OFFSET
1,1
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]).
LINKS
Hemphill, Scott, Pi (gives 1.25 million digits of Pi)
Eric Weisstein's World of Mathematics, Pi Digits.
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).
EXAMPLE
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]
PROG
(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.
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Ross Drewe, Jun 10 2007, Jun 11 2007
STATUS
approved