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

1,1

Comments

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

References

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.

Links

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.

Formula

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

Example

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

Maple

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

Mathematica

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 *)

Prog

(Excel) =if(mod(column(); row())=0; 1; if(column()=1; 1; 0)). Produces the Redheffer matrix.
(PARI) { a(i, j) = (j==1) || (j%i==0); }

Crossrefs

Cf. A008683, A051731.

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

Sequence in context: A204183 A204177 A185917 * A127236 A117947 A175860

Adjacent sequences: A143101 A143102 A143103 * A143105 A143106 A143107

Keyword

nonn,tabl

Author

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

Extensions

Edited and extended by Max Alekseyev, Oct 28 2008

Status

approved