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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 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 [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 [Local copy of first solution to Problem B6.] 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: A279124 A101406 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

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Last modified August 9 00:15 EDT 2022. Contains 356016 sequences. (Running on oeis4.)