OFFSET
0,1
COMMENTS
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).
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
MAPLE
MATHEMATICA
a[n_]:=4^n+2^n; Array[a, 24] (* Frank M Jackson, Dec 28 2017 *)
PROG
(Sage) [2^n + 4^n for n in range(0, 25)]
(Sage) [sigma(4, n)-1for n in range(0, 25)]
(Magma) [ 2^n+4^n: n in [0..25] ];
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Zerinvary Lajos, Jun 04 2009
STATUS
approved