A053725 Number of n X n binary matrices of order dividing 3 (also number of solutions to X^3=I in GL(n,2)).
1, 3, 57, 1233, 75393, 19109889, 6326835201, 6388287561729, 23576681450405889, 120906321631678693377, 1968421511613895105052673, 111055505036706392268074909697, 8965464105556083354144035638870017
Offset: 1
Keywords
References
- V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
Links
- Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Crossrefs
Programs
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PARI
\\ See Morison theorem 2.6 \\ F(n,q,k) is number of solutions to X^k=I in GL(i, GF(q)) for i=1..n. \\ q is power of prime and gcd(q, k) = 1. B(n,q,e)={sum(m=0, n\e, x^(m*e)/prod(k=0, m-1, q^(m*e)-q^(k*e)))} F(n,q,k)={if(gcd(q,k)<>1, error("no can do")); my(D=ffgen(q)^0); my(f=factor(D*(x^k-1))); my(p=prod(i=1, #f~, (B(n, q, poldegree(f[i,1])) + O(x*x^n))^f[i,2])); my(r=B(n,q,1)); vector(n, i, polcoeff(p, i)/polcoeff(r, i))} F(10, 2, 3) \\ Andrew Howroyd, Jul 09 2018