A176758 - OEIS

%I #13 Sep 08 2022 08:45:53

%S 2,8,28,88,264,768,2192,6176,17248,47872,132288,364416,1001600,

%T 2748416,7532800,20627968,56452608,154423296,422276096,1154447360,

%U 3155544064,8624177152,23567831040,64400793600,175970803712,480810303488

%N a(n) = Sum_{k=0..floor((n-1)/2)} (3^k-1)*binomial(n, 2*k+1).

%H G. C. Greubel, <a href="/A176758/b176758.txt">Table of n, a(n) for n = 3..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-4).

%F From _R. J. Mathar_, Jan 29 2012: (Start)

%F a(n) = A002605(n) - 2^(n-1) = 2*A094309(n).

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

%t (* First program *)

%t a[n_, q_]:= Sum[(q^((m-1)/2) - 1)*Binomial[n, m], {m,1,n,2}];

%t Table[a[n, 3], {n, 3, 30}]

%t (* Second program *)

%t A002605[n_]:= (-I*Sqrt[2])^(n-1)*ChebyshevU[n-1, I/Sqrt[2]];

%t Table[(Boole[n==0] - 2^n)/2 + A002605[n], {n, 3, 30}] (* _G. C. Greubel_, Sep 17 2021 *)

%o (Magma) I:=[2,8,28]; [n le 3 select I[n] else 4*Self(n-1) - 2*Self(n-2) +4*Self(n-3): n in [1..31]]; // _G. C. Greubel_, Sep 17 2021

%o (Sage) [(-i*sqrt(2))^(n-1)*chebyshev_U(n-1, i/sqrt(2)) - 2^(n-1) for n in (3..30)] # _G. C. Greubel_, Sep 17 2021

%Y Cf. A002605, A094706.

%K nonn,easy

%O 3,1

%A _Roger L. Bagula_, Apr 25 2010