A127976 - OEIS

%I #38 May 25 2023 16:12:58

%S 1,1,5,9,25,53,125,273,609,1325,2885,6217,13353,28517,60685,128641,

%T 271857,572829,1203925,2524345,5281721,11029461,22991005,47846129,

%U 99420545,206297613,427508325,884842793,1829337929,3777980485

%N a(n) = ((6*n + 10)/27)*2^(n-1) + ((-1)^(n-1))*(6*n + 5)/27.

%C Equals A001045, [1, 1, 3, 5, 11, 21, 43, 85, ...] convolved with A078008, [1, 0, 2, 2, 6, 10, 22, ...]. - _Gary W. Adamson_, May 25 2009

%H Vincenzo Librandi, <a href="/A127976/b127976.txt">Table of n, a(n) for n = 1..1000</a>

%H W. Bosma, <a href="http://www.numdam.org/item?id=JTNB_2001__13_1_27_0">Signed bits and fast exponentiation</a>, J. Th. des Nombres de Bordeaux Vol.13, Fasc. 1, 2001.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-4,-4).

%F From _Joerg Arndt_, May 27 2011: (Start)

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

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

%F E.g.f.: 2*exp(x/2)*(6*x*cosh(3*x/2) + 5*sinh(3*x/2))/27. - _Stefano Spezia_, May 25 2023

%t Table[((6n + 10)/27)2^(n - 1) + ((-1)^(n - 1) )(6n + 5)/27, {n, 1, 100}]

%o (Magma) [((6*n + 10)/27)*2^(n - 1) + ((-1)^(n - 1))*(6*n + 5)/27: n in [1..40]]; // _Vincenzo Librandi_, May 26 2011

%o (PARI) x='x+O('x^30); Vec(x*(1-x)/((1+x)^2*(1-2*x)^2)) \\ _G. C. Greubel_, May 07 2018

%o (PARI) a(n) = ((6*n + 10)/27)*2^(n - 1) + (-1)^(n - 1)*(6*n + 5)/27; \\ _Michel Marcus_, May 09 2018

%Y Cf. A001045, A078008.

%K nonn,easy

%O 1,3

%A _Artur Jasinski_, Feb 09 2007

%E Name edited by _Altug Alkan_, May 09 2018