%I #31 Jan 03 2024 08:45:04

%S 1,9,90,909,9189,92898,939177,9494865,95990922,970446357,9810991629,

%T 99186890706,1002756873105,10137643587801,102489267607866,

%U 1036143151396317,10475171256888693,105901595463208770,1070641008783298233,10823936737447401633,109427535032871733866,1106287454724562457829,11184314186674341431325

%N a(n) = 11*a(n-1) - 9*a(n-2) with a(0)=1, a(1)=9.

%H G. C. Greubel, <a href="/A147841/b147841.txt">Table of n, a(n) for n = 0..990</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-9).

%F a(n) = Sum_{k=0..n} A147703(n,k)*8^k.

%F G.f.: (1-2*x)/(1 -11*x +9*x^2).

%F a(n) = 9*A333344(n-1) = A190872(n+1) - 2*A190872(n) = A333344(n) - A190872(n). - _Kevin Ryde_, Apr 11 2020

%F a(n) = 3^n*(ChebyshevU(n, 11/6) - (2/3)*ChebyshevU(n-1, 11/6)). - _G. C. Greubel_, May 28 2020

%F E.g.f.: exp(11*x/2)*(85*cosh(sqrt(85)*x/2) + 7*sqrt(85)*sinh(sqrt(85)*x/2))/85. - _Stefano Spezia_, Mar 02 2023

%p A147841:= n-> simplify( 3^n*(ChebyshevU(n, 11/6) - (2/3)*ChebyshevU(n-1, 11/6)) ):

%p seq(A147841(n), n=0..25); # _G. C. Greubel_, May 28 2020

%t Table[3^n*(ChebyshevU[n, 11/6] - (2/3)*ChebyshevU[n-1, 11/6]), {n,0,25}] (* _G. C. Greubel_, May 28 2020 *)

%t LinearRecurrence[{11,-9},{1,9},30] (* _Harvey P. Dale_, Feb 28 2023 *)

%o (PARI) a(n) = polcoeff(lift(('x-2)*Mod('x,'x^2-11*'x+9)^n), 1); \\ _Kevin Ryde_, Apr 11 2020

%Y Cf. A147703, A190872, A333344, A333345 (growth power).

%K nonn,easy

%O 0,2

%A _Philippe Deléham_, Nov 14 2008

%E Entries corrected by _Paolo P. Lava_, Nov 18 2008

%E Terms a(18) onward added by _G. C. Greubel_, May 28 2020