A225215 Floor of the Euclidean distance of a point on the (1, 1, 1; 1, 1, 1) 3D walk.

1, 1, 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 31, 32, 32, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 41, 41, 42, 42, 43, 43, 44, 45, 45, 46, 46, 47, 47, 48, 49, 49, 50, 50

Offset: 1

Author

Jon Perry, May 02 2013

Keywords

• nonn
• easy

Comments

Consider a standard 3-dimensional Euclidean lattice. We take 1 step along the positive x-axis, 1 along the positive y-axis, 1 along the positive z-axis, 1 along the positive x-axis, and so on. After 3, 6, 9, 12, 15 etc steps we have returned to the space diagonal (with equal x, y and z coordinates).

This sequence gives the floor of the Euclidean distance to the origin after n steps.

Crossrefs

Cf. A213172, A224985.

Programs

JavaScript

p = new Array(0, 0, 0);
for (a = 1; a < 100; a++) {
p[a%3] += 1;
document.write(Math.floor(Math.sqrt(p[0] * p[0] + p[1] * p[1] + p[2] * p[2])) + ", ");
}

Formula

a(n) ~ n/sqrt(3). - Charles R Greathouse IV, May 02 2013

a(n) = floor(sqrt(A008810(n))), where A008810(n) is the squared Euclidean distance after n steps. - R. J. Mathar, May 02 2013