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.

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.

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

Adjacent sequences: A001518 A001519 A001520 this_sequence A001522 A001523 A001524

Sequence in context: A061481 A017824 A094054 this_sequence A003143 A017983 A017825

nonn,nice,easy

njas

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