%I #11 Dec 17 2017 08:00:58

%S 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,

%T 22,19,21,25,25,27,30,29,25,28,32,34,36,35,36,40,48,47,53,55,57,57,64,

%U 63,64,65,61,53,54,52,46,45,39,41,48,54,58,56,47,47,42,48,47,41,38,36,41

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

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

%C 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).

%C 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]).

%H Hemphill, Scott, <a href="http://www.gutenberg.org/etext/50">Pi</a> (gives 1.25 million digits of Pi)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PiDigits.html">Pi Digits.</a>

%F 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).

%e The first 10 digits of Pi are 3 1 4 1 5 9 2 6 5 3

%e This gives five 2-tuples (x,y pairs): [3 1], [4 1], [5 9], [2 6], [5 3]

%e The x & y vectors are x = [3 4 5 2 5], y = [1 1 9 6 3]

%e 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]

%e 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]

%e The L1 radii from the origin are r = abs(cx) + abs(cy), r = [5 9 4 5 6]

%o (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.

%K nonn,base

%O 1,1

%A _Ross Drewe_, Jun 10 2007, Jun 11 2007