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A083420
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a(n) = 2*4^n - 1.
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45
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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
Up to at least a(7) for n > 0, a(n) is the odd factor of the salient numbers A001676(3+4n) when factored into the product of an even and odd number. 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
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..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) = Sum_{i = 0..n} C(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
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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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2 * 4^Range[0, 31] - 1 (* Alonso del Arte, Oct 17 2017 *)
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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.
Cf. A001676, A014551, A168604, A213243, A213246, A213247, A213248, A279872.
Sequence in context: A169785 A255282 A303449 * A277002 A282898 A036282
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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