A097509 a(n) is the number of times that n occurs as floor(k * sqrt(2)) - k.

3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2

Offset

0,1

Comments

Frequency of n in the sequence A097508. [R. J. Mathar, Sep 19 2010]

Theorem: If the initial term is omitted, this is identical to A276862. For proof, see solution to Problem B6 in the 81st William Lowell Putnam Mathematical Competition (see links). The argument may also imply that A082844 is also the same, apart from two initial terms. - Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Mar 02 2021. Postscript from the same authors, Sep 09 2021: We have proved that the present sequence, A097509 (indexed from 0) matches the definition of our {c_i}.

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Solutions to the 81st William Lowell Putnam Mathematical Competition

Putnam Competitions, The 81st William Lowell Putnam Mathematical Competition, Saturday, February 20, 2021, Problems.

Putnam Competitions, The 81st William Lowell Putnam Mathematical Competition, Saturday, February 20, 2021, Problems [Local copy of Problem B6.]

Putnam Competitions, The 81st William Lowell Putnam Mathematical Competition, Saturday, February 20, 2021, Solutions from Manjul Bhargava, Kiran Kedlaya, and Lenny Ng.

Putnam Competitions, The 81st William Lowell Putnam Mathematical Competition, Saturday, February 20, 2021, Solutions from Manjul Bhargava, Kiran Kedlaya, and Lenny Ng [Local copy of first solution to Problem B6.]

Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See pp. 17-19.

N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Formula

a(n) = A006337(n)-1. - Robert G. Wilson v, Aug 21 2014

Conjecture: a(n+1) = A082844(n). - Benedict W. J. Irwin, Mar 13 2016

A245219 appears to be another sequence identical to this one.

Maple

S:= [seq(floor(n*sqrt(2))-n, n=0..1000)]:
seq(numboccur(i, S), i=0..max(S)); # Robert Israel, Mar 13 2016

Mathematica

f[n_] := Floor[n/Cos[Pi/4]] - n; d = Array[f, 500, 0]; Tally[ Array[ f, 254, 0]][[All, 2]] (* Robert G. Wilson v, Aug 21 2014 *)

Crossrefs

Cf. A006337, A082844, A097508, A276862.

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

Sequence in context: A363445 A363348 A245219 * A095206 A344129 A308006

Adjacent sequences: A097506 A097507 A097508 * A097510 A097511 A097512

Keyword

easy,nonn

Author

Odimar Fabeny, Aug 26 2004

Extensions

More terms from Robert G. Wilson v, Aug 21 2014

Status

approved