%I #28 Sep 04 2019 21:16:37

%S 1,3,5,12,19,45,71,168,265,627,989,2340,3691,8733,13775,32592,51409,

%T 121635,191861,453948,716035,1694157,2672279,6322680,9973081,23596563,

%U 37220045,88063572,138907099,328657725,518408351,1226567328,1934726305,4577611587,7220496869,17083879020,26947261171,63757904493,100568547815

%N Numerators of the lower principal convergents and the lower intermediate convergents to 3^(1/2).

%C The lower principal and intermediate convergents to 3^(1/2), beginning with 1/1, 3/2, 5/3, 12/7, 19/11, form a strictly increasing sequence; with essentially, numerators being this sequence and denominators being A005246.

%C sqrt(floor(a(n)^2/3)+1) = A005246(n+1). Also see A082630. - _Richard R. Forberg_, Nov 14 2013

%C a(n) = U_n(sqrt(6),1) for n odd and a(n) = 3*U_n(sqrt(6),1) for n even, where U_n(sqrt(R),Q) denotes the Lehmer sequence with parameters R and Q. This sequence 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 sequence is a divisibility sequence: if n divides m then a(n) divides a(m). - _Peter Bala_, Sep 03 2019

%D Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

%D Clark Kimberling, "Best lower and upper approximates to irrational numbers," Elemente der Mathematik, 52 (1997) 122-126.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lehmer_sequence">Lehmer sequence</a>

%F a(n) = 4*a(n-2)-a(n-4). G.f.: x*(1+3*x+x^2)/(1-4*x^2+x^4). - _Colin Barker_, Apr 28 2012

%e From _Peter Bala_, Sep 03 2019: (Start)

%e If p(n)/q(n) denotes the n-th convergent to the simple continued fraction alpha = [c(0); c(1), c(2), ...] then a lower semiconvergent is a rational number of the form ( p(2*n) + m*p(2*n+1) )/( q(2*n) + m*q(2*n+1) ) where 0 <= m <= c(2*n+2). The lower semiconvergents include the even-indexed convergents p(2*n)/q(2*n) and give an increasing sequence of approximations to alpha from below.

%e In this case the simple continued fraction expansion sqrt(3) = [1; 1, 2, 1, 2, ...] produces the sequence of convergents (p(n)/q(n))n>=0 = [1/1, 2/1, 5/3, 7/4, 19/11, 26,15, 71/41, ...].

%e Thus the increasing sequence of lower semiconvergents begins 1/1, (1 + 2)/(1 + 1) = 3/2, (1 + 2*2)/(1 + 2*1) = 5/3, (5 + 7)/(3 + 4) = 12/7, (5 + 2*7)/(3 + 2*4) = 19/11, ... with numerators 1, 3, 5, 12, 19, .... (End)

%Y Cf. A002530, A002531, A005246, A082630, A143607, A143609, A143642.

%K nonn,frac,easy

%O 1,2

%A _Clark Kimberling_, Aug 27 2008