A364443 a(n) is the number of integers k of the form x^2 + x*y + y^2 (A003136) with n^2 < k < (n+1)^2.

0, 1, 1, 2, 2, 3, 4, 4, 5, 4, 6, 5, 6, 8, 7, 8, 7, 9, 9, 11, 10, 10, 11, 10, 13, 12, 13, 13, 13, 14, 13, 16, 16, 16, 14, 16, 17, 16, 18, 20, 19, 19, 19, 19, 21, 20, 22, 21, 21, 22, 22, 24, 25, 21, 24, 25, 24, 27, 27, 25, 29, 26, 28, 26, 27, 29, 29, 30, 28, 29, 32, 31

Offset: 0

Author

Hugo Pfoertner, Aug 05 2023

Keywords

• nonn

Comments

a(n) is the number of circles centered at (0,0) that pass through grid points of the hexagonal lattice that intersect the interior of an interval n < x < n+1 on the x-axis.

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..10000
• IBM Research, Circles on a triangular grid, Ponder This Challenge December 2023.
• Hugo Pfoertner, Table of n, a(n) for n = 0..500000

Crossrefs

Cf. A002324, A004016, A077773, A357112, A364729, A364730, A364731, A364732.

Programs

PARI

is_a003136(n) = !n || #qfbsolve(Qfb(1, 1, 1), n, 3);
for (k=0, 75, my (k1=k^2+1, k2=k^2+2*k, m=0); for (j=k1, k2, m+=is_a003136(j)); print1(m,", "))

Python

from sympy import factorint
def A364443(n): return sum(1 for k in range(n**2+1,(n+1)**2) if not any(e&1 for p, e in factorint(k).items() if p % 3 == 2)) # Chai Wah Wu, Aug 07 2023