A020515 a(n) = 4^n - 2^n + 1.

1, 3, 13, 57, 241, 993, 4033, 16257, 65281, 261633, 1047553, 4192257, 16773121, 67100673, 268419073, 1073709057, 4294901761, 17179738113, 68719214593, 274877382657, 1099510579201, 4398044413953, 17592181850113, 70368735789057, 281474959933441, 1125899873288193

Offset: 0

Author

Simon Plouffe

Keywords

• nonn
• easy

Comments

6th cyclotomic polynomial evaluated at powers of 2.

Referred to as Kasami-Welch numbers by Aubry et al. (see links). - Jonathan Vos Post, Sep 15 2009

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..170
• Yves Aubry, Gary Mcguire, and Francois Rodier, A few more functions that are not APN infinitely often, arXiv:0909.2304 [math.AG], 2009.
• Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1.
• Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Crossrefs

Cf. A000079, A002061.

Programs

Magma

[4^n - 2^n + 1: n in [0..40]]; // Vincenzo Librandi, Apr 25 2011

Maple

with(numtheory,cyclotomic):seq(cyclotomic(6,2**i),i=0..24);

Mathematica

Array[4^# - 2^# + 1 &, 30, 0] (* or *)
Cyclotomic[6, 2^Range[0, 29]] (* Paolo Xausa, Apr 02 2024 *)
LinearRecurrence[{7,-14,8},{1,3,13},30] (* Harvey P. Dale, Aug 24 2024 *)

PARI

a(n)=4^n-2^n+1 \\ Charles R Greathouse IV, Jul 02 2013

Python

def A020515(n): return (1<Chai Wah Wu, Mar 11 2025

Formula

From Mohammad K. Azarian, Jan 15 2009: (Start)

G.f.: 1/(1-4*x) - 1/(1-2*x) + 1/(1-x).

E.g.f.: e^(4*x) - e^(2*x) + e^x. (End)

a(n) = A002061(A000079(n)). - Michel Marcus, Apr 06 2016