A396870 a(n) = 4*3^(2*n+1) - floor(2*3^(n+1/2))^2.

3, 8, 11, 99, 332, 1307, 1667, 15003, 44123, 124379, 301211, 256283, 2306547, 9713132, 21143432, 90878747, 221435867, 203504219, 1831537971, 379074443, 3411669987, 30705029883, 58930910396, 530378193564, 859945288136, 1869319912307, 16823879210763, 45751534640339

Offset

0,1

Comments

a(n) is the negated discriminants of the finite fields GF(3^(2*n+1)). Discriminants of finite fields defined in the thesis of Zaitsev and the Alekseenko et al paper to study algebraic curves over finite fields.

a(n) -> oo as n -> oo since each number can only occur finitely times: for each d, the equation x^2 + d = 4*3^n can be partitioned into three elliptic curves x^2 + d = 4*Y^3, x^2 + d = 4*3*Y^3, and x^2 + d = 4*3^2*Y^3, each having only finitely many integer solutions.

Links

Jianing Song, Table of n, a(n) for n = 0..500

E. Alekseenko, S. Aleshnikov, N. Markin and A. Zaytsev, Optimal Curves of Genus 3 over Finite Fields with Discriminant -19, arXiv:0902.1901 [math.AG], 2009-2011. See Definition 1.1.

A.I. Zaitsev, Optimality properties of curves over finite fields, University of Armsterdam. See Section 3.1.

Mathematica

A396870[n_] := 4*3^(2*n + 1) - Floor[2*3^(n + 1/2)]^2;
Array[A396870, 25, 0] (* Paolo Xausa, Jun 09 2026 *)

Prog

(PARI) a(n, p=3) = my(r); sqrtint(4*p^(2*n+1), &r); r

Crossrefs

Cf. A396869.

Cf. discriminants of finite fields whose order is an odd power of p: A201125 (p=2), this sequence (p=3), A396871 (p=5), A396872 (p=7).

Sequence in context: A058565 A170901 A201882 * A356865 A050391 A360572

Adjacent sequences: A396867 A396868 A396869 * A396871 A396872 A396873

Keyword

nonn

Author

Jianing Song, Jun 08 2026

Status

approved