A020522 a(n) = 4^n - 2^n.

0, 2, 12, 56, 240, 992, 4032, 16256, 65280, 261632, 1047552, 4192256, 16773120, 67100672, 268419072, 1073709056, 4294901760, 17179738112, 68719214592, 274877382656, 1099510579200, 4398044413952, 17592181850112, 70368735789056, 281474959933440

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

Comments

Number of walks of length 2*n+2 between any two diametrically opposite vertices of the cycle graph C_8. - Herbert Kociemba, Jul 02 2004

If we consider a(4*k+2), then 2^4 == 3^4 == 3 (mod 13); 2^(4*k+2) + 3^(4*k+2) == 3^k*(4+9) == 3*0 == 0 (mod 13). So a(4*k+2) can never be prime. - Jose Brox, Dec 27 2005

If k is odd, then a(n*k) is divisible by a(n), since: a(n*k) = (2^n)^k + (3^n)^k = (2^n + 3^n)*((2^n)^(k-1) - (2^n)^(k-2) (3^n) + - ... + (3^n)^(k-1)). So the only possible primes in the sequence are a(0) and a(2^n) for n>=1. I've checked that a(2^n) is composite for 3 <= n <= 15. As with Fermat primes, a probabilistic argument suggests that there are only finitely many primes in the sequence. - Dean Hickerson, Dec 27 2005

Let x,y,z be elements from some power set P(n), i.e., the power set of a set of n elements. Define a function f(x,y,z) in the following manner: f(x,y,z) = 1 if x is a subset of y and y is a subset of z and x does not equal z; f(x,y,z) = 0 if x is not a subset of y or y is not a subset of z or x equals z. Now sum f(x,y,z) for all x,y,z of P(n). This gives a(n). - Ross La Haye, Dec 26 2005

Number of monic (irreducible) polynomials of degree 1 over GF(2^n). - Max Alekseyev, Jan 13 2006

Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then a(n) = the number of (x,y) of B for which x does not equal y. - Ross La Haye, Jan 02 2008

For n>1: central terms of the triangle in A173787. - Reinhard Zumkeller, Feb 28 2010

Pronic numbers of the form: (2^n - 1)*2^n, which is the n-th Mersenne number times 2^n, see A000225 and A002378. - Fred Daniel Kline, Nov 30 2013

Indices where records of A037870 occur. - Philippe Beaudoin, Sep 03 2014

Half the total edge length for a minimum linear arrangement of a hypercube of dimension n. (See Harper's paper below for proof). - Eitan Frachtenberg, Apr 07 2017

Number of pairs in GF(2)^{n+1} whose dot product is 1. - Christopher Purcell, Dec 11 2021

Examples

			n=5: a(5) = 4^5 - 2^5 = 1024 - 32 = 992 -> '1111100000'.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..170
• M. Archibald, A. Blecher, A. Knopfmacher, and M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
• Tom Copeland, The Kervaire-Milnor formula
• John Elias, Illustration of initial terms: Twin 2^n hexagonal numbers
• L. H. Harper, Optimal Assignment of Numbers to Vertices, J. SIAM 12(1), p. 131--135, March 1964; alternative link.
• Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
• The Sixtieth William Lowell Putnam Mathematical Competition, Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.
• Index entries for linear recurrences with constant coefficients, signature (6,-8).

Crossrefs

Ratio of successive terms of A028365.

Cf. A000225, A060867, A161168, A006516, A059153, A065442.

Cf. A000079, A265736, A020988, A047849.

Cf. A000079, A000302.

Programs

Haskell

a020522 = (* 2) . a006516  -- Reinhard Zumkeller, Dec 15 2015

Magma

[4^n - 2^n: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

Maple

A020522:=n->4^n-2^n; seq(A020522(n), n=0..50); # Wesley Ivan Hurt, Nov 29 2013

Mathematica

Table[4^n - 2^n, {n, 40}] (* or *) LinearRecurrence[{6, -8}, {0, 2}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)

PARI

a(n)=4^n-2^n \\ Charles R Greathouse IV, Jan 30 2012

Python

def A020522(n): return (1<Chai Wah Wu, Mar 10 2025

Sage

[4^n - 2^n for n in range(0,23)] # Zerinvary Lajos, Jun 05 2009

Formula

From Herbert Kociemba, Jul 02 2004: (Start)

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

a(n) = 6*a(n-1) - 8*a(n-2). (End)

E.g.f.: exp(4*x) - exp(2*x). - Mohammad K. Azarian, Jan 14 2009

From Reinhard Zumkeller, Feb 07 2006, Jaroslav Krizek, Aug 02 2009: (Start)

a(n) = A099393(n)-A000225(n+1) = A083420(n)-A099393(n).

In binary representation, n>0: n 1's followed by n 0's (A138147(n)).

A000120(a(n)) = n.

A023416(a(n)) = n.

A070939(a(n)) = 2*n.

2*a(n)+1 = A030101(A099393(n)). (End)

a(n) = A085812(n) - A001700(n). - John Molokach, Sep 28 2013

a(n) = 2*A006516(n) = A000079(n)*A000225(n) = A265736(A000225(n)). - Reinhard Zumkeller, Dec 15 2015

a(n) = (4^(n/2) - 4^(n/4))*(4^(n/2) + 4^(n/4)). - Bruno Berselli, Apr 09 2018

Sum_{n>0} 1/a(n) = E - 1, where E is the Erdős-Borwein constant (A065442). - Peter McNair, Dec 19 2022

a(n) = A000302(n) - A000079(n). - John Reimer Morales, Aug 04 2025