A013776 a(n) = 2^(4*n+1).

2, 32, 512, 8192, 131072, 2097152, 33554432, 536870912, 8589934592, 137438953472, 2199023255552, 35184372088832, 562949953421312, 9007199254740992, 144115188075855872, 2305843009213693952

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Examples

			G.f. = 2 + 32*x + 512*x^2 + 8192*x^3 + 131072*x^4 + 2097152*x^5 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Tanya Khovanova, Recursive Sequences
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (16).

Crossrefs

Subsequence of A307690.

Cf. A000010, A004171, A016813, A264960, A307690.

Intersection of A000079 and A078164.

Programs

GAP

List([0..20],n->2^(4*n+1)); # Muniru A Asiru, Mar 15 2019

Magma

[2^(4*n+1): n in [0..20]]; // Vincenzo Librandi, Jun 27 2011

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 *)

PARI

a(n)=2<<(4*n) \\ Charles R Greathouse IV, Apr 07 2012

Sage

[2^(4*n+1) for n in (0..20)] # G. C. Greubel, Mar 15 2019

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)

Extensions

Wrong comment deleted by Kevin Ryde, Apr 16 2022