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 A029548 #65 Sep 26 2022 01:50:54
%S A029548 1,32,1023,32704,1045505,33423456,1068505087,34158739328,
%T A029548 1092011153409,34910198169760,1116034330278911,35678188370755392,
%U A029548 1140585993533893633,36463073604713840864,1165677769357309014015
%N A029548 Expansion of 1/(1 - 32*x + x^2).
%C A029548 From _Bruno Berselli_, Nov 21 2011: (Start)
%C A029548 A Diophantine property of these numbers: ((a(n+1) - a(n-1))/2)^2 - 255*a(n)^2 = 1.
%C A029548 More generally, for t(m) = m + sqrt(m^2-1) and u(n) = (t(m)^(n+1) - 1/t(m)^(n+1))/(t(m) - 1/t(m)), we can verify that ((u(n+1) - u(n-1))/2)^2 - (m^2-1)*u(n)^2 = 1. (End)
%C A029548 a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,...,31}. - _Milan Janjic_, Jan 26 2015
%H A029548 Vincenzo Librandi, <a href="/A029548/b029548.txt">Table of n, a(n) for n = 0..600</a>
%H A029548 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A029548 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A029548 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (32,-1).
%F A029548 a(n) = 32*a(n-1) - a(n-2), a(-1)=0, a(0)=1.
%F A029548 a(n) = S(n, 32) with S(n, x) = U(n, x/2) Chebyshev's polynomials of the 2nd kind. See A049310. - _Wolfdieter Lang_, Nov 29 2002
%F A029548 a(n) = (ap^(n+1) - am^(n+1))/(ap - am) with ap=16+sqrt(255) and am=16-sqrt(255).
%F A029548 a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*32^(n-2*k).
%F A029548 a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*31^k. - _Philippe Deléham_, Feb 10 2012
%F A029548 Product {n >= 0} (1 + 1/a(n)) = 1/15*(15 + sqrt(255)). - _Peter Bala_, Dec 23 2012
%F A029548 Product {n >= 1} (1 - 1/a(n)) = 1/32*(15 + sqrt(255)). - _Peter Bala_, Dec 23 2012
%p A029548 seq(simplify(ChebyshevU(n, 16)), n=0..20); # _G. C. Greubel_, Dec 22 2019
%t A029548 Table[GegenbauerC[n, 1, 16], {n,0,20}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008 *)
%t A029548 CoefficientList[Series[1/(1-32x+x^2), {x,0,20}], x] (* _Vincenzo Librandi_, Dec 24 2012 *)
%t A029548 ChebyshevU[Range[0,20], 16] (* _G. C. Greubel_, Dec 22 2019 *)
%o A029548 (Sage) [lucas_number1(n,32,1) for n in range(1, 16)] # _Zerinvary Lajos_, Nov 07 2009
%o A029548 (Sage) [chebyshev_U(n,16) for n in (0..20)] # _G. C. Greubel_, Dec 22 2019
%o A029548 (Magma) I:=[1, 32]; [n le 2 select I[n] else 32*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Dec 24 2012
%o A029548 (PARI) vector( 21, n, polchebyshev(n-1, 2, 17) ) \\ _G. C. Greubel_, Dec 22 2019
%o A029548 (GAP) m:=17;; a:=[1,2*m];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Dec 22 2019
%Y A029548 Cf. A200441, A200442.
%Y A029548 Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), this sequence (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33), A097725 (m=51).
%K A029548 nonn,easy
%O A029548 0,2
%A A029548 _N. J. A. Sloane_