1,2

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).

R. L. Graham, D. E. Knuth and O. Pataschnic, Concrete Mathematics, Addison-Wesley, Reading (1994) 2nd Ed., Ex. 3.46.

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.

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

S. Rabinowitz and P. Gilbert, A nonlinear recurrence yielding binary digits. Math. Mag. 64 (1991), no. 3, 168-171.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Th. Stoll, On Families of Nonlinear Recurrences Related to Digits, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.2.

T. D. Noe, Table of n, a(n) for n=1..200

Stoll, T. On Families of Nonlinear Recurrences Related to Digits, J. Integer Sequences 8, No. 05.3.2, 1-8, 2005.

a(n) = [ sqrt(2)^(n-1) ] + [ sqrt(2)^(n-2) ], n>1. - R. Stephan, Sep 18 2004

Sequence in context: A061481 A017824 A094054 this_sequence A003143 A017983 A139077

Adjacent sequences: A001518 A001519 A001520 this_sequence A001522 A001523 A001524

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from TORSTEN.SILLKE(AT)LHSYSTEMS.COM, Apr 06 2001.