

A129956


L1 ('cityblock') distances from the origin to a 2D pseudorandom walk based on the digits of Pi.


0



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
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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 (cityblock) metric is the simplest and is intrinsically integer valued on integerspaced 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

Table of n, a(n) for n=1..73.
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 2tuples (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

Sequence in context: A303497 A198990 A153356 * A305865 A010774 A272610
Adjacent sequences: A129953 A129954 A129955 * A129957 A129958 A129959


KEYWORD

nonn,base


AUTHOR

Ross Drewe, Jun 10 2007, Jun 11 2007


STATUS

approved



