%I #20 Mar 17 2023 14:53:31

%S 1,3,15,63,255,1023,4095,16383,65535,262143,1048575,4194303,16777215,

%T 67108863,268435455,1073741823,4294967295,17179869183,68719476735,

%U 274877906943,1099511627775,4398046511103,17592186044415,70368744177663

%N a(n) = 0^n + 4^n - 1.

%C A transform of 4^n under the matrix A103452.

%C The square of the cotangent of the arcsin of 1/(2^n). - Al Hakanson (hawkuu(AT)excite.com), Feb 23 2006

%H Vincenzo Librandi, <a href="/A103454/b103454.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).

%F G.f.: (1 - 2*x + 4*x^2)/((1-x)*(1-4*x));

%F a(n) = Sum_{k=0..n} A103452(n, k)*4^k;

%F a(n) = Sum_{k=0..n} (2*0^(n-k) - 1)*0^(k*(n-k))4^k.

%F a(n) = A024036(n), n > 0. - _R. J. Mathar_, Aug 30 2008

%F E.g.f.: 1 - exp(x) + exp(4*x). - _G. C. Greubel_, Jun 21 2021

%F a(n) = 5*a(n-1) - 4*a(n-2). - _Wesley Ivan Hurt_, Mar 17 2023

%t Table[Boole[n==0] +4^n -1, {n,0,40}] (* _G. C. Greubel_, Jun 21 2021 *)

%o (Magma) [0^n+4^n-1: n in [0..30]]; // _Vincenzo Librandi_, Jul 02 2011

%o (Sage) [1]+[4^n -1 for n in [1..40]] # _G. C. Greubel_, Jun 21 2021

%Y Cf. A024036, A103452.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Feb 06 2005