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 1-10 of 10 results.

A132149 Duplicate of A129235.

Original entry on oeis.org

1, 4, 6, 11, 10, 20, 14
Offset: 0

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Author

Keywords

A129234 Triangle read by rows: T(n,k) = n/k + k - 1 if n mod k = 0; otherwise T(n,k)=0 (1 <= k <= n).

Original entry on oeis.org

1, 2, 2, 3, 0, 3, 4, 3, 0, 4, 5, 0, 0, 0, 5, 6, 4, 4, 0, 0, 6, 7, 0, 0, 0, 0, 0, 7, 8, 5, 0, 5, 0, 0, 0, 8, 9, 0, 5, 0, 0, 0, 0, 0, 9, 10, 6, 0, 0, 6, 0, 0, 0, 0, 10, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 12, 7, 6, 6, 0, 7, 0, 0, 0, 0, 0, 12, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 14, 8, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 14
Offset: 1

Views

Author

Gary W. Adamson, Apr 05 2007

Keywords

Comments

Row sums = A129235: (1, 4, 6, 11, 10, 20, 14, ...). Moebius transform of A129234 = A129236. Inverse Moebius transform of A129234 = A129237.

Examples

			First few rows of the triangle:
  1;
  2, 2;
  3, 0, 3;
  4, 3, 0, 4;
  5, 0, 0, 0, 5;
  6, 4, 4, 0, 0, 6;
  7, 0, 0, 0, 0, 0, 7;
  ...

Crossrefs

Cf. A129235, A129236, A129237.

Programs

  • Maple
    T:=proc(n,k) if n mod k = 0 then n/k+k-1 else 0 fi end: for n from 1 to 16 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Apr 17 2007
  • Mathematica
    T[n_,k_]:=If[Mod[n,k]==0,n/k+k-1,0];Table[T[n,k],{n,14},{k,n}]//Flatten (* James C. McMahon, Jan 17 2025 *)
  • PARI
    row(n) = vector(n, k, if (!(n%k), n/k + k - 1, 0)); \\ Michel Marcus, Jan 17 2025

Formula

G.f. = G(t,z) = Sum_{k>=1} t^k*z^k*(k-(k-1)*z^k)/(1-z^k)^2. - Emeric Deutsch, Apr 17 2007

Extensions

Edited by Emeric Deutsch, Apr 17 2007

A143315 Triangle read by rows: T(n, k) = 2*A126988(n, k) - signum(A126988(n, k)).

Original entry on oeis.org

1, 3, 1, 5, 0, 1, 7, 3, 0, 1, 9, 0, 0, 0, 1, 11, 5, 3, 0, 0, 1, 13, 0, 0, 0, 0, 0, 1, 15, 7, 0, 3, 0, 0, 0, 1, 17, 0, 5, 0, 0, 0, 0, 0, 1, 19, 9, 0, 0, 3, 0, 0, 0, 0, 1, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 23, 11, 7, 5, 0, 3, 0, 0, 0, 0, 0, 1, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, Aug 06 2008

Keywords

Comments

Row sums = A129235: (1, 4, 6, 11, 10, 20, 14,...).

Examples

			First few rows of the triangle:
   1;
   3, 1;
   5, 0, 1;
   7, 3, 0, 1;
   9, 0, 0, 0, 1;
  11, 5, 3, 0, 0, 1;
  13, 0, 0, 0, 0, 0, 1;
  15, 7, 0, 3, 0, 0, 0, 1;
  ...

Crossrefs

Cf. A051731, A126988, A129235.

Formula

By columns, replace the 1's in A051731 in succession with (1, 3, 5, 7,...).

Extensions

Definition corrected and a(50) split by Georg Fischer, Jun 08 2023

A129236 A054525 * A129234.

Original entry on oeis.org

1, 1, 2, 2, 0, 3, 2, 1, 0, 4, 4, 0, 0, 0, 5, 2, 2, 1, 0, 0, 6, 6, 0, 0, 0, 0, 0, 7, 4, 2, 0, 1, 0, 0, 0, 8, 6, 0, 2, 0, 0, 0, 0, 0, 9, 4, 4, 0, 0, 1, 0, 0, 0, 0, 10
Offset: 1

Views

Author

Gary W. Adamson, Apr 05 2007

Keywords

Comments

Left border = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6, ...). A129237 = inverse Moebius transform of A129234.

Examples

			First few rows of the triangle:
  1;
  1, 2;
  2, 0, 3;
  2, 1, 0, 4;
  4, 0, 0, 0, 5;
  2, 2, 1, 0, 0, 6;
  6, 0, 0, 0, 0, 0, 7;
  4, 2, 0, 1, 0, 0, 0, 8;
  ...

Crossrefs

Cf. A129234, A129235, A129237.

Formula

A054525 * A129234 as infinite lower triangular matrices.

A129237 A051731 * A129234.

Original entry on oeis.org

1, 3, 2, 4, 0, 3, 7, 5, 0, 4, 6, 0, 0, 0, 5, 12, 6, 7, 0, 0, 6, 8, 0, 0, 0, 0, 0, 7, 15, 10, 0, 9, 0, 0, 0, 8, 13, 0, 8, 0, 0, 0, 0, 0, 9, 18, 8, 0, 0, 11, 0, 0, 0, 0, 10
Offset: 1

Views

Author

Gary W. Adamson, Apr 05 2007

Keywords

Comments

Left border = Sigma(n), A000203: (1, 3, 4, 7, 6, 12, 8, ...). A129236 = Moebius transform of A129234.

Examples

			First few rows of the triangle:
   1;
   3,  2;
   4,  0, 3;
   7,  5, 0, 4;
   6,  0, 0, 0, 5;
  12,  6, 7, 0, 0, 6;
   8,  0, 0, 0, 0, 0, 7;
  15, 10, 0, 9, 0, 0, 0, 8;
  ...

Crossrefs

Cf. A129234, A129235, A129236, A054525, A051731, A000203.

Formula

A051731 * A129234 as infinite lower triangular matrices.

A130307 A051731 * A130296.

Original entry on oeis.org

1, 3, 1, 4, 1, 1, 7, 2, 1, 1, 6, 1, 1, 1, 1, 12, 3, 2, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 15, 3, 2, 2, 1, 1, 1, 1, 13, 2, 2, 1, 1, 1, 1, 1, 1, 18, 3, 2, 2, 2, 1, 1, 1, 1, 1
Offset: 0

Views

Author

Gary W. Adamson, May 20 2007

Keywords

Comments

Row sums = A129235, (1, 4, 6, 11, 10, 20, 14, ...).
Left column = sigma(n), A000203.

Examples

			First few rows of the triangle:
   1;
   3, 1;
   4, 1, 1;
   7, 2, 1, 1;
   6, 1, 1, 1, 1;
  12, 3, 2, 1, 1, 1;
   8, 1, 1, 1, 1, 1, 1;
  ...

Crossrefs

Cf. A130296, A051731, A129235, A000203.

Formula

Inverse Moebius transform of A130296.

A143594 Triangle read by rows, A051731 * (an infinite lower triangular matrix with 1's in the first column and the rest 2's).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 3, 4, 2, 2, 2, 2, 2, 2, 2, 4, 6, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 6, 4, 4, 2, 2, 2, 2, 3, 4, 4, 2, 2, 2, 2, 2, 2, 4, 6, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 10, 8, 6, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Gary W. Adamson, Aug 26 2008

Keywords

Comments

Left column = d(n), A000005: (1, 2, 2, 3, 2, 4, 2,...).
Prime rows have all 2's.
Row sums = A129235: (1, 4, 6, 11, 10, 10,...).

Examples

			First few rows of the triangle =
1;
2, 2;
2, 2, 2;
3, 4, 2, 2;
2, 2, 2, 2, 2;
4, 6, 4, 2, 2, 2;
2, 2, 2, 2, 2, 2, 2;
...

Crossrefs

Cf. A129235, A000005, A051731.

Formula

Triangle read by rows, A051731 * (an infinite lower triangular matrix with 1's in the first column and the rest 2's), i.e. (1; 1,2; 1,2,2;...), 1<=k<=n A051731 = the inverse Mobius transform.

A130267 A051731 * A051340.

Original entry on oeis.org

1, 2, 2, 2, 1, 3, 3, 3, 1, 4, 2, 1, 1, 1, 5, 4, 4, 4, 1, 1, 6, 2, 1, 1, 1, 1, 1, 7, 4, 4, 2, 5, 1, 1, 1, 8, 3, 2, 4, 1, 1, 1, 1, 1, 9, 4, 4, 2, 2, 6, 1, 1, 1, 1, 10
Offset: 0

Views

Author

Gary W. Adamson, May 18 2007

Keywords

Comments

Row sums = A129235: (1, 4, 6, 11, 10, 20, ...).
Left border = A000005, d(n): (1, 2, 2, 3, 2, 4, ...).

Examples

			First few rows of the triangle:
  1;
  2, 2;
  2, 1, 3;
  3, 3, 1, 4;
  2, 1, 1, 1, 5;
  4, 4, 4, 1, 1, 6;
  2, 1, 1, 1, 1, 1, 7;
  ...

Crossrefs

Cf. A051340, A051731, A000005, A129235.

Formula

Inverse Moebius transform of A051340.

A130314 A051731 * A126705.

Original entry on oeis.org

1, 3, 1, 4, 1, 1, 7, 2, 1, 1, 7, 1, 0, 1, 1, 12, 4, 2, 0, 1, 1, 9, 2, 1, 0, 0, 1, 1, 17, 4, 1, 2, 0, 0, 1, 1, 14, 3, 3, 1, 0, 0, 0, 1, 1, 19, 6, 2, 1, 2, 0, 0, 0, 1, 1
Offset: 1

Views

Author

Gary W. Adamson, May 21 2007

Keywords

Comments

Row sums = A129235: (1, 4, 6, 11, 10, 20, ...).

Examples

			First few rows of the triangle:
   1;
   3, 1;
   4, 1, 1;
   7, 2, 1, 1;
   7, 1, 0, 1, 1;
  12, 4, 2, 0, 1, 1;
  ...

Crossrefs

Cf. A051731, A126705, A129235.

Formula

Inverse Moebius transform of A126075.

A134560 Triangle A051731 * A127775 (as infinite lower triangular matrices).

Original entry on oeis.org

1, 1, 3, 1, 0, 5, 1, 3, 0, 7, 1, 0, 0, 0, 9, 1, 3, 5, 0, 0, 11, 1, 0, 0, 0, 0, 0, 13, 1, 3, 0, 7, 0, 0, 0, 15, 1, 0, 5, 0, 0, 0, 0, 0, 17, 1, 3, 0, 0, 9, 0, 0, 0, 0, 19, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 1, 3, 5, 7, 0, 11, 0, 0, 0, 0, 0, 23
Offset: 1

Views

Author

Gary W. Adamson, Oct 31 2007

Keywords

Comments

Obtained also by replacing the 1's in column k of A051731 with (2k-1).

Examples

			First few rows of the triangle:
  1;
  1,  3;
  1,  0,  5;
  1,  3,  0,  7;
  1,  0,  0,  0,  9;
  1,  3,  5,  0,  0, 11;
  1,  0,  0,  0,  0,  0, 13;
  1,  3,  0,  7,  0,  0,  0, 15;
  ...

Crossrefs

Cf. A129235 (row sums), A051731, A127775.
Showing 1-10 of 10 results.

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