A345256 The number of squares, each vertex of which lies on the sinusoid y = n * sin(x), and whose centers coincide with the origin.

0, 0, 0, 2, 2, 2, 4, 6, 8, 10, 12, 14, 18, 20, 22, 26, 30, 32, 36, 42, 42, 48, 54, 56, 62, 70, 72, 78, 86, 90, 96, 106, 110, 114, 126, 132, 136, 148, 156, 160, 170, 180, 184, 196, 208, 212, 222, 236, 240, 252, 266, 272, 282, 298, 306, 314, 332, 342, 348, 366, 380

Offset

1,4

Comments

Graphically, the vertices of all squares are defined as the intersection points of the sinusoid y = n * sin(x) and the same sinusoid rotated 90 degrees around the origin.

Links

Talmon Silver, Table of n, a(n) for n = 1..100

Nicolay Avilov, Problem 2185. Squares and sinusoid (in Russian).

Nicolay Avilov, Problem Kvadrat v sinuse (in Russian).

Talmon Silver, MUMPS-program

Formula

a(n) = (K-1)/4, where K is the number of roots of n*sin(n*sin(x)) + x = 0.

From Merab Leviashvili, Jul 22 2021: (Start)

Writing d=floor(n/(Pi/2)) (mod 4), one has a(n)=

(floor(n/Pi))^2+2*floor(n*sin(n)/(2*Pi)+1/4) if d=0;

(floor(n/Pi+1/2))^2-1-2*floor(n*sin(n)/(2*Pi)+1/4) if d=1;

(floor(n/Pi))^2-1+2*floor(|n*sin(n)/(2*Pi)|+3/4) if d=2;

(floor(n/Pi+1/2))^2-2*floor(|n*sin(n)/(2*Pi)|+3/4) if d=3. (End)

Example

a(3) = 0 since there are no squares with all their vertices on the curve y = 3*sin(x).
a(4) = 2 since there are 2 squares whose vertices lie on the curve y = 4*sin(x). The two squares have approximate coordinates: (1.02; 3.39), (3.39; -1.02), (-1.02; -3.39), (-3.39; 1.02) and (1.98; 3.67), (3.67; -1.98), (-1.98; -3.67), (-3.67; 1.98).

Crossrefs

Sequence in context: A187504 A036654 A262669 * A291299 A241576 A306749

Adjacent sequences: A345253 A345254 A345255 * A345257 A345258 A345259

Keyword

nonn

Author

Nicolay Avilov, Jun 12 2021

Extensions

More terms from Jinyuan Wang, Jun 17 2021

Status

approved