cp's OEIS Frontend

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.

%I A357021 #9 Apr 01 2025 08:10:08
%S A357021 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,
%T A357021 10,10,11,11,11,10,11,12,12,11,12,12,11,12,13,13,12,13,13,12,13,14,14,
%U A357021 14,14,13,14,13,15,15,14,15,15,14,15,16,16,14,16,15
%N 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.
%C A357021 The coordinates (x,y) are defined in an oblique coordinate system with an angle of 120 degrees between the axes, see e.g. A307012.
%C A357021 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)).
%C A357021 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).
%e A357021 R is the resistance between a grid point (x,y) and (0,0) in an infinite triangular lattice of one-ohm resistors.
%e A357021 .
%e A357021    n  x y  r^2   phi      R
%e A357021               (degrees) (ohms)
%e A357021    1  0 0   0          0.0000000000
%e A357021    2  1 0   1    0.000 0.3333333333
%e A357021    3  2 1   3   30.000 0.4359911242
%e A357021    4  2 0   4    0.000 0.4613510850
%e A357021    5  3 1   7   19.107 0.5132889542
%e A357021    6  3 0   9    0.000 0.5362130198
%e A357021    7  4 2  12   30.000 0.5627909282
%e A357021    8  4 1  13   13.898 0.5700986140
%e A357021    9  4 0  16    0.000 0.5891518971
%e A357021   ...
%e A357021   19  7 1  43    7.589 0.6800193341
%e A357021   20  8 4  48   30.000 0.6901322715
%e A357021   21  7 0  49    0.000 0.6920215369
%e A357021   22  8 3  49   21.787 0.6920259223
%e A357021   23  8 2  52   13.898 0.6974842443
%Y A357021 Cf. A003136, A280079, A280317, A305575, A305576, A355565, A355585.
%K A357021 nonn
%O A357021 1,3
%A A357021 _Hugo Pfoertner_, Sep 10 2022