A155157 - OEIS

%I #17 Jan 03 2024 08:47:34

%S 1,9,99,1080,11790,128700,1404900,15336000,167409000,1827450000,

%T 19948590000,217760400000,2377089900000,25948503000000,

%U 283255929000000,3092044320000000,33753002490000000,368450468100000000

%N a(n) = 10*a(n-1) + 10*a(n-2), with a(0)=1, a(1)=9, a(2)=99.

%H G. C. Greubel, <a href="/A155157/b155157.txt">Table of n, a(n) for n = 0..950</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,10).

%F G.f.: (1-x-x^2)/(1-10*x-10*x^2).

%F From _G. C. Greubel_, Mar 20 2021: (Start)

%F a(n) = ([n=0] + 9*A057093(n))/10.

%F a(n) = (1/10)*([n=0] + 9*(-i*sqrt(10))^n*ChebyshevU(n, i*sqrt(10)/2)). (End)

%p 1,seq( simplify(9*(-I*sqrt(10))^n*ChebyshevU(n, I*sqrt(10)/2)/10), n=1..30); # _G. C. Greubel_, Mar 20 2021

%t LinearRecurrence[{10,10},{1,9,99},20] (* _Harvey P. Dale_, Jan 27 2016 *)

%o (Magma) [1]cat[n le 2 select 9*(10*n-9) else 10*(Self(n-1) + Self(n-2)): n in [1..30]]; // _G. C. Greubel_, Mar 20 2021

%o (Sage) [1]+[(9/10)*(-i*sqrt(10))^n*chebyshev_U(n, i*sqrt(10)/2) for n in (1..30)] # _G. C. Greubel_, Mar 20 2021

%Y Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: A155020 (m=2), A155116 (m=3), A155117 (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), A155144 (m=9), this sequence (m=10).

%Y Cf. A057093.

%K nonn

%O 0,2

%A _Philippe Deléham_, Jan 21 2009