A133631 - OEIS

%I #19 Jan 02 2024 08:55:19

%S 1,2,-2,-10,-2,38,46,-106,-290,134,1294,758,-4418,-7450,10222,40022,

%T -866,-160954,-157490,486326,1116286,-829018,-5294162,-1978090,

%U 19198558,27110918,-49683314,-158126986,40606270,673114214,510689134,-2181767722,-4224524258

%N a(n) = a(n-1) - 4*a(n-2), a(0)=1, a(1)=2.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,-4).

%F G.f.: (1+x)/(1-x+4*x^2).

%F a(n) = Sum_{k=0..n} A133607(n,k)*2^k. - _Philippe Deléham_, Dec 29 2007

%F a(n) = 2^n*U(n, 1/4) + 2^(n-1)*U(n-1, 1/4) = A106853(n) + A106853(n-1) where U is the Chebyshev polynomial of the 2nd kind. - _Michael Somos_, Oct 24 2023

%e G.f. = 1 + 2*x - 2*x^2 - 10*x^3 - 2*x^4 + 38*x^5 + 46*x^6 - 106*x^7 + ... - _Michael Somos_, Oct 24 2023

%t a[ n_] := 2^n * ChebyshevU[n, 1/4] + 2^(n-1) * ChebyshevU[n-1, 1/4]; (* _Michael Somos_, Oct 24 2023 *)

%o (PARI) {a(n) = 2^n*polchebyshev(n, 2, 1/4) + 2^(n-1)*polchebyshev(n-1, 2, 1/4)}; /* _Michael Somos_, Oct 24 2023 */

%Y Cf. A106853, A133607.

%K easy,sign

%O 0,2

%A _Philippe Deléham_, Dec 28 2007