

A345256


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


1



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
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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, MUMPSprogram


FORMULA

a(n) = (K1)/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))^212*floor(n*sin(n)/(2*Pi)+1/4) if d=1;
(floor(n/Pi))^21+2*floor(n*sin(n)/(2*Pi)+3/4) if d=2;
(floor(n/Pi+1/2))^22*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



