cp's OEIS Frontend

A169873 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_2^n.

%I A169873 #15 May 02 2024 09:47:28
%S A169873 6,10,18,33,53,97,172,321,603,1153,2227,4353,8553,16897,33491,66561,
%T A169873 132519,264193,527183,1052673,2102943,4202497,8400192,16793601,
%U A169873 33577603,67141633,134264067,268500993,536963592,1073872897,2147669011,4295229441,8590305319,17180393473,34360479823
%N A169873 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_2^n.
%D A169873 J. W. P. Hirschfeld, Linear codes and algebraic curves, pp. 35-53 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.
%H A169873 Robin Visser, <a href="/A169873/b169873.txt">Table of n, a(n) for n = 1..3000</a>
%H A169873 Jean-Pierre Serre, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k55351747/f35.item">Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini</a>, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 397-402.
%H A169873 Gerard van der Geer et al., <a href="http://www.manypoints.org">Tables of curves with many points</a>
%H A169873 Gerard van der Geer and Marcel van der Vlugt, <a href="https://doi.org/10.1090/S0025-5718-99-01143-6">Tables of curves with many points</a>, Math. Comp. 69 (2000) 797-810.
%o A169873 (Sage)
%o A169873 def a(n):
%o A169873     if n==2: return 10
%o A169873     elif (n%2 == 0): return 2^n + 1 + 2^(n/2+2)
%o A169873     elif ((floor(2^(n/2+1))%2 == 0) or (2^n-1).is_square()
%o A169873         or (4*2^n-3).is_square() or (4*2^n-7).is_square()):
%o A169873         if (frac(2^(n/2+1)) > ((sqrt(5)-1)/2)): return 2^n + 2*floor(2^(n/2+1))
%o A169873         else: return 2^n + 2*floor(2^(n/2+1)) - 1
%o A169873     else: return 2^n + 1 + 2*floor(2^(n/2+1))  # _Robin Visser_, Oct 01 2023
%Y A169873 Cf. A005525, A169869-A169883.
%K A169873 nonn
%O A169873 1,1
%A A169873 _N. J. A. Sloane_, Jul 05 2010
%E A169873 More terms from _Robin Visser_, Oct 01 2023