A300269 Number of solutions to 1 +- 8 +- 27 +- ... +- n^3 == 0 (mod n).
1, 0, 2, 4, 4, 0, 20, 48, 80, 0, 94, 344, 424, 0, 1096, 4864, 3856, 0, 16444, 52432, 65248, 0, 182362, 720928, 671104, 0, 4152320, 11156656, 9256396, 0, 34636834, 135397376, 130150588, 0, 533834992, 2773200896, 1857304312, 0, 7065319328, 27541477824, 26817356776
Offset: 1
Keywords
Examples
Solutions for n = 7: ----------------------------------- 1 +8 +27 +64 +125 +216 +343 = 784. 1 +8 +27 +64 +125 +216 -343 = 98. 1 +8 +27 -64 +125 -216 +343 = 224. 1 +8 +27 -64 +125 -216 -343 = -462. 1 +8 +27 -64 -125 +216 +343 = 406. 1 +8 +27 -64 -125 +216 -343 = -280. 1 +8 -27 -64 +125 +216 +343 = 602. 1 +8 -27 -64 +125 +216 -343 = -84. 1 -8 +27 +64 +125 -216 +343 = 336. 1 -8 +27 +64 +125 -216 -343 = -350. 1 -8 +27 +64 -125 +216 +343 = 518. 1 -8 +27 +64 -125 +216 -343 = -168. 1 -8 +27 -64 -125 -216 +343 = -42. 1 -8 +27 -64 -125 -216 -343 = -728. 1 -8 -27 +64 +125 +216 +343 = 714. 1 -8 -27 +64 +125 +216 -343 = 28. 1 -8 -27 -64 +125 -216 +343 = 154. 1 -8 -27 -64 +125 -216 -343 = -532. 1 -8 -27 -64 -125 +216 +343 = 336. 1 -8 -27 -64 -125 +216 -343 = -350.
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..1000
Crossrefs
Programs
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Maple
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^3, m-i^3])) end: a:= n-> b(0, n-1, n): seq(a(n), n=1..60); # Alois P. Heinz, Mar 01 2018
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Mathematica
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^3, m - i^3}}]]; a[n_] := b[0, n - 1, n]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)
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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
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Ruby
def A(n) ary = [1] + Array.new(n - 1, 0) (1..n).each{|i| i3 = 2 * i * i * i a = ary.clone (0..n - 1).each{|j| a[(j + i3) % n] += ary[j]} ary = a } ary[((n * (n + 1)) ** 2 / 4) % n] / 2 end def A300269(n) (1..n).map{|i| A(i)} end p A300269(100)
Extensions
More terms from Alois P. Heinz, Mar 01 2018