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 A041179 #26 Feb 16 2025 08:32:38
%S A041179 1,1,19,20,379,399,7561,7960,150841,158801,3009259,3168060,60034339,
%T A041179 63202399,1197677521,1260879920,23893516081,25154396001,476672644099,
%U A041179 501827040100,9509559365899,10011386405999,189714514673881,199725901079880,3784780734111721
%N A041179 Denominators of continued fraction convergents to sqrt(99).
%C A041179 The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 18 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 A041179 Vincenzo Librandi, <a href="/A041179/b041179.txt">Table of n, a(n) for n = 0..200</a>
%H A041179 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/LehmerNumber.html">MathWorld: Lehmer Number</a>
%H A041179 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,20,0,-1).
%F A041179 G.f.: -(x^2-x-1) / (x^4-20*x^2+1). - _Colin Barker_, Nov 14 2013
%F A041179 a(n) = 20*a(n-2) - a(n-4). - _Vincenzo Librandi_, Dec 12 2013
%F A041179 From _Peter Bala_, May 28 2014: (Start)
%F A041179 The following remarks assume an offset of 1.
%F A041179 Let alpha = ( sqrt(18) + sqrt(22) )/2 and beta = ( sqrt(18) - sqrt(22) )/2 be the roots of the equation x^2 - sqrt(18)*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 A041179 a(n) = Product_{k = 1..floor((n-1)/2)} ( 18 + 4*cos^2(k*Pi/n) ).
%F A041179 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) = 18*a(2*n) + a(2*n - 1). (End)
%t A041179 Denominator[Convergents[Sqrt[99], 30]] (* _Vincenzo Librandi_, Dec 12 2013 *)
%o A041179 (Magma) I:=[1, 1, 19, 20]; [n le 4 select I[n] else 20*Self(n-2)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Dec 12 2013
%Y A041179 Cf. A041178, A010170, A020856, A010550, A002530.
%K A041179 nonn,frac,easy
%O A041179 0,3
%A A041179 _N. J. A. Sloane_
%E A041179 More terms from _Colin Barker_, Nov 14 2013