|
%I #44 Sep 04 2023 16:40:37
%S 6,13,27,55,111,223,447,895,1791,3583,7167,14335,28671,57343,114687,
%T 229375,458751,917503,1835007,3670015,7340031,14680063,29360127,
%U 58720255,117440511,234881023,469762047,939524095,1879048191,3758096383
%N a(n) = 7*2^n - 1.
%C a(n) = A164874(n+2,2); subsequence of A030130. - _Reinhard Zumkeller_, Aug 29 2009
%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^n*charpoly(A,-5). - _Milan Janjic_, Jan 27 2010
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F a(n+1) = 2*a(n) + 1.
%F G.f.: (6-5*x)/((1-x)*(1-2*x)) - _Jaume Oliver Lafont_, Sep 14 2009
%F a(n-1)^2 + a(n) = (7*2^(n-1))^2. - _Vincenzo Librandi_, Aug 08 2010
%F a(n) = (A052940(n+1) + A000225(n+3))/2. - _Gennady Eremin_, Aug 31 2023
%t a=6; lst={a}; k=7; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 16 2008 *)
%t 7*2^Range[0,30]-1 (* _Harvey P. Dale_, May 09 2018 *)
%o (PARI) a(n)=7<<n-1 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Other sequences with recurrence a(n+1) = 2*a(n) + 1 are:
%Y a(0) = 2 gives A153893, a(0)=3 essentially A126646.
%Y a(0) = 4 gives A153894, a(0)=5 essentially A153893.
%Y a(0) = 7 gives essentially A000225.
%Y a(0) = 8 gives A052996 except for some initial terms,
%Y a(0) = 9 is essentially A153894.
%Y a(0) = 10 gives A086225,
%Y a(0) = 11 is essentially A153893.
%Y a(0) = 13 is essentially A086224.
%K nonn,easy
%O 0,1
%A _Marco Matosic_, Jul 27 2003
%E More terms from _David Wasserman_, Feb 22 2005
|
