cp's OEIS Frontend

A053725 Number of n X n binary matrices of order dividing 3 (also number of solutions to X^3=I in GL(n,2)).

%I A053725 #10 Jul 09 2018 19:54:50
%S A053725 1,3,57,1233,75393,19109889,6326835201,6388287561729,
%T A053725 23576681450405889,120906321631678693377,1968421511613895105052673,
%U A053725 111055505036706392268074909697,8965464105556083354144035638870017
%N A053725 Number of n X n binary matrices of order dividing 3 (also number of solutions to X^3=I in GL(n,2)).
%D A053725 V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
%H A053725 Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%o A053725 (PARI) \\ See Morison theorem 2.6
%o A053725 \\ F(n,q,k) is number of solutions to X^k=I in GL(i, GF(q)) for i=1..n.
%o A053725 \\ q is power of prime and gcd(q, k) = 1.
%o A053725 B(n,q,e)={sum(m=0, n\e, x^(m*e)/prod(k=0, m-1, q^(m*e)-q^(k*e)))}
%o A053725 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))}
%o A053725 F(10, 2, 3) \\ _Andrew Howroyd_, Jul 09 2018
%Y A053725 Cf. A053722, A053846, A053856.
%Y A053725 Cf. A053718, A053770, A053771, A053772, A053773, A053774, A053775, A053776, A053777.
%K A053725 nonn
%O A053725 1,2
%A A053725 _Vladeta Jovovic_, Mar 23 2000