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A083420
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a(n) = 2*4^n - 1.
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36
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata..., Fig 11.
Eric Weisstein's World of Mathematics, Rule 220
Index entries for linear recurrences with constant coefficients, signature (5,-4).
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FORMULA
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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
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MAPLE
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seq(2*4^n-1, n = 0..22); # Peter Luschny, Aug 17 2011
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A083421, A000668 (primes in this sequence), A004171, A000244
Sequence in context: A002147 A169785 A255282 * A277002 A036282 A033474
Adjacent sequences: A083417 A083418 A083419 * A083421 A083422 A083423
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KEYWORD
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nonn,easy
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AUTHOR
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Paul Barry, Apr 29 2003
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STATUS
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approved
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