This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A097509 #66 May 24 2025 17:01:59
%S A097509 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,
%T A097509 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,
%U A097509 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
%N A097509 a(n) is the number of times that n occurs as floor(k * sqrt(2)) - k.
%C A097509 Frequency of n in the sequence A097508. [_R. J. Mathar_, Sep 19 2010]
%C A097509 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}.
%H A097509 Robert Israel, <a href="/A097509/b097509.txt">Table of n, a(n) for n = 0..10000</a>
%H A097509 Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, <a href="https://kskedlaya.org/putnam-archive/2020s.pdf">Solutions to the 81st William Lowell Putnam Mathematical Competition</a>
%H A097509 Putnam Competitions, <a href="https://kskedlaya.org/putnam-archive/2020.pdf">The 81st William Lowell Putnam Mathematical Competition, Saturday, February 20, 2021, Problems</a>.
%H A097509 Putnam Competitions, <a href="/A097509/a097509.png">The 81st William Lowell Putnam Mathematical Competition, Saturday, February 20, 2021, Problems</a> [Local copy of Problem B6.]
%H A097509 Putnam Competitions, <a href="https://kskedlaya.org/putnam-archive/2020s.pdf">The 81st William Lowell Putnam Mathematical Competition, Saturday, February 20, 2021, Solutions from Manjul Bhargava, Kiran Kedlaya, and Lenny Ng</a>.
%H A097509 Putnam Competitions, <a href="/A097509/a097509_1.png">The 81st William Lowell Putnam Mathematical Competition, Saturday, February 20, 2021, Solutions from Manjul Bhargava, Kiran Kedlaya, and Lenny Ng</a> [Local copy of first solution to Problem B6.]
%H A097509 Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, <a href="https://arxiv.org/abs/2402.08331">Beatty Sequences for a Quadratic Irrational: Decidability and Applications</a>, arXiv:2402.08331 [math.NT], 2024. See pp. 17-19.
%H A097509 N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)
%F A097509 a(n) = A006337(n)-1. - _Robert G. Wilson v_, Aug 21 2014
%F A097509 Conjecture: a(n+1) = A082844(n). - _Benedict W. J. Irwin_, Mar 13 2016
%F A097509 A245219 appears to be another sequence identical to this one.
%p A097509 S:= [seq(floor(n*sqrt(2))-n, n=0..1000)]:
%p A097509 seq(numboccur(i,S),i=0..max(S)); # _Robert Israel_, Mar 13 2016
%t A097509 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 *)
%o A097509 (Python)
%o A097509 from math import isqrt
%o A097509 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
%Y A097509 Cf. A006337, A082844, A097508, A276862.
%Y A097509 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
%K A097509 easy,nonn
%O A097509 0,1
%A A097509 _Odimar Fabeny_, Aug 26 2004
%E A097509 More terms from _Robert G. Wilson v_, Aug 21 2014