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%I #154 Jan 03 2024 06:48:47
%S 1,7,31,127,511,2047,8191,32767,131071,524287,2097151,8388607,
%T 33554431,134217727,536870911,2147483647,8589934591,34359738367,
%U 137438953471,549755813887,2199023255551,8796093022207,35184372088831,140737488355327,562949953421311
%N a(n) = 2*4^n - 1.
%C Sum of divisors of 4^n. - _Paul Barry_, Oct 13 2005
%C Subsequence of A000069; A132680(a(n)) = A005408(n). - _Reinhard Zumkeller_, Aug 26 2007
%C 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
%C 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
%C 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
%C Largest odd factors of A147590. - _César Aguilera_, Jan 07 2020
%H Vincenzo Librandi, <a href="/A083420/b083420.txt">Table of n, a(n) for n = 0..1000</a>
%H Roudy El Haddad, <a href="https://arxiv.org/abs/2101.09089">Recurrent Sums and Partition Identities</a>, arXiv:2101.09089 [math.NT], 2021.
%H Roudy El Haddad, <a href="https://doi.org/10.7546/nntdm.2022.28.2.167-199">A generalization of multiple zeta value. Part 1: Recurrent sums</a>. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
%H A. J. Macfarlane, <a href="http://www.damtp.cam.ac.uk/user/ajm/Papers2016/GFsForCAsOfEvenRuleNo.ps">Generating functions for integer sequences defined by the evolution of cellular automata...</a>, Fig 11.
%H Robert Schneider, <a href="https://doi.org/10.1007/s40993-016-0039-5">Partition zeta functions</a>, Research in Number Theory, 2(1):9, 2016.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rule220.html">Rule 220</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).
%F G.f.: (1+2*x)/((1-x)*(1-4*x)).
%F E.g.f.: 2*exp(4*x)-exp(x).
%F With a leading zero, this is a(n) = (4^n - 2 + 0^n)/2, the binomial transform of A080925. - _Paul Barry_, May 19 2003
%F From _Benoit Cloitre_, Jun 18 2004: (Start)
%F 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.
%F a(n) = 5*a(n-1) - 4*a(n-2).
%F a(n) = (-4^n/2)*B(2*n, 1/2)/B(2*n). (End)
%F a(n) = A099393(n) + A020522(n) = A000302(n) + A024036(n). - _Reinhard Zumkeller_, Feb 07 2006
%F a(n) = Stirling2(2*(n+1), 2). - _Zerinvary Lajos_, Dec 06 2006
%F a(n) = 4*a(n-1) + 3 with n > 0, a(0) = 1. - _Vincenzo Librandi_, Dec 30 2010
%F a(n) = A001576(n+1) - 2*A001576(n). - _Brad Clardy_, Mar 26 2011
%F a(n) = 6*A002450(n) + 1. - _Roderick MacPhee_, Jul 06 2012
%F a(n) = A000203(A000302(n)). - _Michel Marcus_, Jan 20 2014
%F a(n) = Sum_{i = 0..n} binomial(2n+2, 2i). - _Wesley Ivan Hurt_, Mar 14 2015
%F a(n) = (1/4^n) * Sum_{k = 0..n} binomial(2*n+1,2*k)*9^k. - _Peter Bala_, Feb 06 2019
%F a(n) = A147590(n)/A000079(n). - _César Aguilera_, Jan 07 2020
%F 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
%p seq(2*4^n-1, n = 0..22); # _Peter Luschny_, Aug 17 2011
%t 2 * 4^Range[0, 31] - 1 (* _Alonso del Arte_, Oct 17 2017 *)
%o (Magma) [2*4^n-1 : n in [0..30]]; // _Wesley Ivan Hurt_, Mar 14 2015
%o (PARI) a(n)=2*4^n-1 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Haskell)
%o a083420 = subtract 1 . (* 2) . (4 ^) -- _Reinhard Zumkeller_, Dec 22 2015
%Y Cf. A083421, A000668 (primes in this sequence), A004171, A000244.
%Y Cf. A001676, A014551, A168604, A213243, A213246, A213247, A213248, A279872.
%Y Cf. A000302.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Apr 29 2003
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