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

0,2

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).

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

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 *)

Prog

(Magma) [4^n - 2^n + 1: n in [0..40]]; // Vincenzo Librandi, Apr 25 2011
(PARI) a(n)=4^n-2^n+1 \\ Charles R Greathouse IV, Jul 02 2013

Crossrefs

Cf. A000079, A002061.

Sequence in context: A095934 A151220 A151221 * A049086 A010921 A275634

Adjacent sequences: A020512 A020513 A020514 * A020516 A020517 A020518

Keyword

nonn,easy

Author

Simon Plouffe

Status

approved