A146078 - OEIS

%I #63 Oct 03 2024 23:25:32

%S 1,1,-8,-17,55,208,-287,-2159,424,19855,16039,-162656,-307007,1156897,

%T 3919960,-6492113,-41771753,16657264,392603041,242687665,-3290739704,

%U -5474928689,24141728647,73416086848,-143859470975,-804604252607

%N Expansion of 1/(1-x*(1-9*x)).

%C Row sums of Riordan array (1, x(1-9x)).

%H G. C. Greubel, <a href="/A146078/b146078.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 (1,-9).

%F a(n) = a(n-1) - 9*a(n-2), a(0)=1, a(1)=1.

%F a(n) = Sum_{k=0..n} A109466(n,k)*9^(n-k).

%F From _G. C. Greubel_, Jan 31 2016: (Start)

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

%F E.g.f.: exp(x/2)*(cos(sqrt(35)*x/2) + (1/sqrt(35))*sin(sqrt(35)*x/2)). (End)

%F a(n) = Product_{k=1..n} (1 + 6*cos(k*Pi/(n+1))). - _Peter Luschny_, Nov 28 2019

%F a(n) = 3^n * U(n, 1/6), where U(n, x) is the Chebyshev polynomial of the second kind. - _Federico Provvedi_, Mar 28 2022

%t LinearRecurrence[{1, -9}, {1, 1}, 100] (* _G. C. Greubel_, Jan 30 2016 *)

%o (Sage) [lucas_number1(n,1,9) for n in range(1, 27)] # _Zerinvary Lajos_, Apr 22 2009

%o (PARI) x='x+O('x^30); Vec(1/(1-x+9*x^2)) \\ _G. C. Greubel_, Jan 19 2018

%o (Magma) I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 9*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 19 2018

%Y Cf. A010892, A107920, A106852, A106853, A106854, A145934, A145976, A145978.

%K sign,easy

%O 0,3

%A _Philippe Deléham_, Oct 27 2008