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%I M2534 N1001 #70 Mar 24 2021 15:35:58
%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 Duchêne, 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 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 77.
%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 Aviezri S. Fraenkel, <a href="http://dx.doi.org/10.1016/S0012-365X(00)00138-2">On the recurrence f(m+1)= b(m)*f(m)-f(m-1) and applications</a>, Discrete Mathematics 224 (2000), pp. 273-279.
%H Wen An Liu and Xiao Zhao, <a href="http://dx.doi.org/10.1016/j.dam.2014.08.009">Adjoining to (s,t)-Wythoff's game its P-positions as moves</a>, Discrete Applied Mathematics, 179 (2014), 28-43. See Table 3.
%H N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)
%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
%o (Python)
%o from sympy import integer_nthroot
%o def A001952(n): return 2*n+integer_nthroot(2*n**2,2)[0] # _Chai Wah Wu_, Mar 16 2021
%Y Complement of A001951; equals A001951(n)+2*n.
%Y A bisection of A094077.
%Y Bisection: A187393, A342280.
%Y Cf. A026250, A080764.
%Y The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - _N. J. A. Sloane_, Mar 09 2021
%K nonn,easy,nice
%O 1,1
%A _N. J. A. Sloane_
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