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.

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A382864 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 3, 1, 0, 1, 3, 2, 0, 1, 4, 3, 0, 1, 4, 4, 1, 0, 1, 5, 5, 1, 0, 1, 5, 7, 2, 0, 1, 6, 8, 3, 0, 1, 6, 10, 5, 0, 1, 7, 12, 6, 1, 0, 1, 7, 14, 9, 1, 0, 1, 8, 16, 11, 2, 0, 1, 8, 19, 15, 3, 0, 1, 9, 21, 18, 5, 0, 1, 9, 24, 23, 7
Offset: 0

Views

Author

Seiichi Manyama, Apr 07 2025

Keywords

Examples

			First few rows are:
  1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 2,  1;
  0, 1, 3,  1;
  0, 1, 3,  2;
  0, 1, 4,  3;
  0, 1, 4,  4, 1;
  0, 1, 5,  5, 1;
  0, 1, 5,  7, 2;
  0, 1, 6,  8, 3;
  0, 1, 6, 10, 5;
  0, 1, 7, 12, 6, 1;
  ...

Links

Crossrefs

Row sums give A000009.
Columns 0..10 give A000007, A000012, A004526(n-1), A069905(n-3), A026810(n-6), A026811(n-10), A026812(n-15), A026813(n-21), A026814(n-28), A026815(n-36), A026816(n-45).
Cf. A003056, A008284, A291954, A291960, A291968, A292047, A292049.

Formula

G.f. of column k: x^(k*(k+1)/2) / Product_{j=1..k} (1-x^j).
T(n,k) = |A292047(n,k)| = |A292049(n,k)|.

A143654 Array T(n,k) read by rows: number of binary bracelets with n beads, k of them 0, with 00 prohibited, (n >= 2, 0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 4, 3, 1, 1, 1, 4, 5, 4, 1, 1, 1, 5, 7, 8, 3, 1, 1, 1, 5, 8, 10, 5, 1, 1, 1, 6, 10, 16, 10, 4, 1, 1, 1, 6, 12, 20, 16, 7, 1, 1, 1, 7, 14, 29, 26, 16, 4, 1, 1, 1, 7, 16, 35, 38, 26, 8, 1, 1, 1, 8, 19, 47, 57, 50
Offset: 2

Views

Author

Washington Bomfim, Aug 28 2008

Keywords

Comments

The initial columns give A057427, A057427, A004526, A069905, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516. Row sums give A129526.
A binary bracelet with n beads, k of them 0, with 00 prohibited has from 0 to floor(n/2) beads 0, i.e., 0 <= k <= floor(n/2). If n is even, the bracelet 0101...01 with n/2 beads of each kind does not have 00 and we cannot change any 1 of it to a 0. If n is odd we cannot change a 1 to a 0 in the bracelet 0101...011 with (n-1)/2 beads 0.
The number of binary bracelets with n beads, 0 <= k <= floor(n/2) of them 0 with 00 prohibited, is equal to the number of binary bracelets with n-k beads, k of them 0. See below.
Let B be a binary bracelet with n-k beads, k of them 0. If we insert one 1 (circularly) after a 0 of B, we obtain a bracelet with n-k+1 beads, k of them 0.
If we do this insertion k times, each time after a distinct 0 of B, we obtain a bracelet with n = n-k+k beads, k of them 0, with 00 prohibited.
On the contrary, Let B be a binary bracelet with n beads, k of them 0, with 00 prohibited. If we remove from B one 1 that is after a 0, we obtain a bracelet of n-1 beads, k of them 0. (If not and we undo the removal, the configuration obtained cannot be a bracelet and this is absurd.) If we repeat this removal k times, after each distinct bead 0, we obtain a bracelet with n-k beads, k of them 0.

Examples

			Array begins
1 1
1 1
1 1 1
1 1 1
1 1 2 1
1 1 2 1
1 1 3 2 1
1 1 3 3 1
1 1 4 4 3 1
...
A129526(10) = A057427(10) + A057427(9) + A004526(8) + A069905(7) + A005232(6) +
A032279(5) = 1+1+4+4+3+1 = 14.

Crossrefs

Cf. A057427, A004526, A069905, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516. Row sums of array give A129526.

A343100 Number of partitions of n into 3 parts, at least 2 of which are coprime.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 4, 4, 6, 6, 10, 8, 14, 12, 16, 16, 24, 18, 30, 24, 32, 30, 44, 32, 50, 42, 54, 48, 70, 48, 79, 64, 80, 72, 96, 72, 113, 90, 112, 96, 138, 96, 153, 120, 144, 132, 182, 128, 195, 150, 192, 168, 232, 162, 239, 192, 240, 210, 287, 192, 305, 240, 288, 256
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 05 2021

Keywords

Examples

			a(12) = 8; [10,1,1], [9,2,1], [8,3,1], [7,4,1], [7,3,2], [6,5,1], [5,5,2], [5,4,3].

Links

Crossrefs

Cf. A023023, A069905.

Programs

  • Mathematica
    Table[Sum[Sum[Sign[Floor[1/GCD[i, j]] + Floor[1/GCD[i, n - i - j]] + Floor[1/GCD[j, n - i - j]]], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 100}]

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} sign(floor(1/gcd(i, j)) + floor(1/gcd(i, n-i-j)) + floor(1/gcd(j, n-i-j))).

A343124 Total number of partitions of k*n into 3 parts for k = 1..n.

Original entry on oeis.org

0, 1, 11, 39, 114, 273, 571, 1086, 1925, 3206, 5101, 7800, 11533, 16575, 23252, 31911, 42987, 56943, 74304, 95662, 121682, 153060, 190614, 235200, 287758, 349317, 421001, 503975, 599560, 709125, 834145, 976206, 1137011, 1318314, 1522059, 1750248, 2005011, 2288611
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 05 2021

Keywords

Links

Crossrefs

Cf. A069905.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[1, {i, j, Floor[(k*n - j)/2]}], {j, Floor[k*n/3]}], {k, n}], {n, 50}]

Formula

a(n) = Sum_{k=1..n} Sum_{j=1..floor(k*n/3)} Sum_{i=j..floor((k*n-j)/2)} 1.

A348541 Number of partitions of n into 3 parts (r,s,t) such that n | (r^2 + s^2 + t^2).

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 2, 2, 4, 1, 1, 0, 3, 3, 0, 4, 0, 0, 6, 2, 8, 4, 4, 0, 2, 5, 5, 1, 0, 1, 3, 6, 12, 7, 1, 0, 14, 7, 0, 1, 0, 0, 6, 11, 8, 1, 8, 0, 12, 0, 17, 10, 0, 0, 2, 10, 20, 10, 5, 2, 2, 11, 0, 1, 4, 0, 12, 12, 24, 9, 12, 1, 26, 13, 1, 7, 0, 0, 14, 0, 28, 1, 1
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 21 2021

Keywords

Examples

			a(6) = 2; 6 | (1^2 + 1^2 + 4^2) = 18 and 6 | (2^2 + 2^2 + 2^2) = 12, so a(6) = 2.

Links

Crossrefs

Cf. A069905.

Programs

  • Mathematica
    c[n_] := 1 - Ceiling[n] + Floor[n]; a[n_] := Sum[c[(j^2 + i^2 + (n - i - j)^2)/n], {j, 1, Floor[n/3]}, {i, j, Floor[(n - j)/2]}]; Array[a, 100] (* Amiram Eldar, Oct 22 2021 *)

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} c((j^2 + i^2 + (n-i-j)^2)/n), where c(n) = 1 - ceiling(n) + floor(n).

A348542 Number of partitions of n into 3 parts where at least one part is even.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 2, 5, 4, 8, 6, 12, 9, 16, 12, 21, 16, 27, 20, 33, 25, 40, 30, 48, 36, 56, 42, 65, 49, 75, 56, 85, 64, 96, 72, 108, 81, 120, 90, 133, 100, 147, 110, 161, 121, 176, 132, 192, 144, 208, 156, 225, 169, 243, 182, 261, 196, 280, 210, 300, 225, 320, 240, 341, 256, 363
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 21 2021

Keywords

Links

Crossrefs

Cf. A069905.

Programs

  • Mathematica
    a[n_] := Sum[1 - Mod[i, 2] * Mod[j, 2] * Mod[n - i - j, 2], {j, 1, Floor[n/3]}, {i, j, Floor[(n - j)/2]}]; Array[a, 100] (* Amiram Eldar, Oct 22 2021 *)

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (1-(i mod 2)*(j mod 2)*((n-i-j) mod 2)).

A348543 Number of partitions of n into 3 parts with at least 1 odd part.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 4, 4, 7, 6, 10, 9, 14, 12, 19, 16, 24, 20, 30, 25, 37, 30, 44, 36, 52, 42, 61, 49, 70, 56, 80, 64, 91, 72, 102, 81, 114, 90, 127, 100, 140, 110, 154, 121, 169, 132, 184, 144, 200, 156, 217, 169, 234, 182, 252, 196, 271, 210, 290, 225, 310, 240, 331, 256, 352
Offset: 1

Views

Author

Wesley Ivan Hurt, Oct 21 2021

Keywords

Links

Crossrefs

Cf. A069905.

Programs

  • Mathematica
    a[n_] := Sum[1 - Mod[j + 1, 2] * Mod[i + 1, 2] * Mod[n - i - j + 1, 2], {j, 1, Floor[n/3]}, {i, j, Floor[(n - j)/2]}]; Array[a, 100] (* Amiram Eldar, Oct 22 2021 *)

Formula

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (1-((j+1) mod 2)*((i+1) mod 2)*((n-i-j+1) mod 2)).
G.f.: -x^3*(x^6+x^5+x^4+x^3+x^2+x+1)/(x^12-x^10-x^8+x^4+x^2-1). - Alois P. Heinz, Oct 22 2021
a(n) = a(n-2)+a(n-4)-a(n-8)-a(n-10)+a(n-12). - Wesley Ivan Hurt, Nov 18 2021
Previous Showing 81-87 of 87 results.

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