This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379825 #17 Jan 05 2025 11:01:43
%S A379825 0,1,6,24,72,144,0,-1728,-10368,-41472,-124416,-248832,0,2985984,
%T A379825 17915904,71663616,214990848,429981696,0,-5159780352,-30958682112,
%U A379825 -123834728448,-371504185344,-743008370688,0,8916100448256,53496602689536,213986410758144,641959232274432
%N A379825 a(n) = [x^n] x/(12*x^2 - 6*x + 1).
%H A379825 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12).
%F A379825 a(n) = n! * [x^n] exp(3*x)*sin(sqrt(3)*x)/sqrt(3).
%F A379825 a(n) = (w/6)*((3 - w)^n - (3 + w)^n) where w = sqrt(-3).
%F A379825 a(n) = 6*a(n - 1) - 12*a(n - 2) for n >= 2.
%F A379825 a(n) = 2^n*3^((n - 1)/2)*sin((Pi*n)/6).
%F A379825 a(n) = 2^(n-1)*A057083(n-1) = 2^(n-1)*3^((n-1)/2)*ChebyshevU(n-1, sqrt(3)/2) for n >= 1.
%p A379825 w := sqrt(-3): a := n -> (w/6)*((3 - w)^n - (3 + w)^n):
%p A379825 seq(simplify(a(n)), n = 0..28);
%p A379825 # Alternative:
%p A379825 a := proc(n) option remember; if n < 2 then n else 6*(a(n - 1) - 2*a(n - 2)) fi end:
%p A379825 seq(a(n), n = 0..28);
%t A379825 LinearRecurrence[{6,-12},{0,1},29] (* _James C. McMahon_, Jan 05 2025 *)
%Y A379825 Cf. A057083, A000748, A190965.
%K A379825 sign,easy
%O A379825 0,3
%A A379825 _Peter Luschny_, Jan 04 2025