A097508 a(n) = floor(n*(sqrt(2)-1)).

0, 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 27, 27, 28, 28, 28, 29, 29, 30, 30, 31, 31, 31

Offset

0,6

Comments

Old name was: Differences between floor(n*sqrt(2)) and n.

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Heinz H. Bauschke, Minh N. Dao, and Scott B. Lindstrom, The Douglas-Rachford algorithm for a hyperplane and a doubleton, arXiv:1804.08880 [math.OC], 2018.

Marcel Celaya and Frank Ruskey, Morphic Words and Nested Recurrence Relations, arXiv preprint arXiv:1307.0153 [math.CO], 2013.

Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See pp. 17-18.

Formula

a(n) = (floor(n / cos(45 degrees))) - n.

a(n) = A001951(n) - n. - R. J. Mathar, Sep 19 2010

a(n) = floor((sqrt(2)-1)*n). [Celaya-Ruskey] - N. J. A. Sloane, Nov 14 2013

a(2*n) = 2*a(n) + A197879(n). - Robert Israel, Aug 21 2014

Maple

seq(floor(n*sqrt(2)) - n, n = 0 .. 100); # Robert Israel, Aug 21 2014

Mathematica

Table[Floor[n Sqrt[2]]-n, {n, 0, 80}] (* Harvey P. Dale, Dec 04 2014 *)

Prog

(PARI) a(n)=sqrtint(2*n^2)-n \\ Charles R Greathouse IV, Sep 02 2015
(Magma) [Floor(n*Sqrt(2)) - n: n in [0..100]]; // G. C. Greubel, Mar 27 2018

Crossrefs

Cf. A001951, A197879.

Sequence in context: A172476 A172267 A231151 * A244225 A109964 A247366

Adjacent sequences: A097505 A097506 A097507 * A097509 A097510 A097511

Keyword

easy,nonn

Author

Odimar Fabeny, Aug 26 2004

Extensions

Extended by R. J. Mathar, Sep 19 2010

Definition edited by Robert Israel, Aug 21 2014

Name changed by Michel Dekking, Jul 01 2023

Status

approved