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A079598 a(n) = 2^(4n+1) - 2^(2n).

%I A079598 #40 Aug 11 2025 10:56:25
%S A079598 1,28,496,8128,130816,2096128,33550336,536854528,8589869056,
%T A079598 137438691328,2199022206976,35184367894528,562949936644096,
%U A079598 9007199187632128,144115187807420416,2305843008139952128
%N A079598 a(n) = 2^(4n+1) - 2^(2n).
%C A079598 A sequence rich in perfect numbers, since according to the Euclid-Euler theorem all even perfect numbers are of the form 2^(k-1)*(2^k - 1) and in this sequence k = 2*n + 1.
%C A079598 Let G be a sequence satisfying G(i) = 2*G(i-1) + 8*G(i-2) for arbitrary integers i and without regard to the initial values of G. Then a(n) = (G(i+4*n+2) - G(i)*8^(2*n+1))/(2*G(i+2*n+1)) as long as G(i+2*n+1) != 0. - _Klaus Purath_, Oct 22 2020
%C A079598 In the binary system, the elements of the sequence consist of a total of 4*n+1 bits starting with 2*n+1 ones followed by 2*n zeros. - _Martin Renner_, Mar 22 2022
%C A079598 Michael Stifel and Petrus Bungus erroneously claimed that for n > 0 all these numbers were perfect. See links: Bungus, Hassler, and Stifel. - _Stefano Spezia_, Jul 28 2024
%H A079598 Petrus Bungus, <a href="https://www.digitale-sammlungen.de/de/view/bsb10053307?page=0586">Numerorum Mysteria</a>, Taupinart, (1618). (In Latin). See p. 468.
%H A079598 Uwe Hassler, <a href="https://doi.org/10.56031/2693-9908.1052">Perfect Numbers</a>, Euleriana: 3(2), pp.176-185, (2023). See pp. 177-178.
%H A079598 Michael Stifel, <a href="https://www.e-rara.ch/zut/doi/10.3931/e-rara-10418">Arithmetica integra</a>, Joh. Petreius, (1544). (In Latin). See p. 11.
%H A079598 László Tóth, <a href="https://arxiv.org/abs/2508.04151">Evaluating zeta(s) At Odd Positive Integers Using Automatic Dirichlet Series</a>, arXiv:2508.04151 [math.NT], 2025.
%H A079598 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-64).
%F A079598 a(n+1) = 16*a(n) + 12*2^(2n).
%F A079598 a(n) = Sum_{k=1..2^n} (2*k-1)^3. - _Franz Vrabec_, Jun 24 2006
%F A079598 a(n) = Sum_{k=2*n..4*n} 2^k. - _Martin Renner_, Mar 22 2022
%F A079598 G.f.: ( 1+8*x ) / ( (16*x-1)*(4*x-1) ). - _R. J. Mathar_, Nov 29 2011
%F A079598 E.g.f.: exp(4*x)*(2*exp(12*x) - 1). - _Stefano Spezia_, Jul 28 2024
%p A079598 seq(sum((2*k-1)^3,k=1..2^n),n=0..15);
%p A079598 seq(sum(2^k,k=2*n..4*n),n=0..15);
%o A079598 (PARI) a(n)=2^(4*n+1)-4^n \\ _Charles R Greathouse IV_, Nov 29 2011
%Y A079598 Cf. A000396, A006516.
%K A079598 nonn,easy
%O A079598 0,2
%A A079598 _Marco Matosic_, Jan 28 2003