A357021 - OEIS

A357021

First coordinate x of points in the triangular lattice, sorted first by the distance from the origin and then by the circumferential angle phi restricted to the sector 0 <= phi < Pi/6. y is given in A357022.

0, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 7, 8, 8, 8, 9, 9, 8, 9, 9, 10, 10, 10, 9, 10, 10, 11, 11, 11, 10, 11, 12, 12, 11, 12, 12, 11, 12, 13, 13, 12, 13, 13, 12, 13, 14, 14, 14, 14, 13, 14, 13, 15, 15, 14, 15, 15, 14, 15, 16, 16, 14, 16, 15

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OFFSET

1,3

COMMENTS

The coordinates (x,y) are defined in an oblique coordinate system with an angle of 120 degrees between the axes, see e.g. A307012.

The distance from the origin is given by r = sqrt(x^2 - x*y + y^2), and the circumferential angle is phi = atan(sqrt(3)*y/(2*x - y)).

Using the pairs of terms of this sequence and of A357021(n) as grid indices in an infinite triangular lattice of one-ohm resistors leads to strictly increasing resistances against (0,0) (see A355585). This is similar to the role of A280079 and A280317 used as grid indices in the square lattice (see A355565).

LINKS

Table of n, a(n) for n=1..73.

EXAMPLE

R is the resistance between a grid point (x,y) and (0,0) in an infinite triangular lattice of one-ohm resistors.

n x y r^2 phi R

(degrees) (ohms)

1 0 0 0 0.0000000000

2 1 0 1 0.000 0.3333333333

3 2 1 3 30.000 0.4359911242

4 2 0 4 0.000 0.4613510850

5 3 1 7 19.107 0.5132889542

6 3 0 9 0.000 0.5362130198