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 A094423 #14 Jan 11 2024 07:16:08
%S A094423 1,4,1,144,361,484,19321,28224,128881,2427364,1745041,26501904,
%T A094423 285643801,64995844,4675961161,31354493184,149793121,741117817924,
%U A094423 3178942795681,545370434064,107989070784841,292105630845604
%N A094423 a(n) = A045873(n)^2.
%C A094423 The g.f. is an example of a rational function with nonnegative integer coefficients that is not N-rational.
%H A094423 G. C. Greubel, <a href="/A094423/b094423.txt">Table of n, a(n) for n = 1..1000</a>
%H A094423 Ira M. Gessel, <a href="http://people.brandeis.edu/~gessel/homepage/slides/nonneg.pdf">Rational Functions With Nonnegative Integer Coefficients</a>, slides, 50th Séminaire Lotharingien de Combinatoire, 2003.
%H A094423 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,5,125).
%F A094423 G.f.: x*(1+5*x)/(1+x-5*x^2-125*x^3).
%F A094423 a(n) = A250102(n)/16.
%F A094423 a(n) = (1/16)*( 2*5^n - (1+2*i)^(2*n) - (1-2*i)^(2*n) ) = (1/4)*( Im(1+2*i)^n )^2 = (1/4) * 5^n * sin(n*arctan(2))^2.
%F A094423 From _G. C. Greubel_, Jan 11 2024: (Start)
%F A094423 a(n) = (1/8)*5^n*(1 - ChebyshevU(n, -3/5) - (3/5)*ChebyshevU(n-1, -3/5)).
%F A094423 a(n) = (1/8)*( 5^n - (-1)^n*A066771(n) ).
%F A094423 E.g.f.: (1/8)*exp(-3*x)*(exp(8*x) - cos(4*x)). (End)
%t A094423 LinearRecurrence[{-1,5,125}, {1,4,1}, 40] (* _G. C. Greubel_, Jan 11 2024 *)
%o A094423 (PARI) Vec((x+5*x^2)/(1+x-5*x^2-125*x^3) + O(x^30)) \\ _Michel Marcus_, Aug 28 2015
%o A094423 (Magma) I:=[1,4,1]; [n le 3 select I[n] else -Self(n-1) +5*Self(n-2) +125*Self(n-3): n in [1..41]]; // _G. C. Greubel_, Jan 11 2024
%o A094423 (SageMath)
%o A094423 @CachedFunction
%o A094423 def a(n): # a = A094423
%o A094423 if (n<4): return (0,1,4,1)[n]
%o A094423 else: return -a(n-1) + 5*a(n-2) + 125*a(n-3)
%o A094423 [a(n) for n in range(1,41)] # _G. C. Greubel_, Jan 11 2024
%Y A094423 Cf. A045873, A066771, A250102.
%K A094423 nonn,easy
%O A094423 1,2
%A A094423 _Ralf Stephan_, May 04 2004