A321358 - OEIS

%I #21 Nov 11 2018 08:09:42

%S 3,5,13,45,173,685,2733,10925,43693,174765,699053,2796205,11184813,

%T 44739245,178956973,715827885,2863311533,11453246125,45812984493,

%U 183251937965,733007751853,2932031007405,11728124029613,46912496118445,187649984473773,750599937895085,3002399751580333

%N a(n) = (2*4^n + 7)/3.

%C Difference table:

%C 3,  5, 13,  45,  173,  685,  2733, ...   (this sequence)

%C 2,  8, 32, 128,  512, 2048,  8192, ...    A004171

%C 6, 24, 96, 384, 1536, 6144, 24576, ...    A002023

%H Colin Barker, <a href="/A321358/b321358.txt">Table of n, a(n) for n = 0..1000</a>

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

%F O.g.f.: (3 - 10*x) / ((1 - x)*(1 - 4*x)). - _Colin Barker_, Nov 10 2018

%F E.g.f.: (1/3)*(7*exp(x) + 2*exp(4*x)). - _Stefano Spezia_, Nov 10 2018

%F a(n) = 5*a(n-1) - 4*a(n-2), a(0) = 3, a(1) = 5.

%F a(n) = 4*a(n-1) - 7, a(0) = 3.

%F a(n) = (2/3)*(4^n-1)/3 + 3.

%F a(n) = A171382(2*n) = A155980(2*n+2).

%F a(n) = A193579(n)/3.

%F a(n) = A007583(n) + 2 = A001045(2*n+1) + 2.

%t a[n_]:= (2*4^n + 7)/3; Array[a, 20, 0] (* or *)

%t CoefficientList[Series[1/3 (7 E^x + 2 E^(4 x)), {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* _Stefano Spezia_, Nov 10 2018 *)

%o (PARI) a(n) = (2*4^n + 7)/3; \\ _Michel Marcus_, Nov 08 2018

%o (PARI) Vec((3 - 10*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ _Colin Barker_, Nov 10 2018

%Y Cf. A010701, A010727, A020988, A083594, A002023, A004171, A155980, A171382, A193579.

%K nonn,easy

%O 0,1

%A _Paul Curtz_, Nov 07 2018

%E More terms from _Michel Marcus_, Nov 08 2018