A141538 Decimal expansion of constant arising in enumerating 2 X 2 integer matrices having a prescribed integer eigenvalue.

5, 5, 8, 7, 3, 9, 5, 7, 4, 7, 3, 7, 3, 0, 4, 6, 0, 4, 3, 9, 5, 2, 0, 9, 1, 2, 7, 6, 1, 7, 5, 0, 0, 4, 4, 9, 8, 2, 9, 0, 9, 0, 2, 0, 1, 0, 6, 2, 4, 5, 4, 5, 4, 8, 2, 1, 2, 7, 0, 7, 1, 8, 2, 0, 5, 6, 4, 9, 7, 0, 2, 9, 5, 3, 1, 4, 9, 2, 6, 1, 0, 1, 2, 2, 8, 6, 6, 0, 3, 0, 4, 2, 1, 9, 1, 2, 3, 1, 6, 3, 5, 7, 4, 1, 5

Offset

0,1

Comments

Martin and Wong, Corollary 2, p. 2. Abstract: Random matrices arise in many mathematical contexts and it is natural to ask about the properties that such matrices satisfy. If we choose a matrix with integer entries at random, for example, what is the probability that it will have a particular integer as an eigenvalue, or an integer eigenvalue at all? If we choose a matrix with real entries at random, what is the probability that it will have a real eigenvalue in a particular interval? The purpose of this paper is to resolve these questions, once they are made suitably precise, in the setting of 2 X 2 matrices.

Links

Table of n, a(n) for n=0..104.

Greg Martin and Erick B. Wong, The number of 2x2 integer matrices having a prescribed integer eigenvalue, arXiv:0808.1922 [math.PR], Aug 14 2008.

Formula

(7 * sqrt(2) + 4 + 3*log(1+sqrt(2)))/(3*Pi^2). - Corrected by R. J. Mathar, Aug 20 2008

Example

0.55873957473730460439520912761750044982909020106245454821270718205...

Maple

print((7*sqrt(2)+4+3*log(1+sqrt(2)))/(3*Pi^2)) ; # R. J. Mathar, Aug 20 2008

Mathematica

RealDigits[(7 Sqrt[2] + 4 + 3*Log[1 + Sqrt@2])/(3*Pi^2), 10, 111][[1]] (* Robert G. Wilson v *)

Crossrefs

Sequence in context: A141864 A245189 A217169 * A358257 A003861 A107623

Adjacent sequences: A141535 A141536 A141537 * A141539 A141540 A141541

Keyword

cons,easy,nonn

Author

Jonathan Vos Post, Aug 15 2008

Extensions

Extended by Robert G. Wilson v, Aug 17 2008

Extended by R. J. Mathar, Aug 20 2008

Offset corrected R. J. Mathar, Jan 26 2009

Status

approved