bibkey: sloane2014a049473
authors: N. J. A. Sloane; Clark Kimberling
year: 2014
title: "OEIS A049473, Nearest integer to n/sqrt(2)"
doi: null
url: https://oeis.org/A049473
claim: "%N Nearest integer to n/sqrt(2). %C Let s(n) = zeta(3) - Sum_{k=1..n} 1/k^3.  Conjecture:  for n >=1, s(a(n)) < 1/n^2 < s(a(n)-1), and the difference sequence of A049473 consists solely of 0's and 1, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively.  - _Clark Kimberling_, Oct 05 2014 %N A001953 a(n) = floor((n + 1/2) * sqrt(2)). %N A001954 a(n) = floor((n+1/2)*(2+sqrt(2))); winning positions in the 2-Wythoff game."
strata_touched:
  - D5/S1/Deficit/Beatty/KimberlingRoundedHalfRootTwoDifference
license: citation-only
triage: anchor

OEIS A049473

The NAME of A049473 states:

Nearest integer to n/sqrt(2).

Clark Kimberling’s COMMENT states:

Let s(n) = zeta(3) - Sum_{k=1..n} 1/k^3. Conjecture: for n >=1, s(a(n)) < 1/n^2 < s(a(n)-1), and the difference sequence of A049473 consists solely of 0’s and 1, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively. - Clark Kimberling, Oct 05 2014

The NAME of A001953 states:

a(n) = floor((n + 1/2) * sqrt(2)).

The NAME of A001954 states:

a(n) = floor((n+1/2)*(2+sqrt(2))); winning positions in the 2-Wythoff game.

The AUTHOR of A049473 is N. J. A. Sloane. The author is listed first in the metadata, followed by Clark Kimberling, who wrote the dated conjecture. The year records the 2014 conjecture because the AUTHOR line for this old core sequence supplies no date.

Verified locator

• URL: https://oeis.org/A049473
• A049473 NAME (verbatim): Nearest integer to n/sqrt(2).
• COMMENT (verbatim): Let s(n) = zeta(3) - Sum_{k=1..n} 1/k^3. Conjecture: for n >=1, s(a(n)) < 1/n^2 < s(a(n)-1), and the difference sequence of A049473 consists solely of 0’s and 1, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively. - Clark Kimberling, Oct 05 2014
• A001953 NAME (verbatim): a(n) = floor((n + 1/2) * sqrt(2)).
• A001954 NAME (verbatim): a(n) = floor((n+1/2)*(2+sqrt(2))); winning positions in the 2-Wythoff game.

Only the difference-sequence clause of the COMMENT is settled by the attached formal result; the zeta(3) tail-squeeze clause is not claimed. The classical 2-Wythoff complementarity behind the shifted Beatty positions is cited as literature, not claimed as a new result.