A224985 Floor of the Euclidean distance of a point on the (1, 1, 1; 2, 2, 2) 3D walk.

0, 1, 1, 1, 3, 4, 5, 7, 9, 10, 13, 15, 17, 20, 23, 25, 29, 33, 36, 40, 44, 48, 53, 58, 62, 67, 73, 77, 84, 89, 95, 102, 108, 114, 121, 128, 135, 143, 150, 157, 166, 174, 181, 190, 199, 207, 217, 226, 235, 245, 255, 265, 275, 286, 296, 307, 318, 329, 341, 352, 363, 376

Offset: 0

Author

Jon Perry, Apr 22 2013

Keywords

• nonn

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, 2 along the positive x-axis, and so on.

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

The coordinates are (0,0,0), (1,0,0), (1,1,0), (1,1,1), (3,1,1), (3,3,1), (3,3,3), (6,3,3),... where the x, y and z-coordinates run through A000217. The squared distances are s = 0, 1, 2, 3, 11, 19, 27, 54,... which obey an 11th-order linear recurrence with g.f. -x*(1+4*x^3+x^6) / ( (1+x+x^2)^3*(x-1)^5), a(n) = floor(sqrt(s(n))). - R. J. Mathar, May 02 2013

Crossrefs

Cf. A213172, A225215.

Programs

JavaScript

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