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A143104 Infinite Redheffer matrix read by upwards antidiagonals.

%I A143104 #67 Feb 16 2025 08:33:08
%S A143104 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,
%T A143104 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,
%U A143104 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
%N A143104 Infinite Redheffer matrix read by upwards antidiagonals.
%C A143104 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
%C A143104 The determinant of the n X n Redheffer matrix is given by Mertens's function A002321(n) [Barrett].
%C A143104 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
%C A143104 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
%D A143104 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.
%H A143104 Enrique Pérez Herrero, <a href="/A143104/b143104.txt">Rows n = 1..100 of triangle, flattened</a>
%H A143104 W. B. Barret, R. W. Forcade and A. D. Pollington, <a href="http://dx.doi.org/10.1016/0024-3795(88)90241-8">On the spectral radius of a (0,1) matrix related to Mertens' Function</a>, Lin. Alg. Applic. 107 (1988) 151-159.
%H A143104 Olivier Bordellès and Benoit Cloitre, <a href="http://www.emis.de/journals/JIPAM/images/317_08_JIPAM/317_08.pdf">A matrix inequality for Möbius functions</a>, J. Inequal. Pure and Appl. Math., Volume 10 (2009), Issue 3, Article 62, 9 pp.
%H A143104 Will Dana, <a href="https://sites.math.washington.edu/~morrow/336_15/papers/will.pdf">Eigenvalues of the Redheffer Matrix and their relation to the Mertens Function</a>, (2015)
%H A143104 R. M. Redheffer, <a href="http://dx.doi.org/10.1007/978-3-0348-5936-3_13">Eine explizit lösbare Optimierungsaufgabe</a>, Internat. Schiftenreihe Numer. Math., 36 (1977), 213-216.
%H A143104 T. Tao, <a href="http://terrytao.wordpress.com/2008/07/13/the-mobius-and-nilsequences-conjecture/">The Mobius function is strongly orthogonal to nilsequences</a>
%H A143104 R. C. Vaughan, <a href="http://dx.doi.org/10.1017/S1446788700037654">On the eigenvalues of Redheffer's matrix, II</a>, J. Austral. Math. Soc. (Series A) 60 (1996), 260-273.
%H A143104 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RedhefferMatrix.html">Redheffer Matrix</a>.
%H A143104 Herbert S. Wilf, <a href="http://arxiv.org/abs/math/0408263">The Redheffer matrix of a partially ordered set</a>, arXiv:math/0408263 [math.CO], 2004.
%H A143104 Herbert S. Wilf, <a href="https://doi.org/10.37236/1867">The Redheffer matrix of a partially ordered set</a>, The Electronic Journal of Combinatorics 11(2) (2004), #R10.
%F A143104 a(i,j) = 1 if j=1 or i|j; 0 otherwise.
%F A143104 a(A000217(n)) = a(A000217(n)+1) = 1. - _Enrique Pérez Herrero_, Apr 16 2010
%e A143104 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A143104 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
%e A143104 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0
%e A143104 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
%e A143104 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
%e A143104 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0
%e A143104 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0
%e A143104 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0
%e A143104 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
%e A143104 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
%e A143104 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
%e A143104 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
%e A143104 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
%e A143104 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
%e A143104 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
%e A143104 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
%e A143104 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
%e A143104 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
%e A143104 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
%e A143104 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
%p A143104 A143104 := proc(i,j)
%p A143104     if modp(j,i) =0 or j = 1 then
%p A143104         1;
%p A143104     else
%p A143104         0;
%p A143104     end if;
%p A143104 end proc:
%p A143104 for d from 2 to 10 do
%p A143104     for m from d-1 to 1 by -1 do
%p A143104         n := d-m ;
%p A143104         printf("%d ",A143104(n,m)) ;
%p A143104     end do:
%p A143104 end do: # _R. J. Mathar_, Jul 23 2017
%t A143104 Redheffer[i_, j_] := Boole[Divisible[i, j] || (i == 1)];
%t A143104 T[n_] := n*(n + 1)/2;
%t A143104 S[n_] := Floor[1/2 + Sqrt[2 n]];
%t A143104 j[n_] := 1 + T[S[n]] - n;
%t A143104 i[n_] := 1 + S[n] - j[n];
%t A143104 A143104[n_] := Redheffer[i[n], j[n]]; (* _Enrique Pérez Herrero_, Apr 13 2010 *)
%t A143104 a[i_, j_] := If[j == 1 || Divisible[j, i], 1, 0];
%t A143104 Table[a[i-j+1, j], {i, 1, 14}, {j, 1, i}] // Flatten (* _Jean-François Alcover_, Aug 07 2018 *)
%o A143104 (Excel) =if(mod(column();row())=0;1;if(column()=1;1;0)). Produces the Redheffer matrix.
%o A143104 (PARI) { a(i,j) = (j==1) || (j%i==0); }
%Y A143104 Cf. A008683, A051731.
%Y A143104 Cf. A002033, A144193 .. A144201, A143142. - _Mats Granvik_, Sep 14 2008
%K A143104 nonn,tabl
%O A143104 1,1
%A A143104 _Mats Granvik_, _Roger L. Bagula_ and _Gary W. Adamson_, Jul 24 2008
%E A143104 Edited and extended by _Max Alekseyev_, Oct 28 2008