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A129956
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L1 ('city-block') distances from the origin to a 2D pseudo-random walk based on the digits of Pi.
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0
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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
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1,1
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COMMENTS
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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]).
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LINKS
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Hemphill, Scott, Pi (gives 1.25 million digits of Pi)
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FORMULA
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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).
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EXAMPLE
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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]
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PROG
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(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.
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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