A003053 Order of orthogonal group O(n, GF(2)).

1, 2, 6, 48, 720, 23040, 1451520, 185794560, 47377612800, 24257337753600, 24815256521932800, 50821645356918374400, 208114637736580743168000, 1704875112338069448032256000, 27930968965434591767112450048000, 915241991059360703024740763172864000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..41
• J. H. Conway et al., ATLAS of Finite Groups, Chapter 2.
• F. J. MacWilliams, Orthogonal matrices over finite fields, Amer. Math. Monthly, 76 (1969), 152-164.

Crossrefs

Bisections give A003923 and A090770.

Programs

Maple

h:=proc(n) local m;
if n mod 2 = 0 then m:=n/2;
2^(m^2)*mul( 4^i-1, i=1..m);
else m:=(n+1)/2;
2^(m^2)*mul( 4^i-1, i=1..m-1);
fi;
end;
# This produces a(n+1)

Mathematica

h[n_] := Module[{m}, If[EvenQ[n], m = n/2; 2^(m^2)*Product[4^i-1, {i, 1, m}], m = (n+1)/2; 2^(m^2)*Product[4^i-1, {i, 1, m-1}]]];
a[n_] := h[n-1];
Array[a, 16] (* Jean-François Alcover, Aug 18 2022, after Maple code *)

PARI

a(n) = n--; if (n % 2, m = (n+1)/2; 2^(m^2)*prod(k=1, m-1, 4^k-1), m = n/2; 2^(m^2)*prod(k=1, m, 4^k-1)); \\ Michel Marcus, Jul 13 2017

Python

def size_binary_orthogonal_group(n):
    k = n-1
    if k%2==0:
        m=k//2
        p=2**(m**2)
        for i in range(1,m+1):
            p*=4**i-1
    else:
        m=(k+1)//2
        p=2**(m**2)
        for i in range(1,m):
            p*=4**i-1
    return p
#call and print output for a(n)
print([size_binary_orthogonal_group(n) for n in range(1, 10)])
# Nathan J. Russell, Nov 01 2017

Python

from math import prod
def A003053(n): return (1 << (n//2)**2)*prod((1 << i)-1 for i in range(2,2*((n-1)//2)+1,2)) # Chai Wah Wu, Jun 20 2022

Formula

For formulas see Maple code.

Asymptotics: a(n) ~ c * 2^((n^2-n)/2), where c = (1/4; 1/4)infinity ~ 0.6885375... is expressed in terms of the Q-Pochhammer symbol. - _Cedric Lorand, Aug 07 2017

Extensions

Edited by N. J. A. Sloane, Dec 30 2008

Edited by W. Edwin Clark et al., Jan 15 2015