A161168 a(n) = 2^n + 4^n.

2, 6, 20, 72, 272, 1056, 4160, 16512, 65792, 262656, 1049600, 4196352, 16781312, 67117056, 268451840, 1073774592, 4295032832, 17180000256, 68719738880, 274878431232, 1099512676352, 4398048608256, 17592190238720, 70368752566272

Offset: 0

Author

Zerinvary Lajos, Jun 04 2009

Keywords

• nonn
• easy

Comments

Essentially a duplicate of A063376 and A028402.

a(n) written in base 2: a(0) = 10, a(n) for n >= 1: 110, 10100, 1001000, 100010000, ..., i.e., number 1, (n-1) times 0, number 1, n times 0 (see A163664). a(n) is a bisection of A005418. - Jaroslav Krizek, Aug 14 2009

Central terms of the triangle in A173786. - Reinhard Zumkeller, Feb 28 2010

For n > 0 let 2^(n+1) be the length of the even leg of a primitive Pythagorean triangle (PPT); then the odd leg is constrained to have a length of 4^n-1 and the hypotenuse to have a length of 4^n+1. The resulting triangle has a semiperimeter of 4^n + 2^n. - Frank M Jackson, Dec 28 2017

a(n) is also the number of distinct planar embeddings of the (2n+7)-triangular snake graph. - Eric W. Weisstein, May 21 2024

Links

• Iain Fox, Table of n, a(n) for n = 0..1660
• Eric Weisstein's World of Mathematics, Planar Embedding.
• Eric Weisstein's World of Mathematics, Triangular Snake Graph.
• Index entries for linear recurrences with constant coefficients, signature (6,-8).

Crossrefs

Cf. A005418, A007582, A020522, A028403, A063376, A163664, A173786.

Programs

Magma

[ 2^n+4^n: n in [0..25] ];

Maple

A161168:=n->2^n+4^n: seq(A161168(n), n=0..40); # Wesley Ivan Hurt, Jul 24 2017

Mathematica

a[n_]:=4^n+2^n; Array[a,24] (* Frank M Jackson, Dec 28 2017 *)

PARI

a(n)=2^n+4^n \\ Charles R Greathouse IV, Oct 07 2015

PARI

first(n) = Vec(2*(1 - 3*x)/((1 - 2*x)*(1 - 4*x)) + O(x^n)) \\ Iain Fox, Dec 28 2017

Sage

[2^n + 4^n for n in range(0,25)]

Sage

[sigma(4,n)-1for n in range(0,25)]

Formula

a(n) = 6*a(n-1) - 8*a(n-2); a(0)=2, a(1)=6. - Vincenzo Librandi, Dec 27 2010

G.f.: -2*(3*x-1) / ((2*x-1)*(4*x-1)). - Colin Barker, Mar 19 2013

E.g.f.: e^(2*x) + e^(4*x). - Iain Fox, Dec 28 2017

a(n) = 2*A007582(n). - R. J. Mathar, Feb 26 2018