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a(n) = 2^(4*n + 1) - 1.
2

%I #59 Mar 12 2019 04:48:22

%S 1,31,511,8191,131071,2097151,33554431,536870911,8589934591,

%T 137438953471,2199023255551,35184372088831,562949953421311,

%U 9007199254740991,144115188075855871,2305843009213693951,36893488147419103231,590295810358705651711,9444732965739290427391

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

%H Jens Kruse Andersen, <a href="/A241888/b241888.txt">Table of n, a(n) for n = 0..100</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (17,-16).

%F a(n) = 2^(4*n + 1) - 1 = A000225(4*n + 1) = A013776(n) - 1 = 4*A000225(4*n - 1) + 3.

%F a(n) = 17*a(n-1) - 16*a(n-2). - _Colin Barker_, Aug 31 2014

%F G.f.: (14*x+1) / ((x-1)*(16*x-1)). - _Colin Barker_, Aug 31 2014

%p seq(coeff(series((14*x+1)/((x-1)*(16*x-1)),x,n+1), x, n), n = 0 .. 20); # _Muniru A Asiru_, Mar 12 2019

%t Table[2^(4n + 1) - 1, {n, 0, 29}]

%t CoefficientList[ Series[(14x + 1)/((x - 1) (16x - 1)), {x, 0, 18}], x] (* _Robert G. Wilson v_, Jan 28 2015 *)

%t LinearRecurrence[{17,-16},{1,31},30] (* _Harvey P. Dale_, Mar 13 2016 *)

%o (PARI) vector(40, n, 2^(4*n-3)-1) \\ _Derek Orr_, Aug 11 2014

%o (PARI) Vec((14*x+1)/((x-1)*(16*x-1)) + O(x^100)) \\ _Colin Barker_, Aug 31 2014

%o (GAP) List([0..20],n->2^(4*n+1)-1); # _Muniru A Asiru_, Mar 12 2019

%Y Cf. A000225, A004171, A013776.

%K nonn,easy

%O 0,2

%A _Wassan Letourneur_, Aug 09 2014