A024038 - OEIS

%I #35 Aug 20 2023 10:50:17

%S 1,3,12,55,240,999,4060,16335,65472,262063,1048476,4194183,16777072,

%T 67108695,268435260,1073741599,4294967040,17179868895,68719476412,

%U 274877906583,1099511627376,4398046510663,17592186043932

%N a(n) = 4^n - n^2.

%H Vincenzo Librandi, <a href="/A024038/b024038.txt">Table of n, a(n) for n = 0..500</a>

%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>. [Cached copy]

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-15,13,-4).

%F a(n) = A000325(n)*A006127(n). - _Reinhard Zumkeller_, Apr 10 2010

%F G.f.: (1 - 4*x + 6*x^2 + 3*x^3)/((1 - x)^3*(1 - 4*x)). - _Colin Barker_, May 29 2012

%F E.g.f.: exp(4*x) - x*(1 + x)*exp(x). - _G. C. Greubel_, Aug 18 2023

%t Table[4^n-n^2,{n,0,30}] (* or *) LinearRecurrence[{7,-15,13,-4},{1,3,12,55},30] (* _Harvey P. Dale_, Sep 14 2013 *)

%o (Magma) [ 4^n-n^2: n in [0..30] ]; // _Vincenzo Librandi_, Dec 25 2010

%o (SageMath) [4^n-n^2 for n in range(31)] # _G. C. Greubel_, Aug 18 2023

%Y Cf. A000290, A000302, A000325, A006127.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_