A053170 Number of 3 X 3 integer matrices with elements in the range [ -n,n ] which generate a group of finite order under matrix multiplication.

1584, 13056, 57744, 133440, 307632, 519888, 990288, 1436208, 2108832, 3015648, 4075296, 5244960, 7464816, 9329280, 11739072, 14099616, 17516304, 20539296, 25358400, 29624880, 36250800, 41164992, 47420016, 53422320, 62019168, 70891488, 80028000, 89233248, 100924176, 111503568

Offset

1,1

Comments

Count of 3 X 3 matrices M with det(M) = +- 1 such that M^k = I for some k <= 6. - Sean A. Irvine, Dec 16 2021

To see why k <= 6 covers all cases, consider that if lambda is an eigenvalue of M, lambda^k is an eigenvalue of M^k = I, so lambda^k = 1, therefore all eigenvalues are located on the unity circle. The characteristic polynomial has integer coefficients, so it has 1 or 3 integer roots that are either -1 or +1. As the sum of the roots must also be integer, this leaves as only options for the complex roots the three pairs -1/2 +/- j*sqrt(3)/2, 0 +/- j*1 and 1/2 +/- j*sqrt(3)/2, respectively at angles +/- 120°, +/- 90° and +/- 60° in the complex plane, so there must be a k <= 6 that makes all eigenvalues of M^k equal to 1. If the multiplicity of -1 or 1 is >1, this does not imply yet that M^k must be I, but in this case larger k will also not satisfy the equation (see StackExchange link). - Bert Dobbelaere, Apr 18 2025

Links

Table of n, a(n) for n=1..30.

EuYu, Eigenvalues and power of a matrix (StackExchange)

Sean A. Irvine, Java program (github)

Formula

a(n) = 2*(1 + A053171(n) + A053172(n) + A053173(n) + A053174(n)).

Crossrefs

Cf. A053171, A053172, A053173, A053174.

Sequence in context: A242539 A224945 A306313 * A237916 A159211 A235844

Adjacent sequences: A053167 A053168 A053169 * A053171 A053172 A053173

Keyword

nonn

Author

Ralf W. Grosse-Kunstleve, Mar 26 2000

Extensions

a(11) from Sean A. Irvine, Dec 16 2021

a(12)-a(30) from Bert Dobbelaere, Apr 18 2025

Status

approved