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 A041759 #25 Feb 16 2025 08:32:38
%S A041759 1,1,39,40,1559,1599,62321,63920,2491281,2555201,99588919,102144120,
%T A041759 3981065479,4083209599,159143030241,163226239840,6361740144161,
%U A041759 6524966384001,254310462736199,260835429120200,10166056769303799,10426892198423999
%N A041759 Denominators of continued fraction convergents to sqrt(399).
%C A041759 The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 38 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 27 2014
%H A041759 Vincenzo Librandi, <a href="/A041759/b041759.txt">Table of n, a(n) for n = 0..200</a>
%H A041759 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/LehmerNumber.html">MathWorld: Lehmer Number</a>
%H A041759 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,40,0,-1).
%F A041759 G.f.: -(x^2-x-1) / (x^4-40*x^2+1). - _Colin Barker_, Nov 24 2013
%F A041759 a(n) = 40*a(n-2) - a(n-4) for n > 3. - _Vincenzo Librandi_, Dec 24 2013
%F A041759 From _Peter Bala_, May 27 2014: (Start)
%F A041759 The following remarks assume an offset of 1.
%F A041759 Let alpha = ( sqrt(38) + sqrt(42) )/2 and beta = ( sqrt(38) - sqrt(42) )/2 be the roots of the equation x^2 - sqrt(38)*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 A041759 a(n) = Product_{k = 1..floor((n-1)/2)} ( 38 + 4*cos^2(k*Pi/n) ).
%F A041759 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) = 38*a(2*n) + a(2*n - 1). (End)
%t A041759 Denominator[Convergents[Sqrt[399], 30]] (* _Vincenzo Librandi_, Dec 24 2013 *)
%o A041759 (Magma) I:=[1,1,39,40]; [n le 4 select I[n] else 40*Self(n-2)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Dec 24 2013
%Y A041759 Cf. A041758, A040379, A002530.
%K A041759 nonn,frac,easy
%O A041759 0,3
%A A041759 _N. J. A. Sloane_
%E A041759 More terms from _Colin Barker_, Nov 24 2013