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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.

%I A364443 #24 Jan 16 2024 14:08:13
%S A364443 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 A364443 14,13,16,16,16,14,16,17,16,18,20,19,19,19,19,21,20,22,21,21,22,22,24,
%U A364443 25,21,24,25,24,27,27,25,29,26,28,26,27,29,29,30,28,29,32,31
%N 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.
%C A364443 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 A364443 Hugo Pfoertner, <a href="/A364443/b364443.txt">Table of n, a(n) for n = 0..10000</a>
%H A364443 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 A364443 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a364443.txt">Table of n, a(n) for n = 0..500000</a>
%o A364443 (PARI) is_a003136(n) = !n || #qfbsolve(Qfb(1, 1, 1), n, 3);
%o A364443 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 A364443 (Python)
%o A364443 from sympy import factorint
%o A364443 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 A364443 Cf. A002324, A004016, A077773, A357112, A364729, A364730, A364731, A364732.
%K A364443 nonn
%O A364443 0,4
%A A364443 _Hugo Pfoertner_, Aug 05 2023