A083421 -id:A083421 - cp's OEIS frontend

A083420 a(n) = 2*4^n - 1.

1, 7, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311

Offset: 0

Paul Barry, Apr 29 2003

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.
Eric Weisstein's World of Mathematics, Rule 220
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A083421, A000668 (primes in this sequence), A004171, A000244.
Cf. A001676, A014551, A168604, A213243, A213246, A213247, A213248, A279872.
Cf. A000302.

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

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

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 *)

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

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

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

A140725 Inverse binomial transform of (0 followed by A037481).

Original entry on oeis.org

0, 1, 4, 10, 34, 94, 298, 862, 2650, 7822, 23722, 70654, 212986, 636910, 1914826, 5736286, 17225242, 51642958, 154994410, 464852158, 1394818618, 4183931566, 12552843274, 37656432670, 112973492314, 338912088334, 1016753042218

Offset: 0

Paul Curtz, Jul 12 2008

nonn

From Sean A. Irvine, Jun 07 2025: (Start)
For n>=1, the number of walks of length n-1 starting at vertex 1 (or, by symmetry, vertex 4) in the graph K_{1,1,3}:
    1---2
   /|\ /
  0 | X
   \|/ \
    4---3. (End)

Sean A. Irvine, Walks on Graphs.

Cf. A083421 (bin. transform of (0 followed by A037481)).

Join[{0},LinearRecurrence[{1,6},{1,4},26]] (* or *) a[0]=0;a[n_]:= ((-2)^n+4*3^n)/10;Array[a,27,0] (* James C. McMahon, Jul 13 2025 *)

a(n)= (-1)^n*A091003(n), n>0.
a(n+1)-3*a(n) = (-1)^(n+1)*A000079(n-1), n>0.
|a(n+1)-3*a(n)| = A011782(n).
From R. J. Mathar, Jul 14 2008: (Start)
O.g.f.: (1+3*x)*x / ((1+2*x)*(1-3*x)).
a(n) = ((-2)^n+4*3^n)/10, n>0. (End)
a(n) = a(n-1)+6*a(n-2) for n>2, a(0)=0, a(1)=1, a(2)=4. - Philippe Deléham, Nov 17 2013
a(n) + a(n+1) = A140796(n). - Philippe Deléham, Nov 17 2013
a(n+1) = sum_{k=0..n} A108561(n,k)*(-3)^k. - Philippe Deléham, Nov 17 2013

Edited and extended by R. J. Mathar, Jul 14 2008

cp's OEIS Frontend

A083420 a(n) = 2*4^n - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Formula