A052497 Number of nonsingular n X n matrices over GF(9).

1, 8, 5760, 339655680, 1624314979123200, 629282246371356907929600, 19747506525777609095698646040576000, 50195501537943419769100848121708339934527488000

Offset: 0

Author

Vladeta Jovovic, Mar 16 2000

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..30
• 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

Cf. A027877, A053764.

Cf. A002884, A003792, A053290, A053291, A053292, A053293, A052496, A052498, A132037.

Programs

Magma

[1] cat [&*[(9^n - 9^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 28 2013

Mathematica

Table[Product[(9^n - 9^j), {j, 0, n-1}], {n, 0, 10}] (* G. C. Greubel, May 14 2019 *)

PARI

{a(n) = prod(j=0,n-1, 9^n - 9^j)}; \\ G. C. Greubel, May 14 2019

Sage

[product(9^n - 9^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019

Formula

a(n) = (9^n - 1)*(9^n - 9)*...*(9^n - 9^(n-1)).

a(n) = A053764(n)*A027877(n). - Bruno Berselli, Jan 30 2013

a(n) ~ c * 9^(n^2), where c = A132037. - Amiram Eldar, Jul 06 2025