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

Author

Odimar Fabeny, Aug 26 2004

Keywords

• easy
• nonn

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)

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

Programs

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 *)

Python

from math import isqrt
def A097509(n): return 1-isqrt(m:=n**2<<1)+isqrt(m+(n<<2)+2) if n else 3 # Chai Wah Wu, May 24 2025

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.

Extensions

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