A084173 - OEIS

%I #19 Sep 08 2022 08:45:11

%S 1,3,5,13,27,59,121,249,503,1015,2037,4085,8179,16371,32753,65521,

%T 131055,262127,524269,1048557,2097131,4194283,8388585,16777193,

%U 33554407,67108839,134217701,268435429,536870883,1073741795,2147483617

%N a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4).

%C Original name was: A sum of generalized Jacobsthal numbers.

%H Vincenzo Librandi, <a href="/A084173/b084173.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,2).

%F a(n) = 2^(n+1) - (2*n+1)/2 - (-1)^n/2.

%F G.f.: (4*x^3 - 3*x^2 + 1)/(-2*x^4 + 3*x^3 + x^2 - 3*x + 1).

%F a(n) = A084172(n) + A083579(n).

%t a[n_]:=2^(n+1) - (2*n+1)/2 - (-1)^n/2; Array[a, 50, 0] (* or *)

%t CoefficientList[Series[(4*x^3 - 3*x^2 + 1)/(-2*x^4 + 3*x^3 + x^2 - 3*x + 1), {x, 0, 50}], x] (* or *)

%t LinearRecurrence[{3, -1, -3, 2}, {1, 3, 5, 13}, 50] (* _Stefano Spezia_, Oct 08 2018 *)

%o (Magma) [2^(n+1)-(2*n+1)/2-(-1)^n/2: n in [0..35]]; // _Vincenzo Librandi_, Jul 05 2011

%o (PARI) x='x+O('x^33); Vec((4*x^3-3*x^2+1)/(-2*x^4+3*x^3+x^2-3*x+1)) \\ _Altug Alkan_, Oct 08 2018

%K easy,nonn

%O 0,2

%A _Paul Barry_, May 18 2003