A003053 Order of orthogonal group O(n, GF(2)). (Formerly M1716)

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

Offset

1,2

References

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.

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

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 *)

Prog

(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

Crossrefs

Bisections give A003923 and A090770.

Sequence in context: A365287 A129464 A175430 * A113296 A275462 A063744

Adjacent sequences: A003050 A003051 A003052 * A003054 A003055 A003056

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

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

Status

approved