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A097509
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a(n) is the number of times that n occurs as floor(k * sqrt(2)) - k.
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12
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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
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OFFSET
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0,1
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COMMENTS
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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}.
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LINKS
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FORMULA
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A245219 appears to be another sequence identical to this one.
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MAPLE
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S:= [seq(floor(n*sqrt(2))-n, n=0..1000)]:
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MATHEMATICA
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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 *)
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CROSSREFS
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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
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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