A143142 -id:A143142 - cp's OEIS frontend

A143104 Infinite Redheffer matrix read by upwards antidiagonals.

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

Offset: 1

Mats Granvik, Roger L. Bagula and Gary W. Adamson, Jul 24 2008

Note that Redheffer's matrix (1977) is actually given by A077049: the first row starts with a single 1. We follow the nomenclature of Wilf, Dana, Vaughan and Weisstein, which uses the transpose and sets the first column to all-1. - R. J. Mathar, Jul 22 2017
The determinant of the n X n Redheffer matrix is given by Mertens's function A002321(n) [Barrett].
For n > 1, replacing a(n,n) with 0 in the Redheffer matrix and taking the determinant gives Moebius(n) = A008683(n). The number of permutations with nonzero contribution to this determinant is given by A002033. For first few n, these permutations are shown in the sequences A144193 to A144201. - Mats Granvik, Sep 14 2008
The determinant that is the Moebius function was discovered by reading the blog post "The Mobius function is strongly orthogonal to nilsequences" by Terence Tao. - Mats Granvik, Jan 24 2009

			1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0
1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0
1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0
1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0
1 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 0 0 1 0 0 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 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 0 0 0 1 0 0 0 0 0 0 0
1 0 0 0 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 0 0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

R. C. Vaughan, On the eigenvalues of Redheffer's matrix I, in: Number Theory with an Emphasis on the Markoff Spectrum (Provo, Utah, 1991), 283-296, Lecture Notes in Pure and Appl. Math., 147, Dekker, New-York, 1993.

Enrique Pérez Herrero, Rows n = 1..100 of triangle, flattened
W. B. Barret, R. W. Forcade and A. D. Pollington, On the spectral radius of a (0,1) matrix related to Mertens' Function, Lin. Alg. Applic. 107 (1988) 151-159.
Olivier Bordellès and Benoit Cloitre, A matrix inequality for Möbius functions, J. Inequal. Pure and Appl. Math., Volume 10 (2009), Issue 3, Article 62, 9 pp.
Will Dana, Eigenvalues of the Redheffer Matrix and their relation to the Mertens Function, (2015)
R. M. Redheffer, Eine explizit lösbare Optimierungsaufgabe, Internat. Schiftenreihe Numer. Math., 36 (1977), 213-216.
T. Tao, The Mobius function is strongly orthogonal to nilsequences
R. C. Vaughan, On the eigenvalues of Redheffer's matrix, II, J. Austral. Math. Soc. (Series A) 60 (1996), 260-273.
Eric Weisstein's World of Mathematics, Redheffer Matrix.
Herbert S. Wilf, The Redheffer matrix of a partially ordered set, arXiv:math/0408263 [math.CO], 2004.
Herbert S. Wilf, The Redheffer matrix of a partially ordered set, The Electronic Journal of Combinatorics 11(2) (2004), #R10.

Cf. A008683, A051731.

Cf. A002033, A144193 .. A144201, A143142. - Mats Granvik, Sep 14 2008

=if(mod(column();row())=0;1;if(column()=1;1;0)). Produces the Redheffer matrix.

A143104 := proc(i,j)
    if modp(j,i) =0 or j = 1 then
        1;
    else
        0;
    end if;
end proc:
for d from 2 to 10 do
    for m from d-1 to 1 by -1 do
        n := d-m ;
        printf("%d ",A143104(n,m)) ;
    end do:
end do: # R. J. Mathar, Jul 23 2017

Redheffer[i_, j_] := Boole[Divisible[i, j] || (i == 1)];
T[n_] := n*(n + 1)/2;
S[n_] := Floor[1/2 + Sqrt[2 n]];
j[n_] := 1 + T[S[n]] - n;
i[n_] := 1 + S[n] - j[n];
A143104[n_] := Redheffer[i[n], j[n]]; (* Enrique Pérez Herrero, Apr 13 2010 *)
a[i_, j_] := If[j == 1 || Divisible[j, i], 1, 0];
Table[a[i-j+1, j], {i, 1, 14}, {j, 1, i}] // Flatten (* Jean-François Alcover, Aug 07 2018 *)

PARI
```
{ a(i,j) = (j==1) || (j%i==0); }
```

a(i,j) = 1 if j=1 or i|j; 0 otherwise.

a(A000217(n)) = a(A000217(n)+1) = 1. - Enrique Pérez Herrero, Apr 16 2010

Edited and extended by Max Alekseyev, Oct 28 2008

A144778 Square array T(n,k) read by antidiagonals upwards. Mobius function interleaved with k-1 zeros in each column. Every second value in the first row is either 1 or -1. Redheffer type matrix.

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, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 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, 0, 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, 0, 0, -1

Offset: 1

Mats Granvik, Sep 21 2008

The determinant of this matrix appears to be A059841.

			Table begins:
.1.-1..1.-1..1.-1..1
-1..1..0..0..0..0..0
-1..0..1..0..0..0..0
.0.-1..0..1..0..0..0
-1..0..0..0..1..0..0
.1.-1.-1..0..0..1..0
-1..0..0..0..0..0..1

Cf. A008683, A059841, A143142.

T(n,k) = if(n==1 then if(k mod 2==0 then -1 else 1) else if(n mod k == 0 then A008683(n/k) else 0)).

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))

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A143104 Infinite Redheffer matrix read by upwards antidiagonals.

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A144778 Square array T(n,k) read by antidiagonals upwards. Mobius function interleaved with k-1 zeros in each column. Every second value in the first row is either 1 or -1. Redheffer type matrix.

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Author

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Comments

Examples

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Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Excel