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Floor of the Euclidean distance of a point on the (1, 1, 1; 1, 1, 1) 3D walk.
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%I #16 May 10 2013 03:37:47

%S 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,

%T 16,17,17,18,19,19,20,20,21,21,22,23,23,24,24,25,25,26,27,27,28,28,29,

%U 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

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

%C 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).

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

%F a(n) ~ n/sqrt(3). - _Charles R Greathouse IV_, May 02 2013

%F a(n) = floor(sqrt(A008810(n))), where A008810(n) is the squared Euclidean distance after n steps. - _R. J. Mathar_, May 02 2013

%o (JavaScript)

%o p = new Array(0, 0, 0);

%o for (a = 1; a < 100; a++) {

%o p[a%3] += 1;

%o document.write(Math.floor(Math.sqrt(p[0] * p[0] + p[1] * p[1] + p[2] * p[2])) + ", ");

%o }

%Y Cf. A213172, A224985.

%K nonn,easy

%O 1,4

%A _Jon Perry_, May 02 2013