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 1-4 of 4 results.

A117163 Column 2 of triangle A117162; equals the Moebius transform of A008683 (column 1 of A117162) preceded by a zero.

Original entry on oeis.org

0, 1, -1, -2, 0, -1, 1, 0, 1, -1, 1, 2, 0, -3, 2, 2, 0, -1, 0, 1, 0, -1, 1, 0, 0, -1, 1, 3, 0, -1, -1, -2, 0, 0, 0, 2, 0, -2, 2, 2, 0, 2, -1, 0, -2, -2, 1, -2, -1, 0, 1, 3, 0, -1, -1, 1, 1, 0, 1, -1, 0, -1, 0, 1, 0, 2, -1, 0, 0, 3, -1, -2, 0, -2, 0, 3, -2, 1, -1, -4, -1, -1, 1, -3, 0, 1, 2, 2, 0, 2, -1, 3, 2, -1, 1, 4, 0, 1, -1, 1, 0, -1
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

Keywords

Comments

The inverse Moebius transform of A117164 (column 3 of A117162) equals this sequence preceded by a zero.

Crossrefs

Cf. A117162, A112682, A008683 (column 1); A117164 (column 3).

Formula

Equals the Moebius transform of column 1 preceded by a zero:
[0,1,-1,-2,0,-1,1,0,...] = Moebius([0, 1,-1,-1,0,-1,1,-1,...]).

A117164 Column 3 of triangle A117162; equals the Moebius transform of A117163 (column 2 of A117162) preceded by a zero.

Original entry on oeis.org

0, 0, 1, -1, -2, -1, -1, 2, -1, 3, -1, 2, 2, 1, -2, 1, 2, 1, -1, 0, 1, 1, -1, -2, 2, -2, -1, 2, 3, 1, -1, -3, -2, -2, 3, -1, 2, 1, -5, 0, 2, -1, 2, 0, 4, -1, -2, -1, -1, -4, -3, 2, 3, 1, 2, -4, 1, -2, 0, 0, -1, 1, -1, 1, 1, 2, 2, 0, 0, -4, 3, -1, -2, -2, -1, 1, 5, 3, 1, -4, -3, -3, -1, -2, -3, -2, -3, 1, 2, -4, 1, 2, 3, 4, 2, 3, 4, 1, 4
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

Keywords

Comments

The inverse Moebius transform of column 4 of A117162 equals
this sequence preceded by a zero.

Crossrefs

Cf. A117162, A112682, A008683 (Moebius); A117163 (column 2).

Formula

Equals the Moebius transform of column 2 preceded by a zero:
[0,0,1,-1,-2,-1,-1,2,...] = Moebius([0, 0,1,-1,-2,0,-1,1,...]).

A117165 Triangle of coefficients for the Shift-Moebius transform, read by rows.

Original entry on oeis.org

1, -1, 1, -2, 0, 1, -1, -1, 0, 1, -2, -1, 0, 0, 1, 1, -2, -1, 0, 0, 1, -1, -1, -1, 0, 0, 0, 1, 3, -2, -1, -1, 0, 0, 0, 1, 0, 0, -2, -1, 0, 0, 0, 0, 1, 4, -2, -1, -1, -1, 0, 0, 0, 0, 1, 4, 0, -2, -1, -1, 0, 0, 0, 0, 0, 1, 5, 1, -1, -2, -1, -1, 0, 0, 0, 0, 0, 1, 1, 2, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1, 7, 0, 0, -2, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

Keywords

Comments

Column k = Shift-Moebius transform of a sequence of all zeros except for a single '1' in position k: [0,0,0,..(k-1)zeros..,1,0,0,0,...].
Column 1 is A117166, the Shift-Moebius transform of [1,0,0,0,...].
Column 2 is A117167, the Shift-Moebius transform of [0,1,0,0,...].
Column 3 is A117168, the Shift-Moebius transform of [0,0,1,0,...].
Row sums give A117169, the Shift-Moebius transform of [1,1,1,...].

Examples

			Triangle begins:
1;
-1, 1;
-2, 0, 1;
-1,-1, 0, 1;
-2,-1, 0, 0, 1;
1,-2,-1, 0, 0, 1;
-1,-1,-1, 0, 0, 0, 1;
3,-2,-1,-1, 0, 0, 0, 1;
0, 0,-2,-1, 0, 0, 0, 0, 1;
4,-2,-1,-1,-1, 0, 0, 0, 0, 1;
4, 0,-2,-1,-1, 0, 0, 0, 0, 0, 1;
5, 1,-1,-2,-1,-1, 0, 0, 0, 0, 0, 1;
1, 2,-1,-1,-1,-1, 0, 0, 0, 0, 0, 0, 1;
7, 0, 0,-2,-1,-1,-1, 0, 0, 0, 0, 0, 0, 1;
6, 3,-2,-1,-2,-1,-1, 0, 0, 0, 0, 0, 0, 0, 1;
5, 3, 1,-2,-1,-1,-1,-1, 0, 0, 0, 0, 0, 0, 0, 1; ...

Crossrefs

Cf. A117166 (column 1), A117167 (column 2), A117168 (column 3), A117169 (row sums), A117170 (inverse), A117162, A008683; A117175.

Programs

  • PARI
    {T(n,k)=if(n=c,if((r+n-i)%(c+n-i)==0,moebius((r+n-i)/(c+n-i)),0))))[ n,k])}

Formula

The Shift-Moebius transform of a sequence B is equal to the limit of the iteration: let C_1 = B and for k>1, C_{k+1} = Moebius transform of C_k preceded by k zeros, then shift left k places (to drop the leading k zeros).
Triangle A117162 is a good example, starting with A008683 in column 1 as C_1 and each column k, C_k, is obtained using the above iteration, so that the columns converge to A117166.

A117170 Triangle of coefficients for the Inverse Shift-Moebius transform, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 3, 1, 0, 0, 1, 3, 2, 1, 0, 0, 1, 4, 1, 1, 0, 0, 0, 1, 3, 3, 1, 1, 0, 0, 0, 1, 6, 1, 2, 1, 0, 0, 0, 0, 1, 5, 4, 1, 1, 1, 0, 0, 0, 0, 1, 5, 2, 2, 1, 1, 0, 0, 0, 0, 0, 1, 6, 4, 2, 2, 1, 1, 0, 0, 0, 0, 0, 1, 7, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 7, 6, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

Keywords

Comments

Column k = Inverse-Shift-Moebius transform of all zeros except for a single '1' in position k: [0,0,0,..(k-1)zeros..,1,0,0,0,...].
Column 1 is A117171 and equals Inverse-Shift-Moebius([1,0,0,0,...]).
Column 2 is A117172 and equals Inverse-Shift-Moebius([0,1,0,0,...]).
Column 3 is A117173 and equals Inverse-Shift-Moebius([0,0,1,0,...]).
Row sums give A117174 and equals Inverse-Shift-Moebius([1,1,1,...]).

Examples

			Triangle begins:
1;
1, 1;
2, 0, 1;
2, 1, 0, 1;
3, 1, 0, 0, 1;
3, 2, 1, 0, 0, 1;
4, 1, 1, 0, 0, 0, 1;
3, 3, 1, 1, 0, 0, 0, 1;
6, 1, 2, 1, 0, 0, 0, 0, 1;
5, 4, 1, 1, 1, 0, 0, 0, 0, 1;
5, 2, 2, 1, 1, 0, 0, 0, 0, 0, 1;
6, 4, 2, 2, 1, 1, 0, 0, 0, 0, 0, 1;
7, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
7, 6, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1;
10, 3, 4, 1, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1;
7, 6, 2, 3, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1; ...

Crossrefs

Cf. A117171 (column 1), A117172 (column 2), A117173 (column 3), A117174 (row sums), A117165 (inverse), A117162, A008683; A117176.

Programs

  • PARI
    {T(n,k)=if(n=c,if((r+i)%(c+i)==0,1,0))))[n,k])}
Showing 1-4 of 4 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.