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A083420 a(n) = 2*4^n-1. 34
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Rule 220

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

FORMULA

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

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). Also a(n) = (-4^n/2)*B(2*n, 1/2)/B(2*n). - Benoit Cloitre, Jun 18 2004

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) = A004171(n) - 1. - Irina Gerasimova, Jan 21 2014

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

MAPLE

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

MATHEMATICA

Table[ChebyshevT[2, 2^n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)

CoefficientList[Series[(1 + 2 x)/((1 - x) (1 - 4 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 09 2014 *)

PROG

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

CROSSREFS

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

Sequence in context: A002147 A169785 A255282 * A036282 A033474 A001896

Adjacent sequences:  A083417 A083418 A083419 * A083421 A083422 A083423

KEYWORD

nonn,easy

AUTHOR

Paul Barry, Apr 29 2003

STATUS

approved

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Last modified August 28 03:16 EDT 2015. Contains 261112 sequences.