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A001951 A Beatty sequence: a(n) = floor[n*sqrt 2].
(Formerly M0955 N0356)
65
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, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 50, 52, 53, 55, 56, 57, 59, 60, 62, 63, 65, 66, 67, 69, 70, 72, 73, 74, 76, 77, 79, 80, 82, 83, 84, 86, 87, 89, 90, 91, 93, 94, 96, 97, 98, 100 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Earliest monotonic sequence >0 satisfying the condition : "a(n)+2n is not in the sequence" - Benoit Cloitre, Mar 25 2004

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.

First differences are 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, ..(A006337) . - Philippe Deléham, May 29 2006

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

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

REFERENCES

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

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

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

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).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Pellian representatives, Fib. Quart., 10 (1972), 449-488.

Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190

A. S. Fraenkel, How to beat your Wythoff games' opponent on three fronts, Amer. Math. Monthly, 89 (1982), 353-361 (the case a=2).

Eric Weisstein's World of Mathematics, Beatty Sequence.

Index entries for sequences related to Beatty sequences

MATHEMATICA

f[n_] := Floor[n*Sqrt[2]]; Array[f, 72, 0] (* Robert G. Wilson v, Oct 17 2012 *)

PROG

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

(MAGMA) [Floor(n*Sqrt(2)): n in [0..60]]; // Vincenzo Librandi, Oct 22 2011

(Maxima) makelist(floor(n*sqrt(2)), n, 0, 100); /* Martin Ettl, Oct 17 2012 */

(Haskell)

a001951 = floor . (* sqrt 2) . fromIntegral

-- Reinhard Zumkeller, Sep 14 2014

CROSSREFS

Complement of A001952. Equals A001952(n)-2*n.

Equals A003151(n) - n.

Cf. A022342, A026250.

A bisection of A094077.

Sequence in context: A047381 A097506 A189794 * A039046 A187683 A187351

Adjacent sequences:  A001948 A001949 A001950 * A001952 A001953 A001954

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from David W. Wilson, Sep 20 2000

STATUS

approved

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Last modified October 30 19:36 EDT 2014. Contains 248837 sequences.