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A041059 Denominators of continued fraction convergents to sqrt(35).

%I A041059 #60 Feb 16 2025 08:32:38
%S A041059 1,1,11,12,131,143,1561,1704,18601,20305,221651,241956,2641211,
%T A041059 2883167,31472881,34356048,375033361,409389409,4468927451,4878316860,
%U A041059 53252096051,58130412911,634556225161,692686638072,7561422605881,8254109243953,90102515045411
%N A041059 Denominators of continued fraction convergents to sqrt(35).
%C A041059 The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 10 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 A041059 Vincenzo Librandi, <a href="/A041059/b041059.txt">Table of n, a(n) for n = 0..200</a>
%H A041059 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LehmerNumber.html">Lehmer Number</a>
%H A041059 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,12,0,-1).
%F A041059 G.f.: (1+x-x^2)/(1-12*x^2+x^4). - _Colin Barker_, Jan 01 2012
%F A041059 From _Peter Bala_, May 28 2014: (Start)
%F A041059 The following remarks assume an offset of 1.
%F A041059 Let alpha = ( sqrt(10) + sqrt(14) )/2 and beta = ( sqrt(10) - sqrt(14) )/2 be the roots of the equation x^2 - sqrt(10)*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 A041059 a(n) = Product_{k = 1..floor((n-1)/2)} ( 10 + 4*cos^2(k*Pi/n) ).
%F A041059 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) = 10*a(2*n) + a(2*n - 1). (End)
%t A041059 Denominator[Convergents[Sqrt[35], 30]] (* _Vincenzo Librandi_, Oct 23 2013 *)
%Y A041059 Cf. A010490, A041058, A002539.
%K A041059 nonn,frac,easy
%O A041059 0,3
%A A041059 _N. J. A. Sloane_