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A097509 a(n) is the number of times that n occurs as floor(k * sqrt(2)) - k. 12
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 (list; graph; refs; listen; history; text; internal format)
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
Manjul Bhargava, Kiran Kedlaya, and Lenny Ng, Solutions to the 81st William Lowell Putnam Mathematical Competition
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
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
KEYWORD
easy,nonn
AUTHOR
Odimar Fabeny, Aug 26 2004
EXTENSIONS
More terms from Robert G. Wilson v, Aug 21 2014
STATUS
approved

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Last modified March 28 15:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)