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.

A127468 Triangle read by rows: matrix product A127466*A054525.

Original entry on oeis.org

1, 0, 2, -3, 0, 6, 0, -4, 0, 8, -15, 0, 0, 0, 20, 0, -6, 0, 0, 0, 12, -35, 0, 0, 0, 0, 0, 42, 0, -8, 0, -16, 0, 0, 0, 32, -9, 0, -36, 0, 0, 0, 0, 0, 54, 0, -30, 0, 0, 0, 0, 0, 0, 0, 40, -99, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 0, 12, 0, -24, 0, -24, 0, 0, 0, 0, 0, 48
Offset: 1

Views

Author

Gary W. Adamson, Jan 15 2007

Keywords

Comments

The row sums are n, the index of the row.

Examples

			First few rows of the triangle are:
1;
0, 2;
-3, 0, 6;
0, -4, 0, 8;
-15, 0, 0, 0, 20;
0, -6, 0, 0, 0, 12;
-35, 0, 0, 0, 0, 0, 42;
...

Crossrefs

Cf. A127466, A054525, A002618.

Programs

Formula

T(n,k) = sum_{j=k..n} A127466(n,j) * A054525(j,k).
T(n,n) = A002618(n).

A127470 Triangle equal to the matrix product A127466 * A051731.

Original entry on oeis.org

1, 4, 2, 9, 0, 6, 16, 12, 0, 8, 25, 0, 0, 0, 20, 36, 18, 24, 0, 0, 12, 49, 0, 0, 0, 0, 0, 42, 64, 56, 0, 48, 0, 0, 0, 32, 81, 0, 72, 0, 0, 0, 0, 0, 54, 100, 50, 0, 0, 80, 0, 0, 0, 0, 40, 121, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 144, 108, 96, 72, 0, 72, 0, 0, 0, 0, 0, 48
Offset: 1

Views

Author

Gary W. Adamson, Jan 15 2007

Keywords

Examples

			First few rows of the triangle are:
1;
4, 2;
9, 0, 6;
16, 12, 0, 8;
25, 0, 0, 0, 20;
36, 18, 24, 0, 0, 12;
49, 0, 0, 0, 0, 0, 42;
64, 56, 0, 48, 0, 0, 0, 32;
81, 0, 72, 0, 0, 0, 0, 0, 54;
...

Crossrefs

Cf. A127466, A051731, A127469 (row sums), A061949.

Programs

Formula

T(n,1) = n^2.
T(n,n) = n*phi(n) = A002618(n).

Extensions

Terms corrected by R. J. Mathar, Sep 08 2013

A127467 Mobius transform of A127466.

Original entry on oeis.org

1, 1, 2, 2, 0, 6, 2, 2, 0, 8, 4, 0, 0, 0, 20, 2, 4, 6, 0, 0, 12, 6, 0, 0, 0, 0, 0, 42, 4, 4, 0, 8, 0, 0, 0, 32, 6, 0, 12, 0, 0, 0, 0, 0, 54, 4, 8, 0, 0, 20, 0, 0, 0, 0, 40
Offset: 1

Views

Author

Gary W. Adamson, Jan 15 2007

Keywords

Comments

Left column = phi(n), A000010; right border = n*phi(n), A002618; row sums = A007434, (Mobius transform of the squares).

Examples

			First few rows of the triangle are:
1;
1, 2;
2, 0, 6;
2, 2, 0, 8;
4, 0, 0, 0, 20;
2, 4, 6, 0, 0, 12;
6, 0, 0, 0, 0, 0, 42;
4, 4, 0, 8, 0, 0, 0, 32;
...

Crossrefs

Cf. A127466, A002618, A000010, A054525, A007434.

Formula

Mobius transform of A127466; (A054525 * A127466 as infinite lower triangular matrices).

A127481 Triangle T(n,k) read by rows: T(n,k) = sum_{l=k..n, l|n, k|l} l*phi(k).

Original entry on oeis.org

1, 3, 2, 4, 0, 6, 7, 6, 0, 8, 6, 0, 0, 0, 20, 12, 8, 18, 0, 0, 12, 8, 0, 0, 0, 0, 0, 42, 15, 14, 0, 24, 0, 0, 0, 32, 13, 0, 24, 0, 0, 0, 0, 0, 54, 18, 12, 0, 0, 60, 0, 0, 0, 0, 40, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 28, 24, 42, 32, 0, 36, 0, 0, 0, 0, 0, 48, 14, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gary W. Adamson, Jan 15 2007

Keywords

Examples

			First few rows of the triangle are:
1;
3, 2;
4, 0, 6;
7, 6, 0, 8;
6, 0, 0, 0, 20,
12, 8, 18, 0, 0, 12;
8, 0, 0, 0, 0, 0, 42;
15, 14, 0, 24, 0, 0, 0, 32;
...

Crossrefs

Cf. A054522, A127093, A001157 (row sums), A002618, A127466.

Programs

  • Maple
    A127481 := proc(n,k)
    a :=0 ;
    for l from k to n do
        if modp(n,l) =0 and modp(l,k) =0 then
            a := a+l*numtheory[phi](k) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 06 2013

Formula

T(n,1) = A000203(n).
T(n,n) = A002618(n).
T(n,k) =sum_{l=k..n} A127093(n,l) * A054522(l,k), the matrix product of the infinite lower triangular matrices.
Showing 1-4 of 4 results.

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

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