A169877 - OEIS

%I #10 Aug 17 2023 20:44:40

%S 7,16,38,100,275,784,2280,6724,19964,59536,177989,532900,1596849,

%T 4787344,14356482,43059844,129162891,387459856,1162329651,3486902500,

%U 10460557755,31381413904,94143792483,282430599364,847290450408,2541869016976,7625603007884,22876802020900,68630393933574

%N Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 1 over the field F_3^n.

%H Robin Visser, <a href="/A169877/b169877.txt">Table of n, a(n) for n = 1..2000</a>

%H Max Deuring, <a href="https://doi.org/10.1007/BF02940746">Die Typen der Multiplikatorenringe elliptischer Funktionenkörper</a>, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.

%H Gerard van der Geer et al., <a href="http://www.manypoints.org">Tables of curves with many points</a>

%H 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.

%H W. C. Waterhouse, <a href="https://doi.org/10.24033/asens.1183">Abelian varieties over finite fields</a>, Ann Sci. E.N.S., (4) 2 (1969), 521-560.

%F a(n) = 3^n + 1 + floor(2*3^(n/2)) if 3 does not divide floor(2*3^(n/2)), n is even, or n = 1. Otherwise a(n) = 3^n + floor(2*3^(n/2)) [Deuring-Waterhouse]. - _Robin Visser_, Aug 17 2023

%o (Sage)

%o def a(n):

%o     if (n==1) or (n%2 == 0) or (floor(2*3^(n/2))%3 != 0):

%o         return 3^n + 1 + floor(2*3^(n/2))

%o     else:

%o         return 3^n + floor(2*3^(n/2))  # _Robin Visser_, Aug 17 2023

%Y Cf. A005523, A169869-A169883.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Jul 05 2010

%E More terms from _Robin Visser_, Aug 17 2023