A091087 -id:A091087 - cp's OEIS frontend

A049473 Nearest integer to n/sqrt(2).

0, 1, 1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 18, 19, 20, 21, 21, 22, 23, 23, 24, 25, 25, 26, 27, 28, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 40, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 47

Offset: 0

N. J. A. Sloane

nonn

a(n) = floor(n*sqrt(2)) - floor(n/sqrt(2)). Indeed, the equation {(nearest integer to n/r) = floor(nr) - floor(n/r) for all n>=0} has exactly two solutions: sqrt(2) and -sqrt(2). - Clark Kimberling, Dec 18 2003

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

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A091087.

[0] cat [Round(n/Sqrt(2)): n in [1..100]]; // G. C. Greubel, Jan 27 2018

Round[Range[0,70]/Sqrt[2]] (* Harvey P. Dale, Feb 17 2015 *)

a(n)=round(n/sqrt(2)) \\ Charles R Greathouse IV, Sep 02 2015

A330173 a(n) = n + floor(ns/r) + floor(nt/r), where r = sqrt(2), s = sqrt(2) + 1, t = sqrt(2) + 2.

Original entry on oeis.org

4, 9, 15, 19, 25, 30, 34, 40, 45, 51, 55, 60, 66, 70, 76, 81, 87, 91, 96, 102, 106, 112, 117, 121, 127, 132, 138, 142, 148, 153, 157, 163, 168, 174, 178, 183, 189, 193, 199, 204, 208, 214, 219, 225, 229, 235, 240, 244, 250, 255, 261, 265, 270, 276, 280, 286

Offset: 1

Clark Kimberling, Jan 04 2020

This is one of three sequences that partition the positive integers.  In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked.  Define b(n) and c(n) as the ranks of n/s and n/t.  It is easy to prove that
a(n)=n+[ns/r]+[nt/r],
b(n)=n+[nr/s]+[nt/s],
c(n)=n+[nr/t]+[ns/t], where []=floor.
Taking r = sqrt(2), s = sqrt(2) + 1, t = sqrt(2) + 2 yields
a=A330173, b=A016789, c=A091087.

Cf. A016789, A091087.

r = Sqrt[2]; s = Sqrt[2] + 1; t = Sqrt[2] + 2;
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t]
Table[a[n], {n, 1, 120}]  (* A330173 *)
Table[b[n], {n, 1, 120}]  (* A016789 *)
Table[c[n], {n, 1, 120}]  (* A091087 *)
Table[n+Floor[(n(Sqrt[2]+1))/Sqrt[2]]+Floor[(n(Sqrt[2]+2))/Sqrt[2]],{n,60}] (* Harvey P. Dale, Apr 16 2023 *)

a(n) = n + floor(ns/r) + floor(nt/r), where r = sqrt(2), s = sqrt(2) + 1, t = sqrt(2) + 2.

cp's OEIS Frontend

A049473 Nearest integer to n/sqrt(2).

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