cp's OEIS Frontend

A294226 Length of period of continued fraction expansion of sqrt(3*2^n).

%I A294226 #39 Jul 22 2025 17:12:33
%S A294226 2,2,2,2,2,4,4,8,8,12,16,32,36,60,72,128,136,244,292,508,576,972,1120,
%T A294226 1992,2272,3948,4588,7924,9056,15764,18132,31832,36444,63216,72808,
%U A294226 126456,145332,253112,290968,507096,581952,1012312,1163452,2026504,2327844,4051424,4656388
%N A294226 Length of period of continued fraction expansion of sqrt(3*2^n).
%C A294226 Lim {n->inf} a(2n)/2^n = 0.555...
%C A294226 Lim {n->inf} a(2n+1)/2^n = 0.966...
%C A294226 It seems that Lim {n->inf} a(2n+1)/a(2n) = sqrt(3).
%C A294226 It seems that Lim {n->inf} a(n)/2^n = (Lim {n -> inf} A064932(n)/3^n)/2.
%H A294226 Chai Wah Wu, <a href="/A294226/b294226.txt">Table of n, a(n) for n = 0..80</a> (n = 0..46 from A.H.M. Smeets)
%F A294226 a(n) = A003285(A007283(n)). - _Michel Marcus_, Oct 02 2019
%t A294226 Array[Length@ Last@ ContinuedFraction@ Sqrt[3*2^#] &, 47, 0] (* _Michael De Vlieger_, Oct 25 2017 *)
%o A294226 (Python)
%o A294226 # for odd n
%o A294226 m, p, q = 0, 6, 2
%o A294226 tl, nl, tb, nb = 3, 1, 2, 1
%o A294226 while nl < 10**100000000:
%o A294226     tl = tl * nb + tb * nl
%o A294226     nl = 2 * nl * nb
%o A294226     nb = tl
%o A294226     tb = p * nl
%o A294226 tl = tl *nb + tb * nl
%o A294226 nl = 2 * nl * nb
%o A294226 tel, noe = tl, nl
%o A294226 while m >= 0:
%o A294226     tl = tel*q**m
%o A294226     nl = noe
%o A294226     a0 = tl//nl
%o A294226     t = 0
%o A294226     an = a0
%o A294226     while an != 2*a0:
%o A294226         tl = tl - an*nl
%o A294226         tl, nl = nl, tl
%o A294226         an = tl//nl
%o A294226         t = t + 1
%o A294226     print(2*m+1, t)
%o A294226     m = m+1
%Y A294226 Cf. A003285, A007283, A059927, A064932, A293028.
%K A294226 nonn
%O A294226 0,1
%A A294226 _A.H.M. Smeets_, Oct 25 2017