%I A013648
%S A013648 3,8,15,24,35,48,63,80,99,120,143,168,175,176,195,208,224,255,288,323,
%T A013648 360,399,440,483,528,551,575,624,675,728,783,799,840,899,960,1023,1035,
%U A013648 1088,1155,1224,1247,1295,1368,1403,1443,1520,1599,1680,1763,1848,1872
%N A013648 Numbers n such that period of continued fraction for sqrt(n) contains 
               a single 1.
%C A013648 All terms listed have continued fraction for sqrt(n^2+2n) of the form 
               n, 1, 2n, 1, 2n, 1, 2n, etc. So all the terms of A005563 are here, 
               as well as some additional terms (with even period > 2 and the digit 
               1 in central position) (e.g. sqrt(175)=[13,'4, 2, 1, 2, 4, 26']).
%C A013648 Except for the first term of [A000027], if X=[A000027], Y=[A000012], 
               A= [A013648], we have, for all other terms, Pell's equation: [A000027]^2 
               - [A013648]*[A000012]^2=1; (X^2-A*Y^2=1); example: 2^2-3*1^1=1; 3^2-8*1^2=1 
               and so on. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 11 2009]
%D A013648 Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 
               1984, page 426 (but beware of errors!).
%H A013648 R. Macmillan, <a href="http://www.m-a.org.uk/eb/mg/mg084a.htm">Continued 
               fractions</a>, Math. Gaz. 84, 2000. See p. 34.
%F A013648 Are the numbers C(n+1, 1)*C(n+3, 1)? - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 02 2005
%F A013648 a(n)=2*n+a(n-1)+1 (with a(1)=3) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 06 2009]
%e A013648 For n=2, a(2)=2*2+3+1=8; n=3, a(3)=2*3+8+1=15; n=4, a(4)=2*4+15+1=24 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 06 2009]
%p A013648 seq(2*(n+1)*binomial(n,2)/n, n=2..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Feb 28 2007
%Y A013648 Cf. A040001, A040005, A040011, A040019, A040029, etc.
%Y A013648 Union of A005563 and A102538.
%Y A013648 Cf. A062196.
%Y A013648 Cf. A000012, A000027 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 11 2009]
%Y A013648 Sequence in context: A157606 A086959 A083656 this_sequence A005563 A132411 
               A147998
%Y A013648 Adjacent sequences: A013645 A013646 A013647 this_sequence A013649 A013650 
               A013651
%K A013648 nonn,new
%O A013648 1,1
%A A013648 Clark Kimberling (ck6(AT)evansville.edu)
%E A013648 Additional comments from Francisco Salinas (franciscodesalinas(AT)hotmail.com), 
               Dec 30 2001