A343865 Number of lattice points for which polar coordinates (r,theta) exist such that (0 <= r <= theta <= n).

1, 1, 2, 4, 12, 21, 35, 57, 84, 118, 157, 206, 255, 314, 377, 453, 529, 612, 705, 801, 904, 1014, 1129, 1249, 1375, 1512, 1654, 1801, 1952, 2115, 2280, 2451, 2632, 2816, 3007, 3203, 3404, 3617, 3831, 4055, 4285, 4517, 4759, 5004, 5254, 5514, 5780, 6052, 6333

Offset

0,3

Links

Table of n, a(n) for n=0..48.

Luc Rousseau, An illustration of a(4)=12.

Formula

a(n) ~ Pi*n*(n-2*Pi) + 4*Pi^3/3 = area covered by the radius vector of the Archimedean spiral (r = theta), when 0 <= n-2*Pi <= theta <= n;

a(n) ~ Pi*n^2.

Example

Exhaustive lists of lattice points satisfying the conditions for the first few n's:
n=0:
    (0,0) is (r=0 <= theta=0 <= 0)
n=1:
    (0,0) is (r=0 <= theta=0 <= 1)
n=2:
    (0,0) is (r=0 <= theta=0 <= 2)
    (0,1) is (r=1 <= theta=Pi/2=1.5707... <= 2)
n=3:
    (0,0) is (r=0 <= theta=0 <= 3)
    (0,1) is (r=1 <= theta=Pi/2=1.5707... <= 3)
    (-1,1) is (r=sqrt(2)=1.4142... <= theta=3*Pi/4=2.3561... <= 3)
    (-2,1) is (r=sqrt(5)=2.2360... <= theta=arccos(-2/sqrt(5))=2.6679... <= 3)

Prog

(PARI)
atan2(x, y)=if(x==0&&y==0, 0, if(x>0&&y==0, 0, if(x==0&&y>0, 0.5*Pi, if(x<0&&y==0, Pi, if(x==0&&y<0, 1.5*Pi, if(y>0, acos(x/sqrt(x^2+y^2)), 2*Pi-acos(x/sqrt(x^2+y^2))))))))
f(n, n0, t0)=n+t0-n0-(t0>n0)*2*Pi
b(x, y, n, n0)=my(r=sqrt(x^2+y^2), t0=atan2(x, y)); r<=f(n, n0, t0)
a(n)=my(n0=n%(2*Pi), c=0); for(x=-n, n, for(y=-n, n, if(b(x, y, n, n0), c++))); c
for(n=0, 50, print1(a(n), ", "))

Crossrefs

Cf. A000328.

Sequence in context: A319213 A099603 A319615 * A062767 A173650 A303030

Adjacent sequences: A343862 A343863 A343864 * A343866 A343867 A343868

Keyword

nonn

Author

Luc Rousseau, May 02 2021

Status

approved