A364443 - OEIS

%I #24 Jan 16 2024 14:08:13

%S 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,

%T 14,13,16,16,16,14,16,17,16,18,20,19,19,19,19,21,20,22,21,21,22,22,24,

%U 25,21,24,25,24,27,27,25,29,26,28,26,27,29,29,30,28,29,32,31

%N 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.

%C 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.

%H Hugo Pfoertner, <a href="/A364443/b364443.txt">Table of n, a(n) for n = 0..10000</a>

%H IBM Research, <a href="https://research.ibm.com/haifa/ponderthis/challenges/December2023.html">Circles on a triangular grid</a>, Ponder This Challenge December 2023.

%H Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a364443.txt">Table of n, a(n) for n = 0..500000</a>

%o (PARI) is_a003136(n) = !n || #qfbsolve(Qfb(1, 1, 1), n, 3);

%o 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,", "))

%o (Python)

%o from sympy import factorint

%o 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

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

%K nonn

%O 0,4

%A _Hugo Pfoertner_, Aug 05 2023