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A001951 A Beatty sequence: a(n) = floor(n*sqrt(2)).
(Formerly M0955 N0356)
71
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

For n > 0: A080764(a(n)) = 1. - Reinhard Zumkeller, Jul 03 2015

We can generate A001951 and A001952 without using sqrt(2).  First write the even positive integers in a row:

2   4   6   8   10   12   14 . . .

Then put 1 under 2 and add:

2   4   6   8   10   12   14 . . .

1

3

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

3; that is 2; and add, getting

2   4   6   8   10   12   14 . . .

1   2

3   6

Next, under the 6 in row 1, put the least positive integer not yet in rows 2 and 3; that is 4, and add:

2   4   6   8   10   12   14 . . .

1   2   4

3   6   10

Continue in this manner. - Clark Kimberling, Oct 17 2016

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.

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).

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

Wen An Liu and Xiao Zhao, Adjoining to (s,t)-Wythoff's game its P-positions as moves, Discrete Applied Mathematics, Aug 27 2014; See Table 3.

Eric Weisstein's World of Mathematics, Beatty Sequence.

Index entries for sequences related to Beatty sequences

FORMULA

a(n) = A000196(A001105(n)). - Jason Kimberley, Oct 26 2016

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))", "))

(PARI) a(n)=sqrtint(2*n^2) \\ Charles R Greathouse IV, Oct 19 2016

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

(MAGMA) [Isqrt(2*n^2):n in[0..60]]; // Jason Kimberley, Oct 28 2016

(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; a bisection of A094077.

Cf. A022342, A026250, A080764.

Sequence in context: A258833 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 December 6 06:58 EST 2016. Contains 278775 sequences.