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

1,1

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

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 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

Sequence in context: A198990 A367606 A153356 * A305865 A342204 A010774

Adjacent sequences: A129953 A129954 A129955 * A129957 A129958 A129959

Keyword

nonn,base

Author

Ross Drewe, Jun 10 2007, Jun 11 2007

Status

approved