OFFSET
1,2
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.
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, Dec 30 2008
Edited by W. Edwin Clark et al., Jan 15 2015
STATUS
approved