cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A300328 Triangle read by rows: row n gives the number of solutions to +-1 +- 2 +- 3 +- ... +- n == k (mod n).

Original entry on oeis.org

2, 0, 4, 4, 2, 2, 8, 0, 8, 0, 8, 6, 6, 6, 6, 0, 20, 0, 24, 0, 20, 20, 18, 18, 18, 18, 18, 18, 64, 0, 64, 0, 64, 0, 64, 0, 60, 56, 56, 58, 56, 56, 58, 56, 56, 0, 204, 0, 204, 0, 208, 0, 204, 0, 204, 188, 186, 186, 186, 186, 186, 186, 186, 186, 186, 186
Offset: 1

Views

Author

Seiichi Manyama, Mar 03 2018

Keywords

Examples

			First few rows are:
   2;
   0,  4;
   4,  2,  2;
   8,  0,  8,  0;
   8,  6,  6,  6,  6;
   0, 20,  0, 24,  0, 20;
  20, 18, 18, 18, 18, 18, 18;
  64,  0, 64,  0, 64,  0, 64, 0;

Links

Crossrefs

Cf. A000079 (row sums), A300190, A300324, A300329.

A300307 Number of solutions to 1 +- 3 +- 6 +- ... +- n*(n+1)/2 == 0 mod n.

Original entry on oeis.org

1, 2, 0, 4, 4, 16, 12, 32, 20, 112, 88, 384, 308, 1264, 1056, 4096, 3852, 15120, 13820, 52608, 49824, 190848, 182356, 704512, 671540, 2582128, 2475220, 9615744, 9256428, 35868672, 34636840, 134217728, 130021392, 505292976, 491156304, 1909836416, 1857282536
Offset: 1

Views

Author

Seiichi Manyama, Mar 02 2018

Keywords

Examples

			Solutions for n = 7:
------------------------------
1 +3 +6 +10 +15 +21 +28 =  84.
1 +3 +6 +10 +15 +21 -28 =  28.
1 +3 +6 +10 +15 -21 +28 =  42.
1 +3 +6 +10 +15 -21 -28 = -14.
1 +3 -6 +10 -15 +21 +28 =  42.
1 +3 -6 +10 -15 +21 -28 = -14.
1 +3 -6 +10 -15 -21 +28 =   0.
1 +3 -6 +10 -15 -21 -28 = -56.
1 -3 +6 -10 -15 +21 +28 =  28.
1 -3 +6 -10 -15 +21 -28 = -28.
1 -3 +6 -10 -15 -21 +28 = -14.
1 -3 +6 -10 -15 -21 -28 = -70.

Links

Crossrefs

Cf. A000079, A000217, A000325, A058498, A300190, A300218.

Programs

  • Ruby
    def A(n)
  ary = [1] + Array.new(n - 1, 0)
  (1..n).each{|i|
    it = i * (i + 1)
    a = ary.clone
    (0..n - 1).each{|j| a[(j + it) % n] += ary[j]}
    ary = a
  }
  ary[(n * (n + 1) * (n + 2) / 6) % n] / 2
end
def A300307(n)
  (1..n).map{|i| A(i)}
end
p A300307(100)

Formula

a(2^n) = 2^A000325(n) for n>0 (conjectured).

A300268 Number of solutions to 1 +- 4 +- 9 +- ... +- n^2 == 0 (mod n).

Original entry on oeis.org

1, 0, 2, 4, 6, 0, 10, 48, 32, 0, 94, 344, 370, 0, 1268, 4608, 3856, 0, 13798, 55960, 50090, 0, 182362, 721952, 690496, 0, 2485592, 9586984, 9256746, 0, 34636834, 135335936, 130150588, 0, 493452348, 1908875264, 1857293524, 0, 7049188508, 27603824928
Offset: 1

Views

Author

Seiichi Manyama, Mar 01 2018

Keywords

Examples

			Solutions for n = 7:
-------------------------------
1 +4 +9 +16 +25 +36 +49 =  140.
1 +4 +9 +16 +25 +36 -49 =   42.
1 +4 +9 -16 -25 -36 +49 =  -14.
1 +4 +9 -16 -25 -36 -49 = -112.
1 +4 -9 +16 -25 -36 +49 =    0.
1 +4 -9 +16 -25 -36 -49 =  -98.
1 -4 +9 -16 +25 -36 +49 =   28.
1 -4 +9 -16 +25 -36 -49 =  -70.
1 -4 -9 +16 +25 -36 +49 =   42.
1 -4 -9 +16 +25 -36 -49 =  -56.

Links

Crossrefs

Number of solutions to 1 +- 2^k +- 3^k +- ... +- n^k == 0 (mod n): A300190 (k=1), this sequence (k=2), A300269 (k=3).
Cf. A083527, A215573.

Programs

  • 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^2, m-i^2]))
    end:
a:= n-> b(0, n-1, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 01 2018
  • 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^2, m - i^2}}]];
a[n_] := b[0, n - 1, n];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)
  • 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
  • Ruby
    def A(n)
  ary = [1] + Array.new(n - 1, 0)
  (1..n).each{|i|
    i2 = 2 * i * i
    a = ary.clone
    (0..n - 1).each{|j| a[(j + i2) % n] += ary[j]}
    ary = a
  }
  ary[(n * (n + 1) * (2 * n + 1) / 6) % n] / 2
end
def A300268(n)
  (1..n).map{|i| A(i)}
end
p A300268(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

Views

Author

Seiichi Manyama, Mar 01 2018

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

Crossrefs

Number of solutions to 1 +- 2^k +- 3^k +- ... +- n^k == 0 (mod n): A300190 (k=1), A300268 (k=2), this sequence (k=3).
Cf. A113263, A195938.

Programs

  • 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
  • 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 *)
  • 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
  • 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

A300218 Number of solutions to 1 +- 3 +- 5 +- ... +- (2*n-1) == 0 mod n.

Original entry on oeis.org

1, 2, 2, 4, 4, 12, 10, 36, 30, 104, 94, 344, 316, 1172, 1096, 4132, 3856, 14572, 13798, 52432, 49940, 190652, 182362, 699416, 671092, 2581112, 2485534, 9586984, 9256396, 35791472, 34636834, 134221860, 130150588, 505290272, 490853416, 1908874584, 1857283156
Offset: 1

Views

Author

Seiichi Manyama, Feb 28 2018

Keywords

Examples

			Solutions for n = 7:
----------------------------
1 +3 +5 +7 +9 +11 +13 =  49.
1 +3 +5 -7 +9 +11 +13 =  35.
1 +3 -5 +7 -9 +11 +13 =  21.
1 +3 -5 -7 -9 +11 +13 =   7.
1 -3 +5 +7 +9 -11 +13 =  21.
1 -3 +5 -7 +9 -11 +13 =   7.
1 -3 -5 +7 +9 +11 -13 =   7.
1 -3 -5 +7 -9 -11 +13 =  -7.
1 -3 -5 -7 +9 +11 -13 =  -7.
1 -3 -5 -7 -9 -11 +13 = -21.

Links

Crossrefs

Cf. A063776, A156700, A300190.

Programs

  • Maple
    b:= proc(n, i, m) option remember; `if`(i<1, `if`(n=0, 1, 0),
      add(b(irem(n+j, m), i-2, m), j=[i, m-i]))
    end:
a:= n-> b(n-1, 2*n-3, n):
seq(a(n), n=1..40);  # Alois P. Heinz, Mar 01 2018
  • Mathematica
    Table[With[{s = Range[1, (2 n - 1), 2]}, Count[Map[Total[s #] &, Take[Tuples[{-1, 1}, Length@ s], -2^(n - 1)]], ?(Divisible[#, n] &)]], {n, 22}] (* _Michael De Vlieger, Mar 01 2018 *)

A300329 Number of solutions to +-1 +- 2 +- 3 +- ... +- n == n-1 (mod n).

Original entry on oeis.org

2, 4, 2, 0, 6, 20, 18, 0, 56, 204, 186, 0, 630, 2340, 2182, 0, 7710, 29120, 27594, 0, 99858, 381300, 364722, 0, 1342176, 5162220, 4971008, 0, 18512790, 71582716, 69273666, 0, 260300986, 1010580540, 981706806, 0, 3714566310, 14467258260, 14096302710, 0
Offset: 1

Views

Author

Seiichi Manyama, Mar 03 2018

Keywords

Examples

			Solutions for n = 7:
--------------------------------------------------------------
1 +2 +3 +4 -5 -6 +7 =   6,         -1 +2 +3 -4 +5 -6 +7 =   6,
1 +2 +3 +4 -5 -6 -7 =  -8,         -1 +2 +3 -4 +5 -6 -7 =  -8,
1 +2 +3 -4 +5 +6 +7 =  20,         -1 +2 -3 +4 +5 +6 +7 =  20,
1 +2 +3 -4 +5 +6 -7 =   6,         -1 +2 -3 +4 +5 +6 -7 =   6,
1 +2 -3 -4 -5 -6 +7 =  -8,         -1 -2 +3 -4 -5 -6 +7 =  -8,
1 +2 -3 -4 -5 -6 -7 = -22,         -1 -2 +3 -4 -5 -6 -7 = -22,
1 -2 +3 -4 -5 +6 +7 =   6,         -1 -2 -3 +4 -5 +6 +7 =   6,
1 -2 +3 -4 -5 +6 -7 =  -8,         -1 -2 -3 +4 -5 +6 -7 =  -8,
1 -2 -3 +4 +5 -6 +7 =   6,
1 -2 -3 +4 +5 -6 -7 =  -8.

Links

Crossrefs

Cf. A300190, A300328.

Programs

  • PARI
    a(n) = my (v=vector(n, k, k==1)); for (p=1, n, v = vector(n, k, v[1+(k-1+p)%n]+v[1+(k-1-p)%n])); v[1+(n-1)%n] \\ Rémy Sigrist, Mar 03 2018

A300361 a(n) = 2^A000325(n).

Original entry on oeis.org

2, 2, 4, 32, 4096, 134217728, 288230376151711744, 2658455991569831745807614120560689152, 452312848583266388373324160190187140051835877600158453279131187530910662656
Offset: 0

Views

Author

Seiichi Manyama, Mar 03 2018

Keywords

Links

Crossrefs

Cf. A000079, A000325, A300190, A300307.

Programs

  • Magma
    [2^(2^n-n): n in [1..10]]; // Altug Alkan, Mar 04 2018
  • Mathematica
    Table[2^(2^n-n),{n,0,10}] (* Harvey P. Dale, Apr 08 2020 *)
  • PARI
    {a(n) = 2^(2^n-n)}

Formula

a(n) = A000079(A000325(n)).

A350272 Triangle T(n,k), n >= 1, 0 <= k <= n-1, read by rows, where T(n,k) is the number of solutions to 1 +- 2 +- 3 +- ... +- n == k (mod n).

Original entry on oeis.org

1, 0, 2, 2, 0, 2, 4, 0, 4, 0, 4, 4, 4, 2, 2, 0, 8, 0, 12, 0, 12, 10, 8, 10, 10, 8, 8, 10, 32, 0, 32, 0, 32, 0, 32, 0, 30, 28, 30, 28, 26, 30, 30, 26, 28, 0, 104, 0, 100, 0, 104, 0, 104, 0, 100, 94, 92, 94, 94, 92, 92, 94, 94, 92, 92, 94, 344, 0, 344, 0, 336, 0, 344, 0, 344, 0, 336, 0
Offset: 1

Views

Author

Seiichi Manyama, Dec 22 2021

Keywords

Comments

a(n) is even for n > 1.

Examples

			Triangle begins:
   1;
   0,   2;
   2,   0,  2;
   4,   0,  4,   0;
   4,   4,  4,   2,  2;
   0,   8,  0,  12,  0,  12;
  10,   8, 10,  10,  8,   8, 10;
  32,   0, 32,   0, 32,   0, 32,   0;
  30,  28, 30,  28, 26,  30, 30,  26, 28;
   0, 104,  0, 100,  0, 104,  0, 104,  0, 100;

Crossrefs

Row sums give A131577.
Column 0 gives A300190.

Programs

  • Ruby
    def A(n)
  ary = Array.new(n, 0)
  [1, -1].repeated_permutation(n - 1){|i|
    ary[(2..n).inject(1){|s, j| s + i[j - 2] * j} % n] += 1
  }
  ary
end
def A350272(n)
  (1..n).map{|i| A(i)}.flatten
end
p A350272(10)
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.