|
%I #41 Dec 19 2022 10:15:53
%S 2,6,20,72,272,1056,4160,16512,65792,262656,1049600,4196352,16781312,
%T 67117056,268451840,1073774592,4295032832,17180000256,68719738880,
%U 274878431232,1099512676352,4398048608256,17592190238720,70368752566272
%N a(n) = 2^n + 4^n.
%C Essentially a duplicate of A063376 and A028402.
%C 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
%C Central terms of the triangle in A173786. - _Reinhard Zumkeller_, Feb 28 2010
%C 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
%H Iain Fox, <a href="/A161168/b161168.txt">Table of n, a(n) for n = 0..1660</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8).
%F a(n) = 6*a(n-1) - 8*a(n-2); a(0)=2, a(1)=6. - _Vincenzo Librandi_, Dec 27 2010
%F G.f.: -2*(3*x-1) / ((2*x-1)*(4*x-1)). - _Colin Barker_, Mar 19 2013
%F E.g.f.: e^(2*x) + e^(4*x). - _Iain Fox_, Dec 28 2017
%F a(n) = 2*A007582(n). - _R. J. Mathar_, Feb 26 2018
%p A161168:=n->2^n+4^n: seq(A161168(n), n=0..40); # _Wesley Ivan Hurt_, Jul 24 2017
%t a[n_]:=4^n+2^n; Array[a,24] (* _Frank M Jackson_, Dec 28 2017 *)
%o (Sage) [2^n + 4^n for n in range(0,25)]
%o (Sage) [sigma(4,n)-1for n in range(0,25)]
%o (Magma) [ 2^n+4^n: n in [0..25] ];
%o (PARI) a(n)=2^n+4^n \\ _Charles R Greathouse IV_, Oct 07 2015
%o (PARI) first(n) = Vec(2*(1 - 3*x)/((1 - 2*x)*(1 - 4*x)) + O(x^n)) \\ _Iain Fox_, Dec 28 2017
%Y Cf. A005418, A007582, A020522, A028403, A063376, A163664, A173786.
%K nonn,easy
%O 0,1
%A _Zerinvary Lajos_, Jun 04 2009
|
