A357112 -id:A357112 - cp's OEIS frontend

A035019 Sizes of successive shells in hexagonal (or A_2) lattice.

1, 6, 6, 6, 12, 6, 6, 12, 6, 12, 12, 6, 6, 12, 12, 6, 12, 12, 12, 6, 18, 12, 12, 12, 12, 6, 12, 12, 6, 12, 12, 6, 12, 24, 12, 12, 6, 12, 6, 12, 12, 12, 12, 6, 12, 12, 12, 24, 12, 6, 18, 12, 12, 12, 12, 12, 18, 12, 12, 12, 12, 12, 12, 6, 12, 18, 12, 12, 12, 12

Offset: 0

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.

T. D. Noe, Table of n, a(n) for n = 0..10000
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Cf. A003136, A004016, A038590 (partial sums), A357112.

S:=series(JacobiTheta2(0,q)*JacobiTheta2(0,q^3)+JacobiTheta3(0,q)*JacobiTheta3(0,q^3),q,1001):
subs(0=NULL,[seq(coeff(S,q,j),j=0..1000)]); # Robert Israel, Jul 29 2016

s = EllipticTheta[2, 0, q]*EllipticTheta[2, 0, q^3] + EllipticTheta[3, 0, q]* EllipticTheta[3, 0, q^3] + O[q]^1000; CoefficientList[s, q] /. 0 -> Nothing (* Jean-François Alcover, Sep 19 2016, after Robert Israel *)

Nonzero coefficients in expansion of theta_3(q)*theta_3(q^3) + theta_2(q)*theta_2(q^3).

The corresponding powers of q are A003136. - Robert Israel, Jul 29 2016

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.

Original entry on oeis.org

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

Hugo Pfoertner, Aug 05 2023

nonn

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.

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

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

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,", "))

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

cp's OEIS Frontend

A035019 Sizes of successive shells in hexagonal (or A_2) lattice.

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