A300190
Number of solutions to 1 +- 2 +- 3 +- ... +- n == 0 (mod n).
Original entry on oeis.org
1, 0, 2, 4, 4, 0, 10, 32, 30, 0, 94, 344, 316, 0, 1096, 4096, 3856, 0, 13798, 52432, 49940, 0, 182362, 699072, 671092, 0, 2485534, 9586984, 9256396, 0, 34636834, 134217728, 130150588, 0, 490853416, 1908874584, 1857283156, 0, 7048151672, 27487790720
Offset: 1
Solutions for n = 7:
--------------------------
1 +2 +3 +4 +5 +6 +7 = 28.
1 +2 +3 +4 +5 +6 -7 = 14.
1 +2 -3 +4 -5 -6 +7 = 0.
1 +2 -3 +4 -5 -6 -7 = -14.
1 +2 -3 -4 +5 +6 +7 = 14.
1 +2 -3 -4 +5 +6 -7 = 0.
1 -2 +3 +4 -5 +6 +7 = 14.
1 -2 +3 +4 -5 +6 -7 = 0.
1 -2 -3 -4 -5 +6 +7 = 0.
1 -2 -3 -4 -5 +6 -7 = -14.
Number of solutions to 1 +- 2^k +- 3^k +- ... +- n^k == 0 (mod n): this sequence (k=1),
A300268 (k=2),
A300269 (k=3).
-
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, m-i]))
end:
a:= n-> b(0, n-1, n):
seq(a(n), n=1..60); # Alois P. Heinz, Mar 01 2018
-
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}}]];
a[n_] := b[0, n - 1, n];
Array[a, 60] (* Jean-François Alcover, Apr 29 2020, after Alois P. Heinz *)
-
def A(n)
ary = [1] + Array.new(n - 1, 0)
(1..n).each{|i|
i1 = 2 * i
a = ary.clone
(0..n - 1).each{|j| a[(j + i1) % n] += ary[j]}
ary = a
}
ary[(n * (n + 1) / 2) % n] / 2
end
def A300190(n)
(1..n).map{|i| A(i)}
end
p A300190(100)
A300269
Number of solutions to 1 +- 8 +- 27 +- ... +- n^3 == 0 (mod n).
Original entry on oeis.org
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
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.
Number of solutions to 1 +- 2^k +- 3^k +- ... +- n^k == 0 (mod n):
A300190 (k=1),
A300268 (k=2), this sequence (k=3).
-
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
-
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 *)
-
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
-
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)
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