A143104 -id:A143104 - cp's OEIS frontend

A144199 Square array (8 X 8) read by rows.

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Sep 13 2008

Permutation contributing to the determinant that gives the 4th value of the Möbius function. Determinant of matrix is equal to 1. The array is asymmetrical.

			The array is:
0,0,0,0,0,0,0,1
1,0,0,0,0,0,0,0
0,0,1,0,0,0,0,0
0,0,0,1,0,0,0,0
0,0,0,0,1,0,0,0
0,0,0,0,0,1,0,0
0,0,0,0,0,0,1,0
0,1,0,0,0,0,0,0

Herbert S. Wilf, The Redheffer matrix of a partially ordered set, arXiv:math/0408263 [math.CO], 2004.

Cf. A143104, A002033, A144198, A144200.

A144200 Square array (8 X 8) read by rows.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Mats Granvik, Sep 13 2008

Permutation contributing to the determinant that gives the 8th value of the Möbius function. Determinant of matrix is equal to 1. The array is asymmetrical.

			The array is:
0,0,0,0,0,0,0,1
0,1,0,0,0,0,0,0
0,0,1,0,0,0,0,0
1,0,0,0,0,0,0,0
0,0,0,0,1,0,0,0
0,0,0,0,0,1,0,0
0,0,0,0,0,0,1,0
0,0,0,1,0,0,0,0

Herbert S. Wilf, The Redheffer matrix of a partially ordered set, arXiv:math/0408263 [math.CO], 2004.

Cf. A143104, A002033, A144199, A144201.

A144201 Square array (8 X 8) read by rows.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Mats Granvik, Sep 13 2008

Permutation contributing to the determinant that gives the 8th value of the Möbius function. Determinant of matrix is equal to -1. The array is asymmetrical.

			The array is:
0,0,0,0,0,0,0,1
1,0,0,0,0,0,0,0
0,0,1,0,0,0,0,0
0,1,0,0,0,0,0,0
0,0,0,0,1,0,0,0
0,0,0,0,0,1,0,0
0,0,0,0,0,0,1,0
0,0,0,1,0,0,0,0

Herbert S. Wilf, The Redheffer matrix of a partially ordered set, arXiv:math/0408263 [math.CO], 2004.

Cf. A143104, A002033, A144200.

A144194 Square array (6 X 6) read by rows.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Sep 13 2008

Permutation contributing to the determinant that gives the 6th value of the Möbius function. Determinant of matrix is equal to -1. The array is symmetrical.

			The array is:
0,0,0,0,0,1
0,1,0,0,0,0
0,0,1,0,0,0
0,0,0,1,0,0
0,0,0,0,1,0
1,0,0,0,0,0

J. Davies, Permutation Calculator
Herbert S. Wilf, The Redheffer matrix of a partially ordered set, arXiv:math/0408263 [math.CO], 2004.

Cf. A143104, A002033, A144193, A144195.

A174856 Square array read by antidiagonals up. Redheffer type matrix. T(1,1)=1 and T(n,1) = A049240.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Mats Granvik, Mar 31 2010

The first column is equal to 0 when n is a square greater than 1. The rest of the array is equal to A143104. The determinant of this array is A002819.

			The array begins:
  1,1,1,1,1,1,1,1,1,1
  1,1,0,0,0,0,0,0,0,0
  1,0,1,0,0,0,0,0,0,0
  0,1,0,1,0,0,0,0,0,0
  1,0,0,0,1,0,0,0,0,0
  1,1,1,0,0,1,0,0,0,0
  1,0,0,0,0,0,1,0,0,0
  1,1,0,1,0,0,0,1,0,0
  0,0,1,0,0,0,0,0,1,0
  1,1,0,0,1,0,0,0,0,1

Cf. A143104, A002819, A174854, A174852.

t[1, 1] = 1; t[n_, 1] := Boole[!IntegerQ[Sqrt[n]]]; t[n_, k_] := Boole[n == 1 || Mod[n, k] == 0]; Table[t[n - k + 1, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 05 2013 *)

A144966 Square array T(n,k) read by antidiagonals up. A055615 interleaved with k-1 zeros in each column. Redheffer type matrix.

Original entry on oeis.org

1, -2, 1, -3, 1, 1, 0, 0, 0, 1, -5, -2, 1, 0, 1, 6, 0, 0, 0, 0, 1, -7, -3, 0, 1, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 1, -11, -5, -3, -2, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -13, 6, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 14, 0, 0, 0, -2, 0

Offset: 1

Mats Granvik, Sep 27 2008

Determinant of this array appears to be the triangular numbers. The number of permutations that contribute to the result of the zero corner determinant that give the natural numbers, appears to be given by A002033.

			Determinant of:
1
is equal to 1.
Determinant of:
1,1
-2,0
is equal to 2.
Determinant of:
1,1,1
-2,1,0
-3,0,0
is equal to 3.
Determinant of:
1,1,1,1
-2,1,0,0
-3,0,1,0
0,-2,0,0
is equal to 4.
Determinant of:
1,1,1,1,1
-2,1,0,0,0
-3,0,1,0,0
0,-2,0,1,0
-5,0,0,0,0
is equal to 5.
Determinant of:
1,1,1,1,1,1
-2,1,0,0,0,0
-3,0,1,0,0,0
0,-2,0,1,0,0
-5,0,0,0,1,0
6,-3,-2,0,0,0
is equal to 6.

Cf. A143104, A143142.

=if(mod(row(); column())=0; lookup(row()/column(); A000027; A055615); if(row()=1; 1; 0))

A145110 Number of elements in the Redheffer matrix that contribute to the Moebius function.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 7, 13, 11, 14

Offset: 1

Mats Granvik, Oct 02 2008

a(11) to a(30) are conjectured to be 11, 23, 13, 18, 19, 25, 17, 29, 19, 30, 26, 26, 23, 45, 27, 30, 32, 39, 29, 48. [From Mats Granvik, Dec 21 2008]

			Contribution from _Mats Granvik_, Dec 21 2008: (Start)
The contributing elements for a(8) are:
0,0,0,0,0,0,0,1
1,1,0,0,0,0,0,0
0,0,1,0,0,0,0,0
1,1,0,1,0,0,0,0
0,0,0,0,1,0,0,0
0,0,0,0,0,1,0,0
0,0,0,0,0,0,1,0
1,1,0,1,0,0,0,0
where rows 8, 4 and 2 are included because they are divisors of 8. The sum of this array is 13, therefore a(8) = 13. (End)

Cf. A008683, A143104.

a(9)-a(10) from Mats Granvik, Oct 05 2008

Moved conjectured terms a(11)-a(30) to Comments. - N. J. A. Sloane, May 22 2014

A145379 Square array read by antidiagonals upwards: T(n,k) = 1 if n divides k, T(n,k) = 0 otherwise, n >= 1, k >= 2.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Oct 09 2008

This array is used in the proof of the Moebius function as a zero corner determinant of the Redheffer matrix. This table is the transpose of A051731 where the first row has been deleted.

			Table begins:
1,1,1,1,1,1,1,...
1,0,1,0,1,0,1,...
0,1,0,0,1,0,0,...
0,0,1,0,0,0,1,...
0,0,0,1,0,0,0,...
0,0,0,0,1,0,0,...
0,0,0,0,0,1,0,...
...

Cf. A008683, A051731, A143104.

Excel
```
=if(mod(column()+1;row())=0;1;0)
```

cp's OEIS Frontend

A144199 Square array (8 X 8) read by rows.

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A144200 Square array (8 X 8) read by rows.

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A144201 Square array (8 X 8) read by rows.

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A144194 Square array (6 X 6) read by rows.

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A174856 Square array read by antidiagonals up. Redheffer type matrix. T(1,1)=1 and T(n,1) = A049240.

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Mathematica

A144966 Square array T(n,k) read by antidiagonals up. A055615 interleaved with k-1 zeros in each column. Redheffer type matrix.

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Excel

A145110 Number of elements in the Redheffer matrix that contribute to the Moebius function.

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A145379 Square array read by antidiagonals upwards: T(n,k) = 1 if n divides k, T(n,k) = 0 otherwise, n >= 1, k >= 2.

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Excel