cp's OEIS Frontend

A169878 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_3^n.

%I A169878 #15 May 02 2024 09:47:24
%S A169878 8,20,48,118,306,838,2372,6886,20244,60022,178830,534358,1599374,
%T A169878 4791718,14364057,43072966,129185618,387499222,1162397834,3487020598,
%U A169878 10460762306,31381768198,94144406138,282431662246,847292291373,2541872205622,7625608530780,22876811586838,68630410502264
%N A169878 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_3^n.
%D A169878 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 A169878 Robin Visser, <a href="/A169878/b169878.txt">Table of n, a(n) for n = 1..2000</a>
%H A169878 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 A169878 Gerard van der Geer et al., <a href="http://www.manypoints.org">Tables of curves with many points</a>
%H A169878 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 A169878 (Sage)
%o A169878 def a(n):
%o A169878     if n==2: return 20
%o A169878     elif (n%2 == 0): return 3^n + 1 + 4*3^(n/2)
%o A169878     elif ((floor(2*3^(n/2))%3 == 0) or (3^n-1).is_square()
%o A169878         or (4*3^n-3).is_square() or (4*3^n-7).is_square()):
%o A169878         if (frac(2*3^(n/2)) > ((sqrt(5)-1)/2)): return 3^n + 2*floor(2*3^(n/2))
%o A169878         else: return 3^n + 2*floor(2*3^(n/2)) - 1
%o A169878     else: return 3^n + 1 + 2*floor(2*3^(n/2))  # _Robin Visser_, Oct 01 2023
%Y A169878 Cf. A005525, A169869-A169883.
%K A169878 nonn
%O A169878 1,1
%A A169878 _N. J. A. Sloane_, Jul 05 2010
%E A169878 More terms from _Robin Visser_, Oct 01 2023