A129956 L1 ('city-block') distances from the origin to a 2D pseudo-random walk based on the digits of Pi.

5, 9, 4, 5, 6, 4, 5, 8, 4, 7, 11, 13, 13, 18, 13, 17, 15, 15, 18, 20, 15, 21, 24, 25, 22, 18, 22, 19, 21, 25, 25, 27, 30, 29, 25, 28, 32, 34, 36, 35, 36, 40, 48, 47, 53, 55, 57, 57, 64, 63, 64, 65, 61, 53, 54, 52, 46, 45, 39, 41, 48, 54, 58, 56, 47, 47, 42, 48, 47, 41, 38, 36, 41

Offset: 1

Author

Ross Drewe, Jun 10 2007, Jun 11 2007

Keywords

• nonn
• base

Comments

The distance from the starting point has physical applications, e.g., in aggregation models.

All distance metrics generate sequences which coincide at the zero points. The L1 (city-block) metric is the simplest and is intrinsically integer valued on integer-spaced lattices (as used here).

The r sequence is not affected by the dimension ordering (i.e., whether each pair of values taken from the digits of Pi represents [x,y] or [y,x]).

Examples

			The first 10 digits of Pi are 3 1 4 1 5 9 2 6 5 3
This gives five 2-tuples (x,y pairs): [3 1], [4 1], [5 9], [2 6], [5 3]
The x & y vectors are x = [3 4 5 2 5], y = [1 1 9 6 3]
Adjusting to zero mean gives x = [ -1.5 -0.5 0.5 -2.5 0.5], y = [ -3.5 -3.5 4.5 1.5 -1.5]
The cumulative x,y position vectors are cx = [ -1.5 -2 -1.5 -4 -3.5], cy = [ -3.5 -7 -2.5 -1 -3.5]
The L1 radii from the origin are r = abs(cx) + abs(cy), r = [5 9 4 5 6]
		

Links

• Hemphill, Scott, Pi (gives 1.25 million digits of Pi)
• Eric Weisstein's World of Mathematics, Pi Digits.

Programs

MATLAB

function r = find_L1_radius(pidigits, k); d = pidigits(1:2*k); t = reshape(d, 2, length(d)/2); x = t(1, :); y = t(2, :); cx = cumsum(x - 4.5); cy = cumsum(y - 4.5); r = abs(cx) + abs(cy); return; % pidigits is a MATLAB row vector of at least 2*k digits of Pi (including the initial '3'); % k is the number of 2D radii to calculate.

Formula

r(n) = abs(cx(n)) + abs(cy(n)), where cx = cum_sum([odd digits of Pi] - 4.5) and cy = cum_sum([even digits of Pi] - 4.5).