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

Offset: 1

Author

Hugo Pfoertner, Sep 10 2022

Keywords

• nonn

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 A357022(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).

Examples

			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
   7  4 2  12   30.000 0.5627909282
   8  4 1  13   13.898 0.5700986140
   9  4 0  16    0.000 0.5891518971
  ...
  19  7 1  43    7.589 0.6800193341
  20  8 4  48   30.000 0.6901322715
  21  7 0  49    0.000 0.6920215369
  22  8 3  49   21.787 0.6920259223
  23  8 2  52   13.898 0.6974842443
		

Crossrefs

Cf. A003136, A280079, A280317, A305575, A305576, A355565, A355585.