o A300269   ary[((n * (n + 1)) ** 2 / 4) % n] / 2
%o A300269 end
%o A300269 def A300269(n)
%o A300269   (1..n).map{|i| A(i)}
%o A300269 end
%o A300269 p A300269(100)
%o A300269 (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 A300269 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 A300269 Cf. A113263, A195938.
%K A300269 nonn
%O A300269 1,3
%A A300269 _Seiichi Manyama_, Mar 01 2018
%E A300269 More terms from _Alois P. Heinz_, Mar 01 2018