0,2

Sum of divisors of 4^n. - Paul Barry, Oct 13 2005

Subsequence of A000069; A132680(a(n)) = A005408(n). - Reinhard Zumkeller, Aug 26 2007

If x = a(n), y = A000079(n+1) and z = A087289(n), then x^2 + 2*y^2 = z^2. - Vincenzo Librandi, Jun 09 2014

It seems that a(n) divides A001676(3+4n). Several other entries apparently have this sequence embedded in them, e.g., A014551, A168604, A213243, A213246-8, and A279872. - Tom Copeland, Dec 27 2016

To elaborate on Librandi's comment from 2014: all these numbers, even if prime in Z, are sure not to be prime in Z[sqrt(2)], since a(n) can at least be factored as ((2^(2n + 1) - 1) - (2^(2n) - 1)*sqrt(2))((2^(2n + 1) - 1) + (2^(2n) - 1)*sqrt(2)). For example, 7 = (3 - sqrt(2))(3 + sqrt(2)), 31 = (7 - 3*sqrt(2))(7 + 3*sqrt(2)), 127 = (15 - 7*sqrt(2))(15 + 7*sqrt(2)). - Alonso del Arte, Oct 17 2017

Largest odd factors of A147590. - César Aguilera, Jan 07 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Roudy El Haddad, Recurrent Sums and Partition Identities, arXiv:2101.09089 [math.NT], 2021.

Roudy El Haddad, A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.

A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata with even rule numbers, Fig 11.

Robert Schneider, Partition zeta functions, Research in Number Theory, 2(1):9, 2016.

Samuel S. Wagstaff, Jr., Two Mersenne Prime Conjectures, J. Int. Seq. 28 (2025), 25.7.2. See p. 1.

Eric Weisstein's World of Mathematics, Rule 220

Index entries for linear recurrences with constant coefficients, signature (5,-4).

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

E.g.f.: 2*exp(4*x)-exp(x).

With a leading zero, this is a(n) = (4^n - 2 + 0^n)/2, the binomial transform of A080925. - Paul Barry, May 19 2003

From Benoit Cloitre, Jun 18 2004: (Start)

a(n) = (-16^n/2)*B(2n, 1/4)/B(2n) where B(n, x) is the n-th Bernoulli polynomial and B(k) = B(k, 0) is the k-th Bernoulli number.

a(n) = 5*a(n-1) - 4*a(n-2).

a(n) = (-4^n/2)*B(2*n, 1/2)/B(2*n). (End)

a(n) = A099393(n) + A020522(n) = A000302(n) + A024036(n). - Reinhard Zumkeller, Feb 07 2006

a(n) = Stirling2(2*(n+1), 2). - Zerinvary Lajos, Dec 06 2006

a(n) = 4*a(n-1) + 3 with n > 0, a(0) = 1. - Vincenzo Librandi, Dec 30 2010

a(n) = A001576(n+1) - 2*A001576(n). - Brad Clardy, Mar 26 2011

a(n) = 6*A002450(n) + 1. - Roderick MacPhee, Jul 06 2012

a(n) = A000203(A000302(n)). - Michel Marcus, Jan 20 2014

a(n) = Sum_{i = 0..n} binomial(2n+2, 2i). - Wesley Ivan Hurt, Mar 14 2015

a(n) = (1/4^n) * Sum_{k = 0..n} binomial(2*n+1,2*k)*9^k. - Peter Bala, Feb 06 2019

a(n) = A147590(n)/A000079(n). - César Aguilera, Jan 07 2020

a(n) = numerator(zeta_star({2}_(n + 1))/zeta(2*n + 2)) where zeta_star is the multiple zeta star values and ({2}_n) represents (2, ..., 2) where the multiplicity of 2 is n. - Roudy El Haddad, Feb 22 2022

seq(2*4^n-1, n = 0..22); # Peter Luschny, Aug 17 2011

2 * 4^Range[0, 31] - 1 (* Alonso del Arte, Oct 17 2017 *)

(Magma) [2*4^n-1 : n in [0..30]]; // Wesley Ivan Hurt, Mar 14 2015

(PARI) a(n)=2*4^n-1 \\ Charles R Greathouse IV, Sep 24 2015

(Haskell)

a083420 = subtract 1 . (* 2) . (4 ^) -- Reinhard Zumkeller, Dec 22 2015

(Python)

def A083420(n): return (1<<(n<<1|1))-1 # Chai Wah Wu, Mar 10 2025

Cf. A083421, A000668 (primes in this sequence), A004171, A000244.

Cf. A001676, A014551, A168604, A213243, A213246, A213247, A213248, A279872.

Cf. A000302.

Sequence in context: A169785 A255282 A303449 * A277002 A282898 A036282

Adjacent sequences: A083417 A083418 A083419 * A083421 A083422 A083423

nonn,easy

Paul Barry, Apr 29 2003

approved