%I A001521 M0569 N0206
%S A001521 1,2,3,4,6,9,13,19,27,38,54,77,109,154,218,309,437,618,874,1236,1748,2472,
%T A001521 3496,4944,6992,9888,13984,19777,27969,39554,55938,79108,111876,158217,
%U A001521 223753,316435,447507,632871,895015,1265743,1790031,2531486,3580062,5062972
%N A001521 a(1) = 1; thereafter a(n+1) = [sqrt(2a(n)(a(n)+1))].
%C A001521 Graham and Pollak give an elementary proof of the following result: For 
               given m, define a_n by a_1 = m and a_{n+1} = [ sqrt{2*a_n*(a_n + 
               1)} ], n >= 1. Then a_n = [ tau_m (2^{(n-1)/2} + 2^{(n-2)/2}) ] where 
               tau_m is the m-th smallest element of {1, 2, 3, ... } union { sqrt{2}, 
               2sqrt{2}, 3sqrt{2}, ... }. For m=1 it follows as a curious corollary 
               that a_{2n+1} - 2a_{2n-1} is exactly the n-th bit in the binary expansion 
               of sqrt{2} (A004539).
%D A001521 R. L. Graham, D. E. Knuth and O. Pataschnic, Concrete Mathematics, Addison-Wesley, 
               Reading (1994) 2nd Ed., Ex. 3.46.
%D A001521 R. L. Graham and H. O. Pollak, Note on a nonlinear recurrence related 
               to sqrt(2), Mathematics Magazine, Volume 43, Pages 143-145, 1970. 
               Zbl 201.04705.
%D A001521 R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), 
               no. 8, 697-712.
%D A001521 S. Rabinowitz and P. Gilbert, A nonlinear recurrence yielding binary 
               digits. Math. Mag. 64 (1991), no. 3, 168-171.
%D A001521 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001521 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001521 Th. Stoll, On Families of Nonlinear Recurrences Related to Digits, Journal 
               of Integer Sequences, Vol. 8 (2005), Article 05.3.2.
%H A001521 T. D. Noe, <a href="b001521.txt">Table of n, a(n) for n=1..200</a>
%H A001521 Stoll, T. <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Stoll/
               stoll56.pdf">On Families of Nonlinear Recurrences Related to Digits</
               a>, J. Integer Sequences 8, No. 05.3.2, 1-8, 2005.
%F A001521 a(n) = [ sqrt(2)^(n-1) ] + [ sqrt(2)^(n-2) ], n>1. - R. Stephan, Sep 
               18 2004
%Y A001521 Sequence in context: A061481 A017824 A094054 this_sequence A003143 A017983 
               A139077
%Y A001521 Adjacent sequences: A001518 A001519 A001520 this_sequence A001522 A001523 
               A001524
%K A001521 nonn,nice,easy
%O A001521 1,2
%A A001521 N. J. A. Sloane (njas(AT)research.att.com).
%E A001521 Additional comments from TORSTEN.SILLKE(AT)LHSYSTEMS.COM, Apr 06 2001.