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 A041219 #29 Feb 16 2025 08:32:38
%S A041219 1,1,21,22,461,483,10121,10604,222201,232805,4878301,5111106,
%T A041219 107100421,112211527,2351330961,2463542488,51622180721,54085723209,
%U A041219 1133336644901,1187422368110,24881784007101,26069206375211,546265911511321,572335117886532,11992968269241961
%N A041219 Denominators of continued fraction convergents to sqrt(120).
%C A041219 The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 20 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - _Peter Bala_, May 28 2014
%H A041219 Vincenzo Librandi, <a href="/A041219/b041219.txt">Table of n, a(n) for n = 0..200</a>
%H A041219 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/LehmerNumber.html">MathWorld: Lehmer Number</a>
%H A041219 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,22,0,-1).
%F A041219 From _Colin Barker_, Jul 15 2012: (Start)
%F A041219 a(n) = 22*a(n-2) - a(n-4).
%F A041219 G.f.: (1+x-x^2)/(1-22*x^2+x^4). (End)
%F A041219 From _Peter Bala_, May 28 2014: (Start)
%F A041219 The following remarks assume an offset of 1.
%F A041219 Let alpha = sqrt(5) + sqrt(6) and beta = sqrt(5) - sqrt(6) be the roots of the equation x^2 - sqrt(20)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
%F A041219 a(n) = Product_{k = 1..floor((n-1)/2)} ( 20 + 4*cos^2(k*Pi/n) ).
%F A041219 Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 20*a(2*n) + a(2*n - 1). (End)
%t A041219 Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[120], n]]], {n, 1, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 23 2011 *)
%t A041219 Denominator[Convergents[Sqrt[120],30]] (* _Harvey P. Dale_, Mar 14 2013 *)
%t A041219 CoefficientList[Series[(1 + x - x^2)/(1 - 22 x^2 + x^4), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 24 2013 *)
%Y A041219 Cf. A041218, A002530.
%K A041219 nonn,frac,easy,less
%O A041219 0,3
%A A041219 _N. J. A. Sloane_