A065498 Number of invertible n X n matrices mod 6 (i.e., over the ring Z_6).

1, 2, 288, 1886976, 489104179200, 4755360379856486400, 1695944421638473850132889600, 21967113634648374162210646578639667200, 10286692771039109536373764545035369981946101760000, 173770439600109774111384717714984362383506603790098046648320000

Offset: 0

Author

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 25 2001

Keywords

• nonn
• easy

Links

• Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
• Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Crossrefs

Column k=6 of A316622.

Cf. A002884, A048651, A053290, A065128, A100220.

Programs

Mathematica

a[n_] := 6^(n^2)*Product[(1 - 1/2^k)*(1 - 1/3^k), { k, 1, n} ]; Table[ a[n], {n, 0, 9} ]

Formula

a(n) = 6^(n^2) * Product_{k=1..n} ((1 - 1/2^k)(1 - 1/3^k)).

a(n) = A002884(n)*A053290(n). - Geoffrey Critzer, Jan 26 2018

a(n) ~ c * 6^(n^2), where c = A048651 * A100220 = 0.161757743053... . - Amiram Eldar, Jul 06 2025

Extensions

More terms from Robert G. Wilson v, Nov 28 2001