A164550 - OEIS

%I #9 Sep 08 2022 08:45:47

%S 1,7,39,213,1161,6327,34479,187893,1023921,5579847,30407319,165704373,

%T 903004281,4920912567,26816462559,146136037653,796366838241,

%U 4339792916487,23649656984199,128878563155733,702322407981801

%N a(n) = 6*a(n-1) - 3*a(n-2) for n > 1; a(0) = 1, a(1) = 7.

%C Binomial transform of A164549.

%C Inverse binomial transform of A154235.

%H G. C. Greubel, <a href="/A164550/b164550.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 (6,-3).

%F a(n) = ((3+2*sqrt(6))*(3+sqrt(6))^n + (3-2*sqrt(6))*(3-sqrt(6))^n)/6.

%F G.f.: (1+x)/(1-6*x+3*x^2).

%F a(n) = 3^((n-1)/2)*(sqrt(3)*ChebyshevU(n, sqrt(3)) + ChebyshevU(n-1, sqrt(3))). - _G. C. Greubel_, Jul 16 2021

%t LinearRecurrence[{6,-3}, {1,7}, 31] (* _G. C. Greubel_, Jul 16 2021 *)

%o (Magma) [ n le 2 select 6*n-5 else 6*Self(n-1)-3*Self(n-2): n in [1..21] ];

%o (Sage) [3^((n-1)/2)*(sqrt(3)*chebyshev_U(n, sqrt(3)) + chebyshev_U(n-1, sqrt(3))) for n in (0..30)] # _G. C. Greubel_, Jul 16 2021

%Y Cf. A154235, A164549.

%K nonn

%O 0,2

%A _Klaus Brockhaus_, Aug 15 2009