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.
R. M. Redheffer, Eine explizit lösbare Optimierungsaufgabe, Internat. Schiftenreihe Numer. Math., 36 (1977), 213-216.
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.
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
KEYWORD
nonn,tabl
AUTHOR
EXTENSIONS
Edited and extended by Max Alekseyev, Oct 28 2008
STATUS
approved