%I M0955 N0356 #162 Oct 02 2024 14:35:24

%S 0,1,2,4,5,7,8,9,11,12,14,15,16,18,19,21,22,24,25,26,28,29,31,32,33,

%T 35,36,38,39,41,42,43,45,46,48,49,50,52,53,55,56,57,59,60,62,63,65,66,

%U 67,69,70,72,73,74,76,77,79,80,82,83,84,86,87,89,90,91,93,94,96,97,98,100

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

%C Earliest monotonic sequence greater than 0 satisfying the condition: "a(n) + 2n is not in the sequence". - _Benoit Cloitre_, Mar 25 2004

%C Also the integer part of the hypotenuse of isosceles right triangles. The real part of these numbers is irrational. For proof see Jones and Jones.

%C First differences are 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, ... (A006337 with a 1 in front). - _Philippe Deléham_, May 29 2006

%C It appears that the distance between the a(n)-th triangular number and the nearest square is not greater than floor(a(n)/2). - _Ralf Stephan_, Sep 14 2013

%C These are the nonnegative integers m satisfying sin(m*Pi/r)*sin((m+1)*Pi/r) <= 0, where r = sqrt(2). In general, the Beatty sequence of an irrational number r > 1 consists of the numbers m satisfying sin(m*x)*sin((m+1)*x) <= 0, where x = Pi/r. Thus the numbers m satisfying sin(m*x)*sin((m+1)*x) > 0 form the Beatty sequence of r/(1-r). - _Clark Kimberling_, Aug 21 2014

%C For n > 0: A080764(a(n)) = 1. - _Reinhard Zumkeller_, Jul 03 2015

%C From _Clark Kimberling_, Oct 17 2016: (Start)

%C We can generate A001951 and A001952 without using sqrt(2).

%C First write the even positive integers in a row:

%C 2 4 6 8 10 12 14 . . .

%C Then put 1 under 2 and add:

%C 2 4 6 8 10 12 14 . . .

%C 1

%C 3

%C Next, under 4, put the least positive integer that is not yet in rows 2 and 3;

%C it is 2; and add:

%C 2 4 6 8 10 12 14 . . .

%C 1 2

%C 3 6

%C Next, under the 6 in row 1, put the least positive integer not yet in rows 2 and 3;

%C it is 4, and add:

%C 2 4 6 8 10 12 14 . . .

%C 1 2 4

%C 3 6 10

%C Continue in this manner. (End)

%C This sequence contains an infinite number of powers of 2 (proof in Crux Mathematicorum link). See A103341. - _Bernard Schott_, Mar 08 2019

%C The terms of this sequence generate the multiplicative group of positive rational numbers (observation by Stephen M. Gagola, Jr.; see References). - _Allen Stenger_, Aug 05 2023

%C a(n) is also the number of distinct straight cylinders with integer radius and height having the same surface as a sphere with radius n. - _Felix Huber_, Sep 20 2024

%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 Stephen M. Gagola Jr., Solution of Problem 12282, Am. Math. Monthly, 130 (2023), pp. 682-683.

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

%D Gareth A. Jones and J. Mary Jones, Elementary Number Theory, Springer, 1998; pp. 221-222.

%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).

%D Roland Sprague, Recreations in Mathematics, Blackie and Son, (1963).

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition (1997), Entry sqrt(2), p. 18.

%H Vincenzo Librandi, <a href="/A001951/b001951.txt">Table of n, a(n) for n = 0..10000</a>

%H L. Carlitz, Richard Scoville and Verner E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/10-5/carlitz1.pdf">Pellian representatives</a>, Fib. Quart., 10 (1972), 449-488.

%H Ed Doolittle, <a href="https://cms.math.ca/crux/backfile/Crux_v14n03_Mar.pdf">Problem 19</a>, 26th I.M.O. Finland proposed by Romania, Crux Mathematicorum, p. 70, Vol. 14, Mar. 88.

%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 Aviezri 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), no. 1-3, 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, Aug 27 2014; See Table 3.

%H Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, <a href="https://arxiv.org/abs/2402.08331">Beatty Sequences for a Quadratic Irrational: Decidability and Applications</a>, arXiv:2402.08331 [math.NT], 2024. See pp. 17-18.

%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>

%F a(n) = A000196(A001105(n)). - _Jason Kimberley_, Oct 26 2016

%F a(n) = floor(csc(1/(sqrt(2)*n))) for n > 0, since sqrt(2)*n < csc(1/(sqrt(2)*n)) < sqrt(2)*n + 1/(3*sqrt(2)*n) < floor(sqrt(2)*n) + 1 for n > 0. - _Jianing Song_, Sep 07 2021

%F a(n) = A194102(n) - A194102(n-1) for n > 0. - _M. F. Hasler_, Apr 23 2022

%p a:=n->floor(n*sqrt(2)): seq(a(n),n=0..80); # _Muniru A Asiru_, Mar 09 2019

%t Floor[Range[0, 72] Sqrt[2]] (* _Robert G. Wilson v_, Oct 17 2012 *)

%o (PARI) f(n) = for(j=1,n,print1(floor(sqrt(2*j^2))","))

%o (PARI) a(n)=sqrtint(2*n^2) \\ _Charles R Greathouse IV_, Oct 19 2016

%o (Magma) [Floor(n*Sqrt(2)): n in [0..60]]; // _Vincenzo Librandi_, Oct 22 2011

%o (Magma) [Isqrt(2*n^2):n in[0..60]]; // _Jason Kimberley_, Oct 28 2016

%o (Maxima) makelist(floor(n*sqrt(2)), n, 0, 100); /* _Martin Ettl_, Oct 17 2012 */

%o (Haskell)

%o a001951 = floor . (* sqrt 2) . fromIntegral

%o -- _Reinhard Zumkeller_, Sep 14 2014

%o (Python)

%o from sympy import integer_nthroot

%o def A001951(n): return integer_nthroot(2*n**2,2)[0] # _Chai Wah Wu_, Mar 16 2021

%Y Complement of A001952. Equals A001952(n) - 2*n for n>0.

%Y Equals A003151(n) - n; a bisection of A094077.

%Y Bisections: A022842, A342281.

%Y Cf. A022342, A026250, A080764, A103341.

%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

%Y Partial sums: A194102.

%K nonn,nice,easy

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _David W. Wilson_, Sep 20 2000