A213172 Floor of the Euclidean distance of a point on the (1, 2, 3; 4, 5, 6) 3D walk.

0, 1, 2, 3, 6, 9, 12, 16, 21, 26, 32, 38, 45, 52, 61, 69, 78, 88, 99, 110, 121, 133, 146, 159, 173, 188, 203, 218, 234, 251, 268, 286, 305, 324, 343, 364, 384, 406, 428, 450, 473, 497, 521, 546, 571, 597, 624, 651, 679, 707, 736, 765, 795, 826, 857

Offset: 0

Author

Jon Perry, Apr 14 2013

Keywords

• nonn

Comments

Consider a standard 3-dimensional Euclidean lattice. We take 1 step along the positive x-axis, 2 along the positive y-axis, 3 along the positive z-axis, 4 along the positive x-axis, and so on. This sequence gives the floor of the Euclidean distance to the origin after n steps.

The (x,y,z) coordinates are (1,0,0), (1,2,0), (1,2,3), (5,2,3), (5,7,3), (5,7,9), (12,7,9) etc, where the x values run through A000326, the y-values through A005449, and the z-values through A045943. The squared Euclidean distances are s(n) = 1, 5, 14, 38, 83, 155, 274, 450,..., which obey the recurrence s(n) = 3*s(n-1) -3*s(n-2) +3*s(n-3) -6*s(n-4) +6*s(n-5) -3*s(n-6) +3*s(n-7) -3*s(n-8) +s(n-9), s(n) = (3*n^2+9*n+10)^2/108 +4*A099837(n+3)/27 -2*(-1)^n*A165202(n)/9, with a = floor(sqrt(s(n))). - R. J. Mathar, May 02 2013

Examples

			For a(4) we are at [5,2,3], so a(n) = floor(sqrt(25+4+9)) = 6.
		

Crossrefs

Cf. A054925, A224985, A225215.

Programs

JavaScript

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

Formula

a(n) ~ n^2 sqrt(3)/6. - Charles R Greathouse IV, May 02 2013