A225215 -id:A225215

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A213172

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

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

LINKS

Table of n, a(n) for n=0..54.

FORMULA

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

EXAMPLE

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

PROG

(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]))+", ");

}

CROSSREFS

Cf. A054925, A224985, A225215.

KEYWORD

nonn

AUTHOR

Jon Perry, Apr 14 2013

STATUS

approved

A224985

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

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

LINKS

Table of n, a(n) for n=0..61.

PROG

(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])) + ", ");

}