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A001952 A Beatty sequence: a(n) = floor(n*(2 + sqrt(2))).
(Formerly M2534 N1001)

%I M2534 N1001

%S 3,6,10,13,17,20,23,27,30,34,37,40,44,47,51,54,58,61,64,68,71,75,78,

%T 81,85,88,92,95,99,102,105,109,112,116,119,122,126,129,133,136,139,

%U 143,146,150,153,157,160,163,167,170,174,177,180,184,187,191,194,198

%N A Beatty sequence: a(n) = floor(n*(2 + sqrt(2))).

%C It appears that the distance between the a(n)-th triangular number and the nearest square is greater than floor(a(n)/2). - _Ralf Stephan_, Sep 14 2013

%C A080764(a(n)) = 0. - _Reinhard Zumkeller_, Jul 03 2015

%D Eric DuchĂȘne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, Urban Larsson, Wythoff Visions, Games of No Chance, Vol. 5; MSRI Publications, Vol. 70 (2017), pages 101-153.

%D Fraenkel, Aviezri S., On the recurrence f(m+1)=b(m)f(m)-f(m-1) and applications. Discrete Math. 224 (2000), no. 1-3, 273-279.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 77.

%D Wen An Liu and Xiao Zhao, Adjoining to (s,t)-Wythoff's game its P-positions as moves, Discrete Applied Mathematics, Aug 27 2014; DOI: 10.1016/j.dam.2014.08.009. See Table 3.

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

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

%H T. D. Noe, <a href="/A001952/b001952.txt">Table of n, a(n) for n = 1..10000</a>

%H L. Carlitz, R. Scoville and V. E. Hoggatt, Jr. <a href="http://www.fq.math.ca/Scanned/10-5/carlitz1.pdf">Pellian representatives</a>, Fibonacci Quarterly, 10, issue 5, 1972, 449-488.

%H Ian G. Connell, <a href="http://dx.doi.org/10.4153/CMB-1959-024-3">A generalization of Wythoff's game</a>, Canad. Math. Bull. 2 (1959) 181-190

%H J. N. Cooper and A. W. N. Riasanovsky, <a href="http://www.math.sc.edu/~cooper/Sigma.pdf">On the Reciprocal of the Binary Generating Function for the Sum of Divisors</a>, 2012; <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Cooper/cooper3.html">J. Int. Seq. 16 (2013) #13.1.8</a>

%H A. S. Fraenkel, <a href="http://www.jstor.org/stable/2321643">How to beat your Wythoff games' opponent on three fronts</a>, Amer. Math. Monthly, 89 (1982), 353-361 (the case a=2).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BeattySequence.html">Beatty Sequence.</a>

%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>

%t Table[Floor[n*(2 + Sqrt[2])], {n, 60}] (* _Stefan Steinerberger_, Apr 15 2006 *)

%t Array[Floor[#(2+Sqrt[2])]&,60] (* _Harvey P. Dale_, Dec 01 2015 *)

%o (Haskell)

%o a001952 = floor . (* (sqrt 2 + 2)) . fromIntegral

%o -- _Reinhard Zumkeller_, Jul 03 2015

%o (PARI) a(n)=2*n+sqrtint(2*n^2) \\ _Charles R Greathouse IV_, Jan 05 2016

%Y Complement of A001951; equals A001951(n)+2*n.

%Y A bisection of A094077.

%Y Cf. A026250, A080764.

%K nonn,easy,nice

%O 1,1

%A _N. J. A. Sloane_

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Last modified August 17 11:06 EDT 2019. Contains 326057 sequences. (Running on oeis4.)