%I #40 Aug 21 2022 04:40:59

%S 1,1,5,5,9,9,13,13,17,17,21,21,25,25,29,29,33,33,37,37,41,41,45,45,49,

%T 49,53,53,57,57,61,61,65,65,69,69,73,73,77,77,81,81,85,85,89,89,93,93,

%U 97,97,101,101,105,105,109,109,113,113,117,117,121,121,125,125,129,129

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

%H Vincenzo Librandi, <a href="/A168277/b168277.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F a(n) = 4*n - a(n-1) - 6, with n>1, a(1)=1.

%F a(n) = A163980(n-1), n>1. - _R. J. Mathar_, Nov 25 2009

%F G.f.: x*(1 + 3*x^2)/( (1+x)*(x-1)^2 ). - _R. J. Mathar_, Jul 15 2013

%F a(n) = A168276(n) - 1. - _Vincenzo Librandi_, Sep 17 2013

%F a(n) = a(n-1) +a(n-2) -a(n-3). - _Vincenzo Librandi_, Sep 17 2013

%F E.g.f.: (-1 + 3*exp(x) + 2*(x - 1)*exp(2*x))*exp(-x). - _G. C. Greubel_, Jul 16 2016

%F Sum_{n>=1} 1/a(n)^2 = Pi^2/8 + G, where G is Catalan's constant (A006752). - _Amiram Eldar_, Aug 21 2022

%t CoefficientList[Series[(1 + 3 x^2) / ((1 + x) (x - 1)^2), {x, 0, 80}], x] (* _Vincenzo Librandi_, Sep 16 2013 *)

%t Table[2 n - (-1)^n - 2, {n, 70}] (* _Bruno Berselli_, Sep 17 2013 *)

%t LinearRecurrence[{1,1,-1},{1,1,5},70] (* _Harvey P. Dale_, Aug 25 2015 *)

%o (Magma) [n eq 1 select 1 else 4*n-Self(n-1)-6: n in [1..70]]; // _Vincenzo Librandi_, Sep 16 2013

%o (PARI) a(n)=2*n-(-1)^n-2 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A016813, A163980, A168276.

%Y Cf. A006752, A111003 (Pi^2/8).

%K nonn,easy

%O 1,3

%A _Vincenzo Librandi_, Nov 22 2009

%E New definition from _Bruno Berselli_, Sep 17 2013