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URL: https://oeis.org/A013776

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A013776
a(n) = 2^(4*n+1).
12
2, 32, 512, 8192, 131072, 2097152, 33554432, 536870912, 8589934592, 137438953472, 2199023255552, 35184372088832, 562949953421312, 9007199254740992, 144115188075855872, 2305843009213693952
OFFSET
0,1
COMMENTS
a(n) ~ -Pi*E(2*n)/B(2*n), E(n) Euler number, B(n) Bernoulli number. - Peter Luschny, Oct 28 2012
Equivalently, powers of 2 with final digit 2. - Muniru A Asiru, Mar 15 2019
As phi(a(n)) = (2^n)^4 is a perfect biquadrate (where phi is the Euler totient A000010), this is a subsequence of A078164 and A307690. - Bernard Schott, Mar 28 2022
FORMULA
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 16*a(n-1), n > 0, a(0) = 2.
G.f.: 2/(1 - 16*x). (End)
From Peter Bala, Nov 29 2015: (Start)
a(n) = Sum_{k = 0..n} binomial(2*k,k)*binomial(4*n + 2 - 2*k, 2*n + 1 - k).
Bisection of A264960. (End)
a(n) = A000079(A016813(n)). - Michel Marcus, Nov 30 2015
a(n) = Sum_{k = 0..2*n} binomial(4*n + 2, 2*k + 1) = A004171(2*n). - Peter Bala, Nov 25 2016
E.g.f.: 2*exp(16*x). - G. C. Greubel, Jun 30 2019
From Bernard Schott, Apr 15 2022: (Start)
Sum_{n>=0} 1/a(n) = 8/15.
Sum_{n>=0} (-1)^n/a(n) = 8/17. (End)
EXAMPLE
G.f. = 2 + 32*x + 512*x^2 + 8192*x^3 + 131072*x^4 + 2097152*x^5 + ...
MAPLE
[2^(4*n+1)$n=0..20]; # Muniru A Asiru, Apr 10 2019
MATHEMATICA
2^(4*Range[0, 20]+1) (* G. C. Greubel, Mar 15 2019 *)
NestList[16#&, 2, 20] (* Harvey P. Dale, Jul 28 2019 *)
PROG
(Magma) [2^(4*n+1): n in [0..20]]; // Vincenzo Librandi, Jun 27 2011
(PARI) a(n)=2<<(4*n) \\ Charles R Greathouse IV, Apr 07 2012
(GAP) List([0..20], n->2^(4*n+1)); # Muniru A Asiru, Mar 15 2019
(SageMath) [2^(4*n+1) for n in (0..20)] # G. C. Greubel, Mar 15 2019
CROSSREFS
Subsequence of A307690.
Intersection of A000079 and A078164.
Sequence in context: A163952 A246213 A022028 * A174491 A022019 A010045
KEYWORD
nonn,easy
EXTENSIONS
Wrong comment deleted by Kevin Ryde, Apr 16 2022
STATUS
approved