A350459 Number of positive rational points on the unit circle with denominator <= n and numerator >= 1.

0, 0, 0, 0, 1, 1, 1, 1, 1, 4, 4, 4, 5, 5, 10, 10, 11, 11, 11, 19, 19, 19, 19, 19, 29, 32, 32, 32, 33, 44, 44, 44, 44, 47, 60, 60, 61, 61, 66, 82, 83, 83, 83, 83, 100, 100, 100, 100, 100, 122, 127, 134, 135, 135, 156, 156, 156, 159, 159, 183, 184, 184, 184, 184, 220, 220, 220, 227, 227, 254

Offset: 1

Author

Benoit Cloitre, Jan 01 2022

Keywords

• nonn

Comments

A rational point (x,y) is of the form (a/c, b/d) with (a,b,c,d) integers. Sequence gives the number of quadruples (a,b,c,d) satisfying a >= b >= 1, 1 <= c <= n, 1 <= d <= n and such that a^2/c^2 + b^2/d^2 = 1.

Crossrefs

Cf. A046109.

Programs

PARI

a(n)=sum(a=1,n,sum(b=1,a,sum(c=1,n,sum(d=1,n,if(a^2/c^2+b^2/d^2-1,0,1)))))

Python

def A350459(n): return sum(1 for d in range(1,n+1) for c in range(1,n+1) for b in range(1,d+1) for a in range(1, b+1) if (a*d)**2 + (b*c)**2 == (c*d)**2) # Chai Wah Wu, Jan 19 2022