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A143104
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Infinite Redheffer matrix read by upwards antidiagonals.
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18
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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
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OFFSET
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1,1
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COMMENTS
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The determinant of the nxn Redheffer matrix is given by Mertens function A002321(n).
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 non-zero contribution to this determinant is given by A002033. For first few n, these permutations are shown in the sequences A144193 to A144201. [From 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. [From Mats Granvik, Jan 24 2009]
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REFERENCES
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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.
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.
Herbert S. Wilf, The Redheffer matrix of a partially ordered set, The Electronic Journal of Combinatorics 11(2) (2004), #R10
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LINKS
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Enrique Pérez Herrero, Rows n = 1..100 of triangle, flattened
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.
T. Tao, The Mobius function is strongly orthogonal to nilsequences
E. W. Weisstein, MathWorld: Redheffer Matrix.
Herbert S. Wilf, The Redheffer matrix of a partially ordered set, arXiv:math/0408263v1
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FORMULA
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a(i,j) = 1 if j=1 or j|i; 0 otherwise.
a(A000217(n))=a(A000217(n)+1)=1 [From Enrique Pérez Herrero, Apr 16 2010]
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MATHEMATICA
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Contribution from Enrique Pérez Herrero, Apr 13 2010: (Start)
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]]; (End)
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PROG
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(Excel cell formula) =if(mod(column(); row())=0; 1; if(column()=1; 1; 0)). Produces the Redheffer matrix.
(PARI) { a(i, j) = (j==1) || (j%i==0); }
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CROSSREFS
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Cf. A008683, A051731.
Cf. A002033, A144193 .. A144201, A143142. [From Mats Granvik, Sep 14 2008]
Sequence in context: A014086 A014163 A166360 * A127236 A117947 A175860
Adjacent sequences: A143101 A143102 A143103 * A143105 A143106 A143107
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KEYWORD
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nonn,tabl
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AUTHOR
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Mats Granvik, Roger L. Bagula and Gary W. Adamson, Jul 24 2008
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EXTENSIONS
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Edited and extended by Max Alekseyev, Oct 28 2008
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STATUS
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approved
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